Tensor product categorifications, Verma modules and the blob 2-category

We construct a dg-enhancement of KLRW algebras that categorifies the tensor product of a universal $\mathfrak{sl}_2$ Verma module and several integrable irreducible modules. When the integrable modules are two-dimensional, we construct a categorical action of the blob algebra on derived categories of these dg-algebras which intertwines the categorical action of $\mathfrak{sl}_2$. From the above we derive a categorification of the blob algebra.


Introduction
Dualities are fundamental tools in mathematics in general and in higher representation theory in particular. For example, Stroppel's version of Khovanov homology [41,42] and Khovanov's HOMFLY-PT homology [19] can be seen as instances of higher Schur-Weyl duality (see also [43] for further explanations). In this paper we construct an instance of higher Schur-Weyl duality between U q psl 2 q and the blob algebra of Martin and Saleur [30] by using a categorification of the tensor product of a Verma module and several twodimensional irreducibles.
1.1. State of the art.
1.1.1. Schur-Weyl duality, U q psl 2 q and the Temperley-Lieb algebra. Schur-Weyl duality connects finite-dimensional modules of the general linear and symmetric groups. In particular, it states that over an algebraically closed field the actions of GL m and S r on the Remark 1.1. In [30], the blob algebra is given a different presentation, where the generator of type B is pictured as a dot on the left-most strand, and is an idempotent. We use the presentation given in [14], which is isomorphic to the one in [30] over Zpq, λq (but not over Zrq˘1, λ˘1s). This presentation is closer to the representation theory of U q psl 2 q and is the one that arises from our categorification construction.
More generally, we consider the category B with objects given by M b V br for various r P N, and hom-spaces given by U q psl 2 q-intertwiners. This category, that we call the blob category, has a very similar diagrammatic description as the blob algebra, where objects are collections of r`1 points on the horizontal line. The hom-spaces are presented by flat tangles connecting these points, with the left-most point of the source always connected to the left-most point of the target, allowing 4-valent intersections between the first two strands. These diagrams are subject to the same relations as the blob algebra. We stress that, in contrast to the Temperley-Lieb category of type A, the blob category is not monoidal w.r.t. juxtaposition of diagrams since the blue strand in the pictures above needs to be on the left-hand side of any diagram.

Webster categorification.
In a seminal paper [49], Webster has constructed categorifications of tensor products of integrable modules for symmetrizable Kac-Moody algebras, generalizing Lauda's [27], Khovanov-Lauda [21,22] and Chuang-Rouquier [6] and Rouquier's [40] categorification of quantum groups, and their integrable modules. Webster further used his categorifications to give a link homology theory categorifying the Witten-Reshetikhin-Turaev invariant of tangles. The construction in [49] involves algebras, called KLRW algebras (or tensor product algebras), that are finite-dimensional algebras presented diagrammatically, generalizing cyclotomic KLR algebras. Categories of finitely generated modules over KLRW algebras come equipped with an action of Khovanov-Lauda-Rouquier's 2-Kac-Moody category, and their Grothendieck groups are isomorphic to tensor products of integrable modules. Link invariants and categorifications of intertwiners are constructed using functors given by the derived tensor product with certain bimodules over KLRW algebras.
1.1.4. Verma categorification: dg-enhancements. In [36,37,34], the second and third authors have given a categorification of (universal, parabolic) Verma modules for (quantized) symmetrizable Kac-Moody algebras. In its more general form [34], the categorification is given as a derived category of dg-modules over a certain dg-algebra, similar to a KLR algebra but containing an extra generator in homological degree 1. This dg-algebra can also be endowed with a collection of different differentials, each of them turning it into a dg-algebra whose homology is isomorphic to a cyclotomic KLR algebra. This can be interpreted as a categorification of the projection of a universal Verma module onto an integrable module. Categorification of Verma modules was used by the second and third authors in [35] to give a quantum group higher representation theory construction of Khovanov-Rozansky's HOMFLY-PT link homology.
1.2. The work in this paper. For λ a formal parameter, let Mpλq be the universal U q psl 2 q-Verma module with highest weight λ, and V pNq :" V pN 1 q b¨¨¨b V pN r q, where V pN j q is the irreducible of highest weight q N j , N j P N. In this paper we combine Webster's categorification with the Verma categorification to give a categorification of Mpλq b V pNq. Then we construct a categorification of the blob algebra by categorifying the intertwiners of Mpλq b V pN q where all the N j are 1.
1.2.1. Dg-enhanced KLRW algebras and categorification of tensor products (Sections 3 and 4). Fix a commutative unital ring k. The KLRW algebra is the k-algebra spanned by planar isotopy classes of braid-like diagrams whose strands are of two types: there are black strands labeled by simple roots of a symmetrizable Kac-Moody algebra g and carrying dots, and there are red strands labeled by dominant integral weights. KLRW algebras are cyclotomic algebras in the sense that they generalize cyclotomic KLR algebras to a string of dominant integral weights, where the "violating condition" [49,Definition 4.3] plays the role of the cyclotomic condition. KLRW algebras were also defined without the violating condition, in which case we call them non-cyclotomic or affine KLRW algebras. In the case of sl 2 , for b P N and N P N r , we denote by T N b (resp. r T N b ) the (resp. affine) KLRW algebra spanned by b black strands (all labeled by the simple root of sl 2 ) and r red strands, labeled in order N 1 , . . . , N r from left to right.
Following a procedure analogous to [37,34], we construct in Section 3 an algebra T λ,N b , with λ a formal parameter, that contains the affine KLRW algebra r T N b as a subalgebra. In a nutshell, T λ,N b is defined by putting a vertical blue strand labeled by λ on the left of the diagrams of r T N b , and adding a new generator that we call a nail (this corresponds with the "tight floating dots" of [37,34]). We draw this new generator as: ote that a nail can only be placed on the left-most strand, which is always blue. The nails are subject to the following local relations: When N " ∅ is the empty sequence, we recover the dg-enhanced nilHecke algebra from [37]. The subalgebra spanned by all diagrams without a nail is isomorphic to the affine KLRW algebra r T N b . As we will see, the algebra T λ,N b can be equipped with three (Z-)gradings: two internal gradings, one as in Webster's original definition and an additional grading (see Definition 3.2), as well as a homological grading. The first two of these gradings categorify the parameters q and λ respectively, and we call them q-and λ-gradings. As usual, the homological grading allows us to categorify relations involving minus signs. We write q k (resp. λ k ) for a grading shift up by k in the q-(resp. λ-)grading, and rks for a grading shift up by k in the homological grading, for k P Z.
We let the nail be in homological degree 1, while diagrams without a nail are in homological degree 0. As in the categorification of Verma modules, if we endow the algebra T λ,N b with a trivial differential, then it becomes a dg-algebra categorifying Mpλq b V pN q (see below). We can also equip T λ,N b with a differential d N , for N ě 0, which acts trivially on diagrams without a nail, while d N¨λ‹ ‚ :" N λ and extending using the graded Leibniz rule. The dg-algebra pT λ, N b , d N q is formal with homology isomorphic to the KLRW algebra T pN,N q b (see Theorem 3.13). The usual framework using the algebra map T λ,N b Ñ T λ,N b`1 that adds a black strand at the right of a diagram gives rise to induction and restriction dg-functors E b and F b between the derived dg-categories D dg pT λ,N b , 0q and D dg pT λ,N b`1 , 0q. The following describes the categorical U q psl 2 q-action: of dg-functors.
As usual in the context of categorification, the notation ' rβ`|N|´2bsq on the right-hand side is an infinite coproduct categorifying multiplication by the rational fraction pλq |N|´2bλ´1 q´| N |`2b q{pq´q´1q interpreted as a Laurent series. Turning on the differential d N gives functors E N b , F N b on D dg p' bě0 T λ,n b , d N q. In this case, the right-hand side in Theorem 4.1 becomes quasi-isomorphic to a finite sum and we recover the usual action on categories of modules over KLRW algebras (see Proposition 4.3).
In [33], the second author introduced the notion of an asymptotic Grothendieck group, which is a notion of a Grothendieck group for (multi)graded categories of objects admitting infinite iterated extensions (like infinite composition series or infinite resolutions) whose gradings satisfy some mild conditions. Denote by Q K ∆ 0 p´q the asymptotic Grothendieck group (tensored over Zppq, λqq with Qppq, λqq). The categorical U q psl 2 q-actions on the derived categories D dg p' bě0 T λ,N b , 0q and D dg p' bě0 T λ,N b , d N q descend to the asymptotic Grothendieck group and we have the main result of Section 4, which reads as following: Theorem 4.7. There are isomorphisms of U q psl 2 q-modules Q K ∆ 0 pT λ,N , 0q -Mpλq b V pN q, and Q K ∆ 0 pT λ,N , d N q -V pNq b V pNq, for all N P N.
In Section 7.1 we prove that in the case of b " 1, N " 1, . . . , 1 and N " 1, the dg-algebra pT λ,1,..., 1 1 , d 1 q is isomorphic to a dg-enhanced zigzag algebra, generalizing [45, §4]. (Sections 5 and 6). We study the case of N " 1, . . . , 1 in more detail. We define several functors on D dg pT λ,N , 0q commuting with the categorical action of U q psl 2 q. As in [49], these are defined as a first step via (dg-)bimodules over the abovementioned dg-enhancements of KLRW-like algebras. To simplify matters, let T λ,r be the dg-enhanced KLRW algebra with r strands labeled 1 and a blue strand labeled λ. The categorical Temperley-Lieb action is realized by a pair of biadjoint functors, constructed in the same way as in [49]. They are given by derived tensoring with the pT λ,r , T λ,r˘2 q-bimodules B i and B i generated respectively by the diagram  and its mirror along a horizontal axis. We stress again that the blue strand is on the left. Moreover, these diagrams are subjected to some local relations (see Section 5.1). Taking the derived tensor product with these bimodules defines the coevaluation and evaluation dg-functors as B i :" B i b L T´: D dg pT λ,r , 0q Ñ D dg pT λ,r`2 , 0q, B i :" B i b L T´: D dg pT λ,r`2 , 0q Ñ D dg pT λ,r , 0q. In Section 6.1 we extend [49] and prove that these functors satisfy the relations of the Temperley-Lieb algebra: Corollaries 6.3 and 6.5. There are natural isomorphisms

The blob 2-category
We define the double braiding functor in the same vein, using the pT λ,r , T λ,r q-bimodule X generated by the diagram  The double braiding functor is then defined as the derived tensor product Ξ :" X b L T´: D dg pT λ,r , 0q Ñ D dg pT λ,r , 0q. The functors B i , B i and Ξ intertwine the categorical U q psl 2 q-action on D dg pT λ,r , 0q: Proposition 6.1. We have natural isomorphisms E˝Ξ -Ξ˝E and F˝Ξ -Ξ˝F, and also E˝B i -B i˝E , F˝B i -B i˝F , and similarly for B i .
The first main result of Section 6 is that the blob algebra acts on D dg pT λ,r , 0q. This follows from the Temperley-Lieb action in Section 6.1 and Corollary 6.11, Proposition 6.14 and Corollary 6.17, summarized below. Corollary 6.11, Proposition 6.14 and Corollary 6.17. The functor Ξ : D dg pT λ,r , 0q Ñ D dg pT λ,r , 0q is an autoequivalence, with inverse given by Ξ´1 :" RHOM T pX,´q : D dg pT λ,r , 0q Ñ D dg pT λ,r , 0q.
There are quasi-isomorphisms One of the main difficulties in establishing the results above is that, in order to compute derived tensor products, we have to take left (resp. right) cofibrant replacements of several dg-bimodules. As observed in [29, §2.3], while the left (resp. right) module structure remains unchanged when passing to the left (resp. right) cofibrant replacement, the right (resp. left) module structure is preserved only in the A 8 sense. As a consequence, constructing natural transformations between compositions of derived tensor product functors often requires to use A 8 -bimodules maps. We have tried to avoid as much as possible to end up in this situation, replacing the potentially unwieldy A 8 -bimodules by quasi-isomorphic dg-bimodules.
Let B r be a certain subcategory (see Section 6.3) of the derived dg-category of pT λ,r , 0q-pT λ,r , 0q-bimodules generated by the dg-bimodules corresponding with the dg-functors identity, Ξ˘1 and B i˝Bi . Given two dg-bimodules in B r , we can compose them in the derived sense by replacing both of them with a bimodule cofibrant replacement (i.e. a cofibrant replacement as dg-bimodule, and not only left or right dg-module), and taking the usual tensor product. This gives a dg-bimodule, isomorphic to the derived tensor product of the two initial dg-bimodules. In particular, it equips Q K ∆ 0 pB r q with a ring structure. We show that B r is a categorification of the blob algebra B r with ground ring extended to Qppq, λqq: Corollary 6.19. There is an isomorphism of Qppq, λqq-algebras This result generalizes to the blob category. However, a technical issue we find here is that dg-categories up to quasi-equivalence do not form a 2-category, but rather an p8, 2qcategory [9]. Concretely in our case, we consider a sub-p8, 2q-category of this p8, 2qcategory, where the objects are the derived dg-categories D dg pT λ,r , 0q for various r P N, and the 1-hom are generated by the dg-functors identity, Ξ˘1, B i and B i . Moreover, these 1hom are stable p8, 1q-categories, and thus their homotopy categories are triangulated (see [28]). In particular, we write Q K ∆ 0 pBq for the category with the same objects as B and with hom-spaces given by the asymptotic Grothendieck groups of the homotopy category of the 1-hom of B. By [9] and [46], we can compute these hom-spaces by considering usual derived categories of dg-bimodules, and we obtain the following, again after extending the ground rings to Qppq, λqq: There is an equivalence of categories 3. The general case: symmetrizable g. The definition of dg-enhanced KLRW algebras in Section 3 generalizes immediately to any symmetrizable g. We indicate this generalization in Section 7.2. We expect that the results of Section 3 and Section 4 extend to this case without difficulty.
1.2.4. Quiver Schur algebras. Quiver Schur algebras were introduced geometrically by Stroppel and Webster in [44] to give a graded version of the cyclotomic q-Schur algebras of Dipper, James and Mathas [7]. Independently, Hu and Mathas [13] constructed a graded Morita equivalent variant of the quiver Schur algebras in [44] as graded quasihereditary covers of cyclotomic KLR algebras for linear quivers. While the construction in [44] is geometric, the construction in [13] is combinatorial/algebraic.
More recently, Khovanov, Qi and Sussan [23] gave a variant of the quiver Schur algebras in [13] for the case of cyclotomic nilHecke algebras, and showed that Grothendieck groups of their algebras can be identified with tensor products of integrable modules of U q psl 2 q. Following similar ideas, in Section 7.3 we construct a dg-algebra, which we conjecture to be the quiver Schur variant of the dg-enhanced KLRW algebra of Section 3 (Conjectures 7.15 and 7.17).
1.2.5. Appendix. We have moved the most computational proofs to Appendix A, leaving only a sketch of some of the proofs in the main text. The reader can also find in Appendix B some explanations and results about homological algebra, A 8 -structures and asymptotic Grothendieck groups.  [14, §3.4] gives rise to a Jones polynomial for tangles of type B (i.e. tangles in the annulus). We expect that by introducing braiding functors as in [49], we obtain a link homology of type B, yielding invariants of links in the annulus akin to ones introduced by Asaeda-Przytycki-Sikora [3] (see also [4,12,38]). Given a link in the annulus, the invariant obtained from our construction would be a dg-endofunctor of the derived dg-category of dg-modules over the dg-enhanced KLRW algebra pT λ,H , 0q. This means that the empty link is sent to the dg-endomorphism space of the identity functor, which coincides with the Hochschild cohomology of T λ,H , and is infinite-dimensional (the center of T λ,H is already infinite-dimensional). By restricting to the subcategory of dg-modules over pT λ,H 0 , 0q, it becomes 1-dimensional since T λ,H 0 k. With this restriction, we conjecture that our "would-be" invariant coincides with the usual annular Khovanov homology.
The following is a work in progress with A. Wilbert. As it is the case of using Webster's machinery [49], computing the tangle invariant of type B using our framework could be unwieldy. A more computation-friendly alternative could be to use dg-bimodules over annular arc algebras constructed using the annular TQFT of [3], as done in [1, §5.3] (see also [8, §5]). Furthermore, evidences show there is a (at least weak) categorical action of the blob algebra on the derived category of dg-modules over these annular arc algebras.
In a different direction, one could try to extend our results to construct a Khovanov invariant for links in handlebodies, in the spirit of the handlebody HOMFLY-PT-link homology of Rose-Tubbenhauer in [39].

1.3.2.
Constructions using homotopy categories. KLRW algebras are given diagrammatically, which is the often an appropriate framework for constructions with an additive flavor. Nevertheless, the various functors realizing the various intertwiners and the braiding need to pass to derived categories of modules. This makes it harder to describe explicitly the 2-categories realizing these symmetries since a bimodule for two of those algebras induces an A 8 -bimodule on the level of derived categories. This was pointed out by Mackaay and Webster in [29], who gave explicit constructions of categorified intertwiners in order to prove the equivalence between the several existing gl n -link homologies. One of the things [29] tells us is how to construct homotopy versions of Webster's categorifications.
A construction using homotopy categories for the results in this paper seems desirable from our point of view. We hope it can be done either by mimicking [29], which can turn out to be a technically challenging problem, or alternatively, by a construction of dg-enhancements for redotted Webster algebras, as considered in [25] and [20] to give a homotopical version of some of the above, but whose low-tech presentation might hide difficulties.

Generalized blob algebras and variants.
The results of [14] were extended in [26], where the first and third authors have computed the endomorphism algebra of the U q pgl m qmodule M p pΛq b V bn for M p pΛq a parabolic universal Verma modules and V the natural module of U q pgl m q, which is always a quotient of an Ariki-Koike algebra. As particular cases (depending on p and the relation between n and m) we obtain Hecke algebras of type B with two parameters, the generalized blob algebra of Martin and Woodcock [31] or the Ariki-Koike algebra itself. With this result in mind it is tantalizing to ask for an extension to gl m of the work in this paper. Modulo technical difficulties the methods in this paper could work for gl m in the case of a parabolic Verma module for a 2-block parabolic subalgebra, which is the case where the generators of the endomorphism algebra satisfy a quadratic relation. Constructing a categorification of the Ariki-Koike algebra or the generalized blob algebra as the blob 2-category in Section 6 looks quite challenging at the moment, in particular for a functor-realization of the cyclotomic relation and the relation τ " 0 (for the generalized blob algebra in the presentation given in [26,Theorem 2.24]).
Acknowledgments. The authors thank Catharina Stroppel for interesting discussions, and for pointing us [30], helping to clarify the confusion with the terminology of "blob algebra" and "Temperley-Lieb algebra of type B". The authors would also like to thank the referee for his/her numerous, detailed and helpful comments. A.L. was supported by the 2. Quantum sl 2 and the blob algebra 2.1. Quantum sl 2 . Recall that quantum sl 2 can be defined as the Qpqq-algebra U q psl 2 q, with generic q, generated by K, K´1, E and F with relations Quantum sl 2 becomes a bialgebra when endowed with comultiplication and with counit εpK˘1q :" 1, εpEq :" εpF q :" 0. There is a Qpqq-linear anti-involutionτ of U q psl 2 q defined on the generators bȳ τ pEq :" q´1K´1F,τ pF q :" q´1EK,τ pKq :" K.
where rns q is the n-th quantum integer rns q :" q n´q´n q´q´1 " q n´1`qn´1´2`¨¨¨`q1´n .
In particular, let V :" V p1q be the fundamental U q psl 2 q-module.
The module V pNq can be equipped with the Shapovalov form which is a non-degenerate bilinear form such that pv N,0 , v N,0 q N " 1 and which isτ -Hermitian: for any v, v 1 P V pNq and u P U q psl 2 q, we have pu¨v, v 1 q N " pv,τpuq¨v 1 q N . A computation shows that where r0s q ! :" 1 and rns q ! :" rns q rn´1s q . . . r2s q r1s q .

Verma modules.
Let β be a formal parameter and write λ :" q β as a formal variable. Let b be the standard upper Borel subalgebra of sl 2 and U q pbq be its quantum version. It is the U q psl 2 q-subalgebra generated by K, K´1 and E. Let K λ be a 1-dimensional Qpλ, qqvector space, with fixed basis element v λ . We endow K λ with an U q pbq-action by declaring that: K˘1v λ :" λ˘1v λ , Ev λ :" 0, extending linearly through the obvious inclusion Qpqq ãÑ Qpq, λq. The universal Verma module Mpλq is the induced module It is irreducible and infinite-dimensional with Qpq, λq-basis v λ,0 :" v λ , v λ,1 , . . . , v λ,i , . . . and where rβ`ks q :" λq k´λ´1 q´k q´q´1 .
The Verma module Mpλq can also be equipped with a Shapovalov form p¨,¨q λ , which is again a non-degenerate bilinear form such that pv λ , v λ q λ " 1 and which isτ -Hermitian: for any v, v 1 P Mpλq and u P U q psl 2 q, we have pu¨v, v 1 q λ " pv,τ puq¨v 1 q λ . One easily calculates that pv λ,i , v λ,j q λ " δ i,j λ i q´i 2 ris q !rβs q rβ´1s q¨¨¨r β´i`1s q .

Tensor products.
Given W and W 1 two U q psl 2 q-modules, their tensor product W bW 1 is again a U q psl 2 q-module with the action induced by ∆. Explicitly, for all w P W and w 1 P W 1 . For N " pN 1 , . . . , N r q P N r we write V pN q :" V pN 1 q b¨¨¨b V pN r q and M b V pNq :" Mpλq b V pN 1 q b¨¨¨b V pN r q. In the particular case N 1 "¨¨¨" N r " 1, we write V r for the r-th folded tensor product V b V b¨¨¨b V .
be the set of weak compositions of b into r`1 parts, that is: Consider also P r,N b :" tpb 0 , b 1 , . . . , b r q P P r b |b i ď N i for 1 ď i ď ru Ă P r b . In addition to the induced basis by the tensor product, the space M b V pNq admits a basis that will be particularly useful for categorification. For ρ " pb 0 , . . . , b r q P P r b , we write v ρ :" F br`¨¨¨F b 1`F b 0 pv λ q b v N 1 ,0˘¨¨¨b v Nr,0˘. Then, M b V pN q has a basis given by ! v ρ |ρ P P r,N b , b ě 0 ) .
In particular, we have that M b V pN q λq |N|´2b has a basis given by tv ρ u ρPP r,N b . One can describe inductively the change of basis from tv ρ u ρPP r,N b to the induced basis as follows: v pb 0 ,...,brq " minpbr,Nrq ÿ for any pb 0 , . . . , b r q P P r b and for any pb 0 , . . . , b r´1 q P P r´1 b and 0 ď n ď N r , with " n k ‰ q :" rnsq! rksq!rn´ksq! .
We can also use these formulas to inductively rewrite a vector v ρ with ρ P P r b in terms of various v κ for κ P P r,N b . Indeed, we have v pb 0 ,...,brq " minpbr,Nrq ÿ for any pb 0 , . . . , b r q P P r b . 2.1.6. Shapovalov forms for tensor products. Following [49, §4.7], we consider a family of bilinear forms p¨,¨q λ,N on tensor products of the form Mpλq b V pN q satisfying the following properties: (1) each form p¨,¨q λ,N is non-degenerate; (2) for any v, v 1 P Mpλq b V pNq and u P U q psl 2 q we have pu¨v, v 1 q λ,N " pv,τ puq¨v 1 q λ,N ; (3) for any f P Qpq, λq and v, v 2.2. The blob algebra. Recall that the blob algebra B r is the Qpλ, qq-algebra with generators u 1 , . . . , u r´1 and ξ, and with the relations of type A: and the blob relations: Note that ξ is invertible, with inverse given by ξ´1 " λ`q´2λ´1´q´2ξ, and that the relations (3)- (6) imply that the generators u 1 , . . . , u r´1 generate a subalgebra isomorphic to the Temperley-Lieb algebra of type A.
The blob algebra has several well-known diagrammatic presentations. The most classical one already appeared in [30], but (a slight modification of) the one in [14] is more convenient for our purposes. This presentation is given by setting where diagrams are taken up to planar isotopy and read from bottom to top, and with local relations "´pλq`λ´1q´1q (8)
Remark 2.2. In the graphical description of B r given in [14] the generator ξ is presented as a double braiding (see [14, Figure 1]). We don't follow that interpretation in our diagrammatics in order to simplify pictures, but we keep the terminology (see §5.3 ahead). Remark 2.3. With respect to [14] our conventions switch pλ, qq and pλ´1, q´1q, which can be interpreted as exchanging the double braiding by the double inverse braiding.
There is an action of B r on M b V r that intertwines with the quantum sl 2 -action. This action can be described locally, identifying the first vertical strand in B r with the identity on Mpλq, and the ith vertical strand with the identity on the i-th copy of V in M b V r .
Then the action is given using the following maps where the formula for ξ is obtained by acting twice with an R-matrix. In our conventions, we have ξ " f˝Θ 21˝f˝Θ where Θ is given by the action of The following will be useful later: Proof. A computational proof is given in Appendix A.
As a matter of fact, this completely determines End Uqpsl 2 q pM b V r q: Theorem 4.9]). There is an isomorphism (13) B r -End Uqpsl 2 q pM b V r q.
The blob category B is the Qpλ, qq-linear category given by ‚ objects are non-negative integers r P N; ‚ Hom B pr, r 1 q is given by Qpλ, qq-linear combinations of string diagrams connecting r`1 points on the bottom to r 1`1 points on the top, with the first strand always connecting the left-most point to the left-most point, where the strings cannot intersect each other except for diagrams like ξ. Diagrams are considered up to planar isotopy and subject to the relations (6), (8) and (9). Let TL be the Temperley-Lieb category of type A, defined diagrammatically. It is a Qpqq-linear monoidal category equivalent to Fundpsl 2 q, the full monoidal subcategory of U q psl 2 q -mod generated by V . Note that B can be endowed with a structure of module category over TL, by gluing diagrams on the right.
Also consider the full subcategory MV Ă U q psl 2 q -mod given by the modules MpλqbV br for all r P N. It is a module category over Fundpsl 2 q by acting on the right with tensor product of U q psl 2 q-modules. Remark 2.7. Note that [14] considers projective Verma modules with integral highest weight. The case of universal Verma modules was studied in [26], albeit not in the categorical setup.

Dg-enhanced KLRW algebras
In [37] and [34] it was explained how to construct a 'dg-enhancement' of cyclotomic nilHecke algebras to pass from a categorification of the integrable module V pNq to a categorification of the Verma module Mpλq. This suggests that one might try to go from a categorification of V pNq b V pN q to a categorification of Mpλq b V pNq by constructing a dg-enhancement of KLRW algebras [49, §4], which we do next.
3.1. Preliminaries and conventions. Before defining the various algebras, we fix some conventions, and we recall some common facts about dg-structures (a reference for this is [15]). First, let k be a commutative unital ring for the remaining of the paper.
where we refer to the Z-grading as homological (or h-degree) and the Z n -grading as g-degree, with a differential d : A Ñ A such that: The homology of pA, d A q is HpA, d A q :" kerpdq{ impdq, which is a ZˆZ n -graded algebra that decomposes as À hPZ,gPZ n H h g pA, d A q :" H h pA g , d A q. A morphism of dg-algebras f : pA, d A q Ñ pA 1 , d A 1 q is a morphism of algebras that preserves the ZˆZ n -grading and commutes with the differentials. Such a morphism induces a morphism f˚: HpA, d A q Ñ HpA 1 , d A 1 q. We say that f is a quasi-isomorphism whenever f˚is an isomorphism. Also, we say that pA, Remark 3.1. Note that, in contrast to [15], the differential decreases the homological degree instead of increasing it.
Similarly, a Z n -graded left dg-module is a ZˆZ n -graded module M with a differential d M such that: M " 0. Homology, maps between dg-modules and quasi-isomorphisms are defined as above, and there are similar notions of Z n -graded right dg-modules and dg-bimodules.
In our convention, a Z m -graded category is a category with a collection of m autoequivalences, strictly commuting with each others. The category pA, d A q -mod of (left) Z n -graded dg-modules over a dg-algebra pA, d A q is a ZˆZ n -graded abelian category, with kernels and cokernels defined as usual. The action of Z is given by the homological shift functor r1s : pA, d A q -mod Ñ pA, d A q -mod acting by: ‚ increasing the degree of all elements in a module M up by 1, i.e. deg h pmr1sq " deg h pmq`1; ‚ switching the sign of the differential d M r1s :"´d M ; ‚ introducing a sign in the left-action r¨pmr1sq :" p´1q deg h prq pr¨mqr1s.
The action of g P Z n is given by increasing the Z n -degree of elements up by g, in the sense that pgMq g 0`g :" pMq g 0 , or in other terms, an element x P M with degree g 0 becomes of degree g 0`g in gM.
There are similar definitions for categories of right dg-modules and dg-bimodules, with the subtlety that the homological shift functor does not twist the right-action: pmr1sq¨r :" pm¨rqr1s.
As usual, a short exact sequence of dg-(bi)modules induces a long exact sequence in homology.
Let f : pM, d M q Ñ pN, d N q be a morphism of dg-(bi)modules. Then, one constructs the mapping cone of f as The mapping cone is a dg-(bi)module, and it fits in a short exact sequence: where ı N and π M r1s are the inclusion and projection N 3.1.2. Hom and tensor functors. Given a left dg-module pM, d M q and a right dg-module pN, d N q, one constructs the tensor product If pN, d N q (resp. pM, d M q) has the structure of a dg-bimodule, then the tensor product inherits a left (resp. right) dg-module structure.
Given a pair of left dg-modules pM, d M q and pN, d N q, one constructs the dg-hom space where HOM A is the ZˆZ n -graded hom space of maps between ZˆZ n -graded A-modules. Again, if pM, d M q (resp. pN, d N q) has the structure of a dg-bimodule, then it inherits a left (resp. right) dg-module structure.
In particular, given a dg-bimodule pB, d B q over a pair of dg-algebras pS, d S q-pR, d R q, we obtain tensor and hom functors which form a adjoint pair pBb pR,d R q´q $ HOM pS,d S q pB,´q. Explicitly, the natural bijection

Diagrammatic algebras.
We always read diagram from bottom to top. We say that a diagram is braid-like when it is given by strands connecting a collection of points on the bottom to a collection of points on the top, without being able the turnback. Suppose these diagrams can have singularities (like dots, 4-valent crossings, or other similar decorations).
A braid-like planar isotopy is an isotopy fixing the endpoints and that does not create any critical point, in particular it means we can exchange distant singularities f and g: Suppose that the diagrams carry a homological degree (associated to singularities), and consider linear combination of such diagrams. Then, a graded braid-like planar isotopy is an isotopy fixing the endpoints, that does not create any critical point and such that we get a sign whenever we exchange two distant singularities f and g: where |f | (resp. |g|) is the homological degree of f (resp. g).

3.2.
Dg-enhanced KLRW algebras. Let N " pN 1 , . . . , N r q. Recall the KLRW algebra [49, §4] (also called tensor product algebra) on b strands T N b is the diagrammatic k-algebra generated by braid-like diagrams on b black strands and r red strands. Red strands are labeled from left to right by N 1 , . . . , N r and cannot intersect each other, while black strands can intersect red strands transversely, they can intersect transversely among themselves and can carry dots. Diagrams are taken up to braid-like planar isotopy, and satisfy local relations (18)-(23) which are given below, together with the violating condition that a black strand in the leftmost region is 0: We write r T N b for the same construction but without the violating condition. The following are the defining (local) relations of T ‚ The nilHecke relations: "`" (19) ‚ The black/red relations: Multiplication is given by vertical concatenation of diagrams if the labels and colors of the strands agree, and is zero otherwise. As explained in [49, §4], the algebra T N b is finite-dimensional and Z-graded (we refer to this grading as q-grading), with In the case of N " pNq the algebra T pN q b contains a single red strand labeled N, and is isomorphic to the cyclotomic nilHecke algebra NH N b .
is the diagrammatic k-algebra carrying an homological degree generated by braid-like diagrams on b black strands, r red strands and a blue strand on the left. Red strands are labeled from left to right by N 1 , . . . , N r and the blue strand is labeled λ. Black strands can carry dots and intersect transversely with black and red strands. Moreover, the left-most black strand can be nailed on the blue strand, giving a 4-valent vertex as follows: λ We put the crossings and the dot in homological degree 0, while the nail is in homological degree 1. These diagrams are taken modulo graded braid-like planar isotopy, and subject to the local relations (18)-(23) of T N b , together with the local relations: Remark 3.3. Note that there can be no black or red strand at the left of the blue strand.
Remark 3.4. Note that since nails are stuck on the left, we can not exchange them using a graded braid-like planar isotopy. Thus, because nails are the only generators carrying a non-zero homological degree, we could consider diagrams up to usual braid-like planar isotopy. However, the homological degree of the nail will play an important role in the categorification of the structure constant rβ`ks q appearing in Mpλq b V pNq, and graded braid-like planar isotopy will play an important role in Section 5.
Clearly, there is an injection of algebra r given by adding a vertical blue strand at the left of a diagram in r with an extra Z 2 -grading, the first one being inherited from r T N b and denoted q, the second is written λ. We declare that deg q,λ¨λ‹ ‚:" p0, 2q, and the elements without a nail are all in degree λ zero and have the same q-degree as in (24), so that the inclusion r preserves the q-grading. One easily checks that it is well-defined.
In the case of N " ∅, the algebra T λ,∅ b contains only a blue strand labeled λ and no red strands, and is isomorphic to the dg-enhanced nilHecke algebra introduced in [37, Definition 2.3]. To match with the notation from [37], we write A b :" T λ,∅ b . We will often endow T λ,N b with a trivial differential, turning it into a Z 2 -graded dg-algebra 3.3. Basis theorem. For any ρ " pb 0 , b 1 , . . . , b r q P P r b , define the idempotent

Polynomial action.
We now define an action of the dg-algebra T λ,N b on Pol r b :" À ρPP r b Pol b ε ρ , the free module over the ring Pol b :" Zrx 1 , . . . , x b s b Ź ‚ pω 1 , . . . , ω b q generated by ε ρ for each ρ P P r b . We recall the action of the symmetric group S b on Pol b used in [37, §2.2]. We view S b as a Coxeter group with generators σ i " pi i`1q. The generator σ i acts on Pol b as follows, For κ, ρ P P r b , an element of 1 κ T λ,N b 1 ρ acts by zero on any Pol b ε ρ 1 for ρ 1 ‰ ρ and sends Pol b ε ρ to Pol b ε κ . It remains to describe the action of the local generators of T λ,N b on a polynomial f P Pol b . First, similarly as in [49,Lemma 4.12], we put Fix ρ " pb 0 , . . . , b r q P P r b . Let NH n be the nilHecke algebra on n-strands (it is described as a diagrammatic algebra with only black strands having dots and relations (18) and (19)). There is a map η ρ : where we recall that A b 0 is isomorphic to the dg-enhanced nilHecke algebra of [37], identifying the nilHecke generators with each other and the the nail with the "leftmost floating dot". The tensor product A b 0 b NH b 1 b¨¨¨NH br acts on Pol r b through η ρ . This action is only non-zero on Pol b ε ρ and it is readily checked that this action coincides with the tensor product of the polynomial actions of , and of the usual action of the nilHecke algebra NH

3.3.2.
Left-adjusted expressions. We recall the notion of a left-adjusted expression as in [37, Section 2.2.1]: a reduced expression σ i 1¨¨¨σ i k of an element w P S r`b is said to be leftadjusted if i 1`¨¨¨`ik is minimal. One can obtain a left-adjusted expression of any element of S r`b by taking recursively its representative in the left coset decomposition As one easily confirms, if we think of permutations in terms of string diagrams, then a left-reduced expression is obtained by pulling every strand as far as possible to the left.
. We now turn to the diagrammatic description of a basis of T λ,N b similar to [34, Section 3.2.3]. For an element ρ P P r b and 1 ď k ď b, we define the tightened nail θ k P 1 ρ T λ,N b 1 ρ as the following element: where the nailed strand is the k-th black strand counting from left to right. This element has degree deg h,q,λ pθ k q " p1,´4pk´1q`2pN 1`¨¨¨`Ni q, 2q. Lemma 3.7. Tightened nails anticommute with each other, up to terms with a smaller number of crossings: Proof. Similar to [34,Lemma 3.12], and omitted.
, . . . , b 0`. . .`b r`1 u, then we have θ 2 k " 0. Now fix κ, ρ P P r b and consider the subset of permutations κ S ρ Ă S r`b , viewed as diagrams with a blue strand, b black strands and r red strands, such that: ‚ the blue strand is always on the left of the diagram, ‚ the strands are ordered at the bottom by 1 ρ and at the top by 1 κ , ‚ for any reduced expression of w P κ S ρ , there are no red/red crossings. Example 3.9. If κ " ρ " p0, 1, 1q, then the set κ S ρ has two elements, namely and Note that the second element is not left-adjusted.
For each w P κ S ρ , l " pl 1 , . . . , l b q P t0, 1u b and a " pa 1 , . . . , a b q P N b we define an element b w,l,a P 1 κ T λ,N m 1 ρ as follows: (1) we choose a left-reduced expression of w in terms of diagrams as above; (2) for each 1 ď i ď b, if l i " 1, then we nail the i-th black strand at the top, counting from the left, on the blue strand by pulling it from its leftmost position; (3) finally, for each 1 ď i ď b, we add a i dots on the i-th black strand at the top.
Example 3.10. We continue the example of κ " ρ " p0, 1, 1q. If we choose l " p1, 0q and a " p0, 1q for w the permutation with a black/black crossing, after left-adjusting it, then we obtain b w,l,a " Proof. By Lemma 3.7, with arguments similar to [34,Proposition 3.13], one shows that this set generates 1 κ T λ,N m 1 ρ as a k-module. The proof consists in an induction on the number of crossings, allowing to apply braid-moves in order to reduce diagrams. In order to show that this set is linearly independent over k, we apply Lemma 3.6.
In the following, we draw T λ,N b 1 ρ with ρ " pb 0 , . . . , b r q as a box diagram Moreover, when we draw something like where N 1 " pN 1 , . . . , N r´1 q, and the isomorphism is given by inclusion.
Proof. The claim follows immediately from Theorem 3.11.
3.4. Dg-enhancement. For each N P N, we want to define a non-trivial differential d N on T λ,N b . First, we collapse the Z 2 -grading into a single Z-grading, which we also call q-degree, through the map Z 2 Ñ Z, pa, bq Þ Ñ a`bN (i.e. specializing λ " q N ). Then, we put , and extending by the graded Leibniz rule w.r.t. the homological grading. A straightforward computation shows that d N respects all the defining relations of T λ,N b , and therefore is well-defined.
Proof. The proof follows by similar arguments as in [34,Theorem 4.4], by using Corollary 3.12. We leave the details to the reader.

A categorification of Mpλq b V pNq
In this section we explain how derived categories of pT λ,N b , 0q-dg-modules categorify the U q psl 2 q-module Mpλq b V pNq. Since the construction is very similar to the one in [37] and [34], we will assume some familiarity with [37] and [34], and we will refer to these papers for several details.
We introduce the notations where we recall that q a λ b p´q is a shift up by pa, bq in the Z 2 -grading, and p´qr1s is a shift up by 1 in the homological grading. We

D
sending the unit 1 P T λ,N to the idempotent 1 b,1 . This map gives rise to derived induction and restriction dg-functors , 0q, so that we can replace derived tensor products (resp. derived homs) by usual tensor products and Id b is the identity dg-functor on D dg pT λ,N b , 0q.
Proof. Consider the map where φ is the projection onto the following summands of Corollary 3.12 (i.e. when i " r and t " b r´1 ). Note that, a priori, this only defines a map of left modules. Fortunately, by applying similar arguments as in [34,Lemma 5.4], it is possible to show that it defines a map of bimodules. Exactness follows from a dimensional argument using Corollary 3.12.
Introducing the differential d N from Section 3.4 in the picture, the map (27) lifts to a map of dg-algebras pT These corresponds with derived induction and (shifted) derived restriction dg-functors along (27), by Corollary 3.12 again.
By Proposition B.2, Theorem 4.1 can be seen as a quasi-isomorphism of mapping cones where h N is given by multiplication by the element where ' r´ksq M :" ' rksq Mr1s. that consists in adding a vertical red strand labeled N r`1 at the right a diagram: , 0q be the corresponding induction dg-functor, and let , 0q be the restriction dg-functor.
Proof. The statement follows from Corollary 3.12.

Categorification theorem.
In this section we suppose that k is a field. Recall the notion of an asymptotic Grothendieck group , 0q is a free Zppq, λqq-module generated by the classes of projective T λ,N b -modules with a trivial differential. Let Q K ∆ 0 p´q :" K ∆ 0 p´qb Zppq,λqq Qppq, λqq.
For each ρ P P r b , there is a projective T λ,N b -module given by Recall the inclusion η ρ : (26). It is well-known (see for example [21, § 2.2.3]) that NH n admits a unique primitive idempotent up to equivalence given by where ϑ n P S n is the longest element, τ w 1 w 2¨¨¨wk :" τ w 1 τ w 2¨¨¨τ w k , with τ i being a crossing between the i-th and pi`1q-th strands, and x i is a dot on the i-th strand. There is a similar result for NH b 0 Ă A b 0 (see [37, §2.5.1]). Moreover, for degree reasons, any primitive idempotent of T λ,N b is the image of a collection of idempotents under the inclusion η ρ for some ρ, and thus is of the form It is also well-known (see for example [ as left NH n -modules. For the same reasons, we obtain . As in [21, §2.5], let ψ : T λ,N Ñ pT λ,N q op be the map that takes the mirror image of diagrams along the horizontal axis. Given a left pT λ,N , 0q-module M, we obtain a right pT λ,N , 0q-module M ψ with action given by m ψ¨r :" p´1q deg h prq deg h pmq ψprq¨m for m P M and r P T λ,N . Then we define the dg-bifunctor p´,´q : D dg pT λ,N , 0qˆD dg pT λ,N , 0q Ñ D dg pk, 0q, pW, W 1 q :" W ψ b L pT λ,N ,0q W 1 .
Proposition 4.5. The dg-bifunctor defined above satisfies: Proof. Straightforward, except for the last point which follows from Proposition 4.4, together with the adjunction I $Ī.
Comparing Proposition 4.5 to Section 2.1.6, we deduce that p´,´q has the same properties on the asymptotic Grothendieck group of pT λ,r , 0q as the Shapovalov form on M b V r .

The categorification theorem. Let E :"
À bě0 E b and F :" À bě0 F b . By Theorem 4.1 and Proposition B.7, we know that Q K ∆ 0 pT λ,N , 0q is an U q psl 2 q-module, with action given by the pair rFs, rEs.
Proof. It is well-known (see for example [36,Lemma 7.2]) that whenever k ą n, then the unit element in NH k can be rewritten as a combination of elements having n consecutive dots somewhere on the left-most strand. Thus, for any ρ 1 P P r b , we obtain that 1 ρ 1 can be rewritten as a combination of elements factorizing through elements in t1 ρ u ρPP r,N b .
We consider Mpλq b V pNq over the ground ring Qppq, λqq instead of Qpq, λq.
Theorem 4.7. There are isomorphisms of U q psl 2 q-modules Proof. We have a Qppq, λqq-linear map , v ρ Þ Ñ rP ρ s. By Lemma 4.6, this map is surjective. It commutes with the action of K˘1 and E because of Corollary 3.12. By Proposition 4.5, the map intertwines the Shapovalov form with the bilinear form induced by the bifunctor p´,´q on Q K ∆ 0 pT λ,N , 0q. Thus, it is a Qppq, λqqlinear isomorphism. Since the map intertwines the Shapovalov form with the bifunctor p´,´q, and commutes with the action of E and K˘1, we deduce by non-degeneracy of the Shapovalov form that it also commutes with the action of F . Thus, it is a map of U q psl 2 q-modules.
The case Q K ∆ 0 pT λ,N , d N q follows from Theorem 3.13 together with [49,Theorem 4.38].

Cups, caps and double braiding functors
Throughout this section, we fix N " p1, 1, . . . , 1q P N r and write T λ,r :" T λ,N , and b T :" b p' rě0 T λ,r q . Also, when we will talk about (bi)modules, we will generally mean Z 2 -graded dg-(bi)module, assuming it is clear from the context. 5.1. Cup and cap functors. Following [49, §7] (see also [48, §4.3]), we define the cup bimodule B i for 1 ď i ď r`1 as the pT λ,r`2 , 0q-pT λ,r , 0q-bimodule generated by the diagrams Here, generated means that elements of B i are given by taking the diagram above and gluing any diagram of T λ,r`2 on the top, and any diagram of T λ,r on the bottom. The diagrams in B i are considered up to graded braid-like planar isotopy, with the cup being in homological degree 0, and subject to the same local relations as the dg-enhanced KLRW algebra (20)- (23) and (25), together with the following extra local relations: " " (30) We set the Z 2 -degree of the generator in (28) as deg q,λˆ˙: " p0, 0q.
Similarly, we define the cap bimodule B i by taking the mirror along the horizontal axis of B i . However, we declare that the cap is in homological degree´1, and with Z 2 -degree given by deg q,λˆ˙: " p´1, 0q.
Note that since the red cap has a´1 homological degree, it anticommutes with the nails when applying a graded planar isotopy.
From this, one defines the coevaluation and evaluation dg-functors as by Proposition 5.1 below. Thus, q´1B i r´1s is right adjoint to B i . Similarly, we obtain that qB i r1s is left-adjoint to B i . The unit and counit of B i % q´1B i r´1s gives a pair of maps of bimodules and similarly qB i r1s % B i gives

Tightened basis.
Take κ " pb 0 , . . . , b r`2 q P P r`2 b and ρ P P r b . Letκ i be given by   Recall the basis κ B ρ of Theorem 3.11. We claim that is a basis for 1 κ B i 1 ρ . We postpone the proof of this for later.   Let pB i be the left pT λ,r`2 , 0q-module given by the dg-module where the differential is given by the arrows, which are the maps given by adding the term in the label at the bottom of , or . Similarly, we define a right cofibrant replacement B i q » ։ B i by taking the symmetric along the horizontal line and shifting everything by q´1p´qr´1s.
Proof. Consider the surjective map T i, ։ B i that closes the elements at the bottom by a cup:

Þ Ñ
This map is indeed surjective since any black strand going to the left of the cap factors through a black strand going to the right, using (30). Then the claim follows by observing that is an exact sequence. Indeed, by Theorem 3.11, we know that adding a black/red crossing is an injective operation, and thus the sequence is exact on q 2 T i, . For the same reason we also have that By Theorem 3.11, we know that if an element can be written as a diagram with a black strand crossing a red strand on the left, and as a different diagram with the same black strand crossing a red strand on the right, then it can be rewritten as a diagram with the same strand going straight, but carrying a dot. These elements correspond exactly with the image of the preceding map in the complex, which is thus exact at the second position. Finally, we observe that Proof. As in Theorem 3.11, one can show that the elements in (31) span the space 1 κ B i 1 ρ , mainly using (30) and (23). Linear independence follows from a dimensional argument, using Proposition 5.1 and Theorem 3.11. The computation of the dimensions can be done at the non-categorified level, and thus is a consequence of (10) of Lemma 2.4.
Therefore, the map ř g ℓ : ' b i ℓ"1 q b i`1´2 ℓ p1ρiT λ,r q » Ý Ñ 1 ρ B i of right modules is an isomorphism, whereρ i and g ℓ are as in Section 5.1.2. In particular, B i is a cofibrant right dg-module.
With Theorem 4.7 in mind, this means that B i acts on Q K ∆ 0 pT λ,N , 0q as the cap of B on M b V r (see (10)), and Proposition 5.1 means that B i acts as the cup (see (11)    for all pb 0 , . . . , b r q P N r`1 . We consider diagrams in X up to graded braid-like planar isotopy with the generators being in homological degree 0, and subject to the relations (20)-(23) and (25), and the extra local relations We set the Z 2 -degree of the generator as deg q,λ¨1 λ‚ :" p0,´1q.
We define the double braiding functor as Ξ :" X b L T´: D dg pT λ,r , 0q Ñ D dg pT λ,r , 0q. 5.3.1. Tightened basis. Let us now describe a basis of the bimodule X, similar as the basis of T λ,r b given in Theorem 3.11. We fix κ and ρ two elements of P r b and recall the set κ S ρ defined in Section 3.3.3. For each w P κ S ρ , l " pl 1 , . . . , l b q P t0, 1u b and a " pa 1 , . . . , a b q P N b we define an element x w,l,a P 1 κ X1 ρ as follows: (1) choose a left-reduced expression of w in terms of diagrams as above, Note that the unbraiding map is a map of pT λ,r b , 0q-pT λ,r b , 0q-bimodules.
Proof. Showing that this set generates 1 κ X1 ρ is similar to [34, Proposition 3.13] and we leave the details to the reader.
To show that the elements px w,l,a q w,l,a are linearly independant we consider a linear combination ř w,l,a α w,l,a x w,l,a " 0 and apply the unbraiding map u. We now pull the first red strand to its original position before the last step of the construction of x w,l,a . This has the effect of adding dots on some black strands because of (21).
We now rewrite u´ř w,l,a α w,l,a x w,la¯" 0 in terms of the tightened basis of T λ,r b . We carefully look at the terms with the highest number of crossings: by pulling the dots at the top, we obtain different elements of the tightened basis of T λ,r b plus terms with a lower number of crossings. From the freeness of the tightened basis of T λ,r b , we deduce that the coefficient of the terms with the highest number of crossings must be zero and we can proceed by a descending induction on the number of crossings.
Proof. The matrix of u in terms of tightened bases can be made in column echelon form with pivots being 1.
Note that Y 1 0 " 0 and Y 0 0 " λ´1pT λ,r b 1 0,ℓ,ρ q. We write and similarly for Y 1,t k , Y 1 0 k and Y 0,t k . Define the cofibrant pT λ,r b , 0q-module pX k given by the mapping cone Note that each ı t k is injective, and therefore so is ı k . Then, consider the left module map γ k : pX k Ñ X k , given by γ k :" γ 1 for all 0 ď t ď k´1.
Lemma 5.7. The map γ k : pX k Ñ X k is surjective.
Proof. The statement can be proved by observing that X k is generated as a left pT λ,r b , 0qmodule by the elements for all 0 ď t ď k´1. The details can be found in Appendix A.2.
is a short exact sequence of left Z 2 -graded pT λ,r , 0q-modules.
Proof. Since we already have a complex with an injection and a surjection, it is enough to show that where gdim is the graded dimension in the form of a Laurent series in N h˘1, λ˘1, q˘1 . This can be shown by induction on k, and the details are in Appendix A.2.
Proof. It is an immediate consequence of Lemma 5.8.
Again, having Theorem 4.7 in mind, it means Ξ acts on Q K ∆ 0 pT λ,N , 0q as the element ξ of B on M b V r (see (12)).

A categorification of the blob algebra
As in [49, §7], the cup and cap functors respect a categorical instance of the Temperley-Lieb algebra relations (3)-(6). We additionally show that the double braiding functor respects a categorical version of the blob relations (8) and (9). Note that Webster also proves that the cup and cap functors intertwine the categorical U q psl 2 q-action, which categorifies the fact that the Temperley-Lieb algebra describes morphisms of U q psl 2 qmodules. We start by proving the same for these functors in the dg-setting as well as for the double braiding functors: Proposition 6.1. We have natural isomorphisms E˝Ξ -Ξ˝E and F˝Ξ -Ξ˝F, and also E˝B i -B i˝E , F˝B i -B i˝F , and similarly for B i .
Proof. Since E and F are given by derived tensor product with a dg-bimodule that is cofibrant both as left and as right module, all compositions are given by usual tensor product of dg-bimodules. Then, the first isomorphism is equivalent to 1 pT λ,r b`1 q, which in turn follows from Theorem 5.5 and Corollary 3.12. The case with F is identical, and so is the proof for B i using Corollary 5.2.
Then, we use all this to show that compositions of the functors B i ,B i and Ξ realize a categorification of B.

Proposition 6.2. There is an isomorphism
Proof. We proveB i´1 b L T B i -T λ,r , the other case follows similarly. Using Proposition 6.1 and the fact that B i˝I -I˝B i for i ă r´1 (where we recall I is the induction along a red strand defined in Section 4.1.2), we can work locally, supposing that i " r´1 and b i " 0. Then, we have thatB i´1 b T pB i looks like q¨‚r1s q 2¨‚ r2s ' q¨‚r1ś which is isomorphic to (29). Note that it is an isomorphism of dg-bimodules, since all the higher composition maps of the A 8 -structure must be zero by degree reasons, concluding the proof. Corollary 6.3. There is a natural isomorphismB i˘1˝Bi -Id.

Proposition 6.4. There is a distinguished triangle
Proof. We haveB Thus, since B i -T λ,r , we have that HpB i b L T B i q -qpT λ,r qr1s ' q´1pT λ,r qr´1s. In order to compute η i , recall (or see Appendix B.3.1) that the unit of the adjunction pB i b L T´q $ pRHOM T pB i ,´qq is given by Thus, η i identifies qpT λ,r qr1s with qpB i qr1s Ă HpB i b L T B i q in homology. Similarly, the counit of the adjunction pB i b L´q $ pRHOM T pB i ,´qq is and thus, ε 1 is the isomorphism B i » Ý Ñ T λ,r . Therefore, ε identifies q´1pT λ,r qr´1s with Because the connecting morphism in Proposition 6.4 is zero, the triangle splits and we have Corollary 6.5. There is a natural isomorphism B i˝Biq Idr1s ' q´1 Idr´1s.

Blob relations.
Proving the blob relations requires some preparation.
6.2.1. Quadratic relation. We define recursively the following element by setting z 0 :" 0, for all t ě 0. Note that z 2 is given by a single crossing since the second term is zero in this case. One easily sees that deg q pz t q " 2´2t. Define a map of left modules is given by multiplication on the bottom by Also define a map of left modules Recall that the unbraiding map (Definition 5.4) u : λX ãÑ T λ,r b , is given by Proof. The proof is a straightforward computation using (20) and (22) together with (32). We leave the details to the reader.
Thus, there is an induced map as left modules.
Theorem 6.7. The map Proof. The statement can be proven by showing that Conepϕq has a trivial homology, and thus is acyclic. This is done in details in Appendix A.3.1.
The next step is to prove that ϕ defines a map of A 8 -bimodules. Luckily, by the following proposition, we do not need to use any A 8 -structure here.
Proof. Tensoring to the left is a right-exact functor, thus Lemma 5.8 gives us an exact sequence It is not hard to see that 1 b ı k is injective, and thus we have a short exact sequence Taking a mapping cone preserves quasi-isomorphisms. Thus, we have a quasi-isomorphism be the map given by composing ϕ with the quasi-isomorphism in (34). We also writẽ ϕ 0 :" p1 b γq˝ϕ 0 . Therefore, by Lemma B.3, proving that ϕ is a map of A 8 -bimodules ends up being the same as proving thatφ 0 is a map of dg-bimodules.
Proof. The statement follows by proving thatφ 0 is a map of dg-bimodules, which is done in details in Appendix A.3.2.
Corollary 6.10. There is an exact sequence of dg-bimodules.  Proof. We have λ´1q´p k´2tq p1 0,k`ℓ,ρ Xqr´1s¸, Then, the map on the top of diagrams, for all 0 ď t ď k´1.
Proof. By Lemma 6.13 and Proposition 5.9, we have Then, we compute gdim HOM T pX1 ρ , X1 ρ 1 q " gdim HOM T pP ρ , P ρ 1 q, using the fact that Ξ decategorifies to the action of ξ. More precisely, as in [49, §4.7], the bifunctor RHOM T p´,´q decategorifies to a sesquilinear version of the Shapovalov form when restricted to a particular subcategory of D dg pT λ,r , 0q, and this sesquilinear form respects pξw, ξw 1 q " pw, w 1 q. Finally, we observe that the map HOM T pP ρ , P ρ 1 q Id X bp´q ãÝ ÝÝÝÝ Ñ HOM T pX1 ρ , X1 ρ 1 q, is injective, since the map P ρ Ñ X1 ρ given by gluing  on the top of diagrams is injective. This can be seen by composing the above map P ρ Ñ X1 ρ with the injection u : X1 ρ Ñ P ρ , and observing it yields an injective map. Therefore, RHOM T pX1 ρ , X1 ρ 1 q -1 ρ T λ,r 1 ρ 1 , and Ξ is an autoequivalence.

Categorification of relation (8).
Lemma 6.15. There is a quasi-isomorphism Proof. Let us write X :" X b T T 1, . Then we have The statement follows by observing that the first map is injective, and its image coincides with the kernel of the second one.
Our goal will be to show the following: For this, we will need to understand the left A 8 -action on B 1 q: qr´1s.

T
We start by constructing a composition map T b B 1 q Ñ B 1 q, by defining it on each generator of T . We extend it by first rewriting elements in T as basis elements and then applying recursively the definition in terms of generating elements (so that it is welldefined). Dots and crossings act on each of the summand by simply adding the three missing vertical strands between the λ-strand and the remaining of the diagram, and gluing on top. For example in q´1pT qr´1s, we have     One can easily verify that this respects the differential in B 1 q. The higher multiplication maps T bB 1 qbT Ñ B 1 q and T bT bB 1 q compute the defect of the map T bB 1 q Ñ B 1 q for being a left T -action. Concretely, it means that we can compute these higher multiplication maps by looking how both side of each defining relation of T act on B 1 q. For example, the relation λ " λ is respected on q´1pT qr´1s up to adding the elements appearing in the right of the following equation: Note that it means the higher maps only involve elements coming from (25). Also, one can easily verify that the other two relations in (25) are already respected for the multiplication map T b q´1pT qr´1s Ñ B 1 q, so that our computation above completely determine T b T b q´1pT qr´1s Ñ B 1 q. There is a similar higher multiplication map T b q´1pT qr´1s b T Ñ B 1 q, which is non-trivial in the case All higher multiplications maps vanish: except for all of these are zero for degree reasons, and the remaining two are zero by the calculations above. Therefore, what remains is isomorphic to λqpT λ,r qr1s ' λ´1q´1pT λ,r qr´1s, as dg-bimodules. We conclude by applying Lemma 6.15.
Corollary 6.17. There is a quasi-isomorphism of dg-functors.
6.3. The blob 2-category. In this section, we suppose k is a field. Let Bpr, r 1 q be the subcategory of dg-functors D dg pT λ,r , 0q Ñ D dg pT λ,r 1 , 0q c.b.l.f. generated by all compositions of Ξ, B i andB i , and identity functor whenever r " r 1 , where c.b.l.f. generated means it is given by certain (potentially infinite) iterated extensions of these objects (see Definition B.9 for a precise definition). As explained in Appendix B.4.4, there is an induced morphism Ý Ý Ñ Hom Qppq,λqq p Q K ∆ 0 pT λ,r , 0q, Q K ∆ 0 pT λ,r 1 , 0qq, sending the equivalence class of an exact dg-functor to its induced map on the asymptotic Grothendieck groups of its source and target (this is similar to the fact that an exact functor between triangulated categories induces a map on their triangulated Grothendieck groups).
Recall the blob category B, but consider it as defined over Qppq, λqq instead of Qpq, λq.
Theorem 6.18. There is an isomorphism Comparing the action of B on M b V r from Section 2.2 with the cofibrant replacement pX from Section 5.4, and pB i and pB i from Section 5.2, we deduce there is a commutative diagram where the arrow f is the obvious surjective one, sending ξ to rΞs, and cup/caps to rB i s/rB i s. Because the diagram commutes and using Theorem 2.5, we deduce that f is injective, and thus it is an isomorphism.
In particular, if we write B r :" Bpr, rq, then we have: There is an isomorphism of Qppq, λqq-algebras Q K ∆ 0 pB r q -B r . By Faonte [9], we know that A 8 -categories form an p8, 2q-category, where the homspaces are given by Lurie's dg-nerves [28] of the dg-categories of A 8 -functors (or equivalently quasi-functors, see Appendix B.2.1). Thus, we can define the following: Definition 6.20. Let B be the p8, 2q-category defined by ‚ objects are non-negative integers r P N (corresponding to D dg pT λ,r , 0q); ‚ Hom B pr, r 1 q is Lurie's dg-nerve of the dg-category Bpr, r 1 q. We refer to B as the blob 2-category.
We define Q K ∆ 0 pBq to be the category with objects being non-negative integers r P N and homs are given by asymptotic Grothendieck groups of the homotopy categories of Hom B pr, r 1 q. These homs are equivalent to Q K ∆ 0 pBpr, r 1 qq. Corollary 6.21. There is an equivalence of categories Q K ∆ 0 pBq -B. 7. Variants and generalizations 7.1. Zigzag algebras. In [45, §4] it was proven that for g " sl 2 the KLRW algebra T 1,...,1 1 with r red strands and only one black strand is isomorphic to a preprojective algebra A ! r of type A. It is a Koszul algebra, whose quadratic dual was used by Khovanov-Seidel in [24] to construct a categorical braid group action.
Let k be a field of any characteristic and let Q r be the following quiver and kQ r its path algebra. We endow kQ r with a ZˆZ 2 -grading by declaring that degpi|i˘1q :" p0, 1, 0q, degpθq :" p1, 0, 2q.
We consider the first grading as homological, and the second and third gradings are called the q-grading and the λ-grading respectively. We denote the straight path that starts on i 1 and ends at i n by pi 1 |i 2 | . . . |i n´1 |i n q and the constant path on i by piq. The set tp0q,¨¨¨, prqu forms a complete set of primitive orthogonal idempotents in kQ r .
Definition 7.1. Let A ! r be algebra given by the quotient of the path algebra kQ r by the relations pi|i´1|iq " pi|i`1|iq, for i ą 0, θp0|1|0q " p0|1|0qθ, We usually consider A ! r as a dg-algebra pA ! r , 0q with zero differential. We can also consider a version of A ! r with a non-trivial differential d given by of which one easily checks that it is well-defined. Furthermore, the isomorphism upgrades to isomorphisms of dg-algebras pA ! r , 0q -pT λ,r 1 , 0q and pA ! r , dq -pT λ,r 1 , d 1 q.
Proof. First, one can show by a straightforward computation that the map defined above respects all defining relations of A ! r . Moreover, by turning any dot in T λ,r 1 to a double crossings using (21), it is not hard to construct an inverse of the map defined above. We leave the details to the reader. Moreover, by Proposition 7.2, the results in Section 6 can be pulled to the derived category of Z 2 -graded pA ! r , 0q-modules, endowing D dg pA ! r , 0q -D dg pT λ,r 1 , 0q with a categorical action of B r . 7.2. Dg-enhanced KLRW algebras: the general case. Fix a symmetrizable Kac-Moody algebra g with set of simple roots I and dominant integral weights µ :" pµ 1 , . . . , µ d q.
7.2.1. Dg-enhanced KLRW algebras: g symmetrizable. Recall that the KLRW algebra [49, §4] T µ b pgq on b strands is the diagrammatic k-algebra generated by braid-like diagrams on b black strands and r red strands. Red strands are labeled from left to right by µ 1 , . . . , µ r and cannot intersect each other, while black strands are labeled by simple roots and can intersect red strands transversally, they can intersect transversally among themselves and can carry dots. Diagrams are taken up to braid-like planar isotopy and satisfy the following local relations: ‚ the KLR local relations (2.5a)-(2.5g) in [49,Definition 2.4]; ‚ the local black/red relations (36)-(39) for all ν P µ and for all α j , α k P I, given below; ‚ a black strand in the leftmost region is 0.
Multiplication is given by concatenation of diagrams that are read from bottom to top, and it is zero if the labels do not match. The algebra T µ b pgq is finite-dimensional and can be endowed with a Z-grading (we refer to [49,Definition 4.4] for the definition of the grading).
In the case of µ " ν the algebra T ν b pgq contains a single red strand labeled ν and is isomorphic to the cyclotomic KLR algebra R ν pbq for g in b strands.
Definition 7.4. Fix a g-weight λ " pλ 1 , . . . , λ |I| q with each λ i being a formal parameter. The dg-enhanced KLRW algebra T λ,µ b pgq is defined as in Definition 3.2, with a blue strand labeled by λ and with the r red strands labeled by µ 1 , . . . , µ r and the black strands labeled by simple roots. The black strands can carry dots and be nailed on the blue strand: α j λ with everything in homological degree 0, except that a nail is in homological degree 1. The diagrams are taken up to graded braid-like planar isotopy, and are required to satisfy the same local relations as T µ b pgq, together with the following extra local relations: for all α j , α k P I. We usually consider T λ,µ b pgq as a Z 1`|I| -graded dg-algebra pT λ,µ b pgq, 0q with trivial differential. In the case of µ " ∅ the algebra T λ,∅ b pgq contains a blue strand labeled λ and is

Note that we have an inclusion T
The results of Section 3 can be generalized to T λ,µ b pgq. In particular, one can prove it is free over k and that it admits a basis similar to the one in Theorem 3.11. Moreover, by using induction and restriction functors that add a black strand, we obtain a categorical action of g on D dg pT λ,µ b pgq, 0q (in the sense of [34]), which categorifies the U q pgq-action on the tensor product of a universal Verma module and several integrable modules.
Fix an integrable dominant weight κ of g and define a differential d κ on T λ,µ b pgq (after specialization of the λ j -grading to q κ j ) by setting Proof. The proof follows by similar arguments as in [34,Theorem 4.4].

7.2.2.
Dg-enhanced KLRW algebras for parabolic subalgebras. Let p Ď g be a parabolic subalgebra with partition I " I f \ I r of the set of simple roots, and pλ, nq " pλ i q iPI , with λ i a formal parameter if i P I r , and λ i " q n i with n i P N if i P I f .
Introduce a differential d λ,n on T λ,µ b pgq (after specialization of the λ j -grading to q n j for each α j P I r ) by setting and d λ,n ptq " 0 for all t P T µ b pgq Ă T λ,µ b pgq, and extending by the graded Leibniz rule w.r.t. the homological grading. As before, a straightforward computation shows that it is well-defined.
Definition 7.7. We define the dg-enhanced p-KLRW algebra as Note that by Proposition 7.6 we have a quasi-isomorphism pT Similarly as above, D dg pT λ,µ b pgq, d λ,n q categorifies the tensor product of a parabolic Verma module and several integrable modules, and comes with a categorical action of g.

7.
3. Dg-enhanced quiver Schur algebras. In order to define a quiver Schur algebra of type A 1 , we follow the approach of [23], which best suits our goals. We actually use a slightly different definition because theirs corresponds to a thick version of KLRW algebra (see [23, §9.2]), and we want to relate it to the version we use.

Cyclic modules and quiver Schur algebras. Recall that NH
is the Ncyclotomic nilHecke algebra on b strands. Fix r ě 0 and N " pN 0 , N 1 , . . . , N r q P N r such that where we recall that x b is a dot on the bth black strand. Then, we consider the cyclic right The quiver Schur algebra (of type A 1 ) is defined as the Z-graded algebra: where END means the (Z-)graded endomorphism ring. The Z-graded algebra Q N b is isomorphic to T N b [49,Proposition 5.33]. The reduced quiver Schur algebra (of type A 1 ) is defined as (this can be shown by observing that if b i ą N i for some i, then Y N ρ is isomorphic to a direct sum of elements in tq´d eg q px N ρ 1 q{2 Y N ρ 1 |ρ 1 P P r,N b u ), and thus to T N b .

Dg-enhanced cyclic modules.
Our goal is to construct a dg-enhancement of Y N ρ over pT λ,H b , d N q, the dg-enhanced KLRW algebra without red strands. We will simply write T λ for ℓ P Z be the algebra defined similarly as T λ b (see Definition 3.2) except that the blue strand is labeled by q ℓ λ, and the nail is in Z 2 -degree: Whenever ℓ ě ℓ 1 and b ď b 1 , there is an inclusion of algebras given by first turning any q ℓ λ-nail into a q ℓ 1 λ-nail by adding dots: so that the blue strand labeled q ℓ λ becomes labeled q ℓ 1 λ, and then adding b 1´b vertical black strands at the right: A straightforward computation shows that the map in (40) is well-defined, and Theorem 3.11 shows that the map is injective. By restriction, the inclusion defines a left action of T q ℓ λ b on any T q ℓ 1 λ b 1module.
Definition 7.8. We define the right T λ b -modules Note that we can endow G N ρ with either a differential of the form d N (as in Section 3.4) or a trivial one, making it a right dg-module over pT λ b , d N q or pT λ b , 0q respectively. Example 7.9. Take for example r " 2. Then, we picture G N ρ in terms of diagrams as: Note that whenever N`ℓ ě 0 we can equip T q ℓ λ b with a differential d N given by and it is compatible with the inclusion in (40). We conjecture the following: Proof. The claim follows from Theorem 3.11.
Proposition 7.12. Suppose ρ and ρ 1 are such that b i " b 1 i for all 0 ď i ď m except i " j and i " j`1 where they respect b j " b 1 j´1 and b j`1 " b 1 j`1`1 . Then there is an inclusion of right dg-modules Proof. We can work locally, and thus we want to prove that We apply Lemma 7.11 on T q´nλ k`1 inside G 2 . The left summand is clearly in G 1 . For the right summand, it is less clear since the nails in T λ b all acts by adding a nail and n dots on the blue strand labeled q´nλ. Thus, we want to show that n n¨¨n k¨¨q´nλ P G 1 . The term of the left is clearly in G 1 since there are n dots next to the nail, so that it can be obtained from a nail in T λ b . The terms on the right are also in G 1 since we can slide the nail and crossings on the left to the top, into T q´nλ

By (19), we have
We obtain an inclusion px n 1¨¨¨x n k qT q´nλ k ãÑ px n 1¨¨¨x n k x n k`1 qT q´nλ k`1 , of q-degree 2n by adding a vertical strand on the right on which we put n dots (again, the fact it is an inclusion follows immediatly from Theorem 3.11). In turns, it gives rise to a map of right (dg-)modules px n 1¨¨¨x n k qT q´nλ In terms of diagrams, we can picture the inclusion above as: This generalizes into the following proposition: Proposition 7.13. Under the same hypothesis as in Proposition 7.12, we obtain a map of right dg-modules

Dg-quiver Schur algebra.
Definition 7.14. We define the dg-quiver Schur algebras as where END dg is the Z 2 -graded (Z-graded in the first case) dg-endomorphism ring (see Section 3.1.2). We also define a reduced version as Conjecture 7.15. There is a quasi-isomorphism Our goal is to construct a graded map of algebras Dots on the ith black strand (resp. black/black crossings on the ith and pi`1qth black strands) on 1 ρ is sent to multiplication on the left (i.e. gluing on top) by a dot on the ith black strand (resp. crossing) on G N ρ . These are indeed maps of right T λ b -modules since the dots and crossing commutes with x n i x n i`1 for all n ě 0. Similarly, a nail on the blue strand labeled λ in T λ,N b is sent to multiplication on the left by a nail on the blue strand labeled q´N r´¨¨¨´N1 λ in G N ρ . For black/red crossing τ i , if the red strand goes from bottom left to top right, then we have 1 ρ 1 τ i 1 ρ where ρ and ρ 1 are as in Proposition 7.12. Then, we associate to it the map G N ρ Ñ G N ρ 1 of Proposition 7.12. If the red strand goes from bottom right to top left, then we have 1 ρ τ i 1 ρ 1 , and we associate to it the map G N ρ 1 Ñ G N ρ of Proposition 7.13.
Proposition 7.16. The map defined above gives rise to maps of Z-graded dg-algebras and of Z 2 -graded dg-algebras Proof. We show the assignment given above is a map of algebras, the commutation with the differentials being obvious since the image by d N 0 of a nail on a blue strand labeled λ consists of N 0 dots on the first black strand; and the image by d N of a nail on a blue strand labeled q´N r´¨¨¨´N1 λ consists of N´N r´¨¨¨´N1 " N 0 dots. Thus, we need to prove the map respects all the defining relations in Definition 3.2. Relations in (18) and (19) are immediate by construction. The relations in (20) follow from commutations of dots. Since the map in Proposition 7.13 is multiplication by n j`1 dots and the map in Proposition 7.12 is an inclusion, we have the relations in (21). For the left side of (22) both black/red crossings are given by an inclusion, and thus commutes with the multiplication on the left by the black/black crossing. For the right side, the black/red crossings give a multiplication by x N j`1 i x N j`1 i`1 , which commutes with the black/black crossing. For (23), one the black/red crossing is an inclusion and the other one is multiplication by x N j`1 i on both side of the equality, so that the relation follows from (19). Finally, the relation in (25) is immediate by construction. We also conjecture that the reduced dg-quiver Schur algebra p red dg Q N b , 0q is dg-Morita equivalent to the non-reduced one p dg Q N b , 0q.

Appendix A. Detailed proofs and computations
We give the detailed computations used to prove various results of the paper.
Proof. We start with the cap. We have We now turn to the cup. It suffices to do the computation for i " r`1 because of the recursive definition of v ρ . By definition, v pb 0 ,...,bnq is sent to´qv pb 0 ,..
Lemma 5.7. The map γ k : pX k Ñ X k is surjective.
Proof. First, we recall the following well-known relation (41) "ẃ hich follows easily from (18) and (19). We also observe (42) 1 λ Consequently, using Theorem 5.5, we deduce that X k is generated as left pT λ,r b , 0q-module by the elements for all 0 ď t ď k´1. In particular, γ k is surjective.
Lemma A.1. Suppose r " 1 and ℓ " 0. As a ZˆZ 2 -graded k-module, X1 k,0 admits a decomposition Proof. It follows from Theorem 5.5 that we have a decomposition concluding the proof.
Lemma 5.8. The sequence is a short exact sequence of left pT λ,r , 0q-modules.
Proof. Since we already have a complex with an injection and a surjection, it is enough to show that gdim X k " gdim Y 0 k´g dim Y 1 k , where gdim is the graded dimension in the form of a Laurent series in N h˘1, λ˘1, q˘1 . We will show this by induction on k. When k " 0, this is immediate. Suppose it is true for k, and we will show it for k`1. Let Note that rk`1s q " qrks q`q´k , (44) rβ´k`1s h q " q´1rβ´ks h q´h λq´k, and rk`1s q rβ´k`1s h q " rks q rβ´ks h q`q´1´k rβ´ks h q´h λq´krk`1s q .
The general case follows from a similar argument, using the fact that X decomposes similarly to T λ,r b whenever r ą 1, that is as in Corollary 3.12, replacing all T by X. We leave the details to the reader.
A.3. Proofs of Section 6. Lemma 6.12. As a right pT λ,r , 0q-module, 1 1,k`ℓ´1,ρ X is generated by the elements Proof. We prove this claim using an induction on k. The case k " 1 is obvious. We suppose it is true for k´1, and thus it is enough to show that we can generate the element: The second term on the right-hand side is generated by the second element in (48). For the first term of the right-hand side, we slide the dot to the left using repeatedly (19): Because of the symmetric of (42), the first term on the right-hand side is generated by the second element in (48). We now prove that every element of the sum on the right-hand side is generated by elements in (48). By applying the induction hypothesis, it suffices to show that for every 1 ď j ď k´2, the elements    Another application of the symmetry of (42) deals with the first term, and every term of the sum is handled trough a descending induction on j, noting that the sum is zero if j " k´2.
For the second term, we apply once again (18) is a quasi-isomorphism.
The goal of this section is to prove Theorem 6.7, which we will achieve by showing that Conepϕ k q is acyclic. We have that Conepϕ k q is given by the complex The map ϕ 1 k´u is injective since u is injective by Corollary 5.6, and the map u b γ k is surjective. We want to first show that ϕ 0 k`1 b ı k is surjective on the kernel of u b γ k . This requires some preparation.
Proof. Let K Ă X b T Y 0 k be the submodule generated by the elements in (58). A straightforward computation shows that K Ă kerpu b γ k q. Thus, we have a complex where the left arrow is an injection and the right arrow is a surjection. Furthermore, by Theorem 5.5, we have that Therefore, by Lemma 5.8 we obtain that the sequence in (59) is exact. In particular, we have K " kerpu b γ k q.
Proof. We will show by backward induction on t that the elements (58) are all in impϕ 0 k1 b ı k q. The case t " k´1 is Lemma A.5. The induction step is essentially similar to the proof of Lemma A.5. In particular we want to show that (58) is in for 0 ď t 1 ď k´1, and  Thus, since t 1 ą t, by induction hypothesis we know that Then, by the same arguments as in Lemma A.5, that is using (49), we obtain (58).
Lemma A.8. The map ϕ 0 k is injective. Proof. Since adding black/red crossings is injective, it is enough because of Lemma A.3 to show that the left T λ,0 k -module map is injective. Since T λ,0 k is isomorphic to the dg-enhanced nilHecke algebra of [37], we know by the results in [37, Proposition 2.5] that there is a decomposition where the box labeled NH k´1 is the nilHecke algebra, and the circle labeled k is the algebra generated by labeled floating dots in the rightmost region (see [37, §2.4]). These floating dots correspond to combinations of nails, dots and crossings, giving elements that are in the (graded w.r.t. the homological degree) center of T λ,0 k . Furthermore, the map

NH k´1
is injective (this can be deduced by sliding all dots to the bottom using (19), and then using a basis theorem as for example in [21,Theorem 2.5] to see the map takes the form of a column echelon matrix with 1 as pivots).
Therefore, after decomposing T λ,0 k , (60) yields a column echelon form matrix with injective maps as pivots, and thus is injective.
Proof. First, recall that ı k is injective (as explained in Section 5.4). Thus, both p1 b ı k q and ϕ 0 k are injective, and we get We observe that imp1 b ı k q X impϕ 0 k q X pX b T Y 0,t k q is generated by The case with a nail is similar, concluding the proof.
Proof of Theorem 6.7. Since ϕ 1 k´u is injective, and u b γ k is surjective, and by Proposition A.7 and Proposition A.9, we conclude that Conepϕ k q is acyclic for all k. Consequently, ϕ is a quasi-isomorphism.
The goal of this section is to prove Theorem 6.9. To this end, we first prove that the mapφ 0 : q 2 pT λ,r b qr1s Ñ X b T X is a map of bimodules.      for all t ě 0.
Proof. Recall thatφ 0 :" p1 b γq˝ϕ 0 . Then, we obtain We prove the statement by induction on k. The claim is clearly true for k " 0 and k " 1. Suppose it is true for k`1, and we will show it is true for k`2.

ϕpk`1q
We prove the statement by induction on t. The case t " 0 follows from (42). Suppose the claim is true for t ě 0. We compute using (63) . which is symmetric with respect to taking the mirror image along the horizontal axis. Therefore, we get the same a crossing at the bottom ofφpt`3q, finishing the proof.    for all t ě 0.
Proof. We prove the statement by induction on t. The case t " 0 follows from (32). We suppose the claim is true for t ě 0. We compute using the mirror of (63), finishing the proof.
Proposition A. 16. The mapφ 0 is a map of dg-bimodules.
Proof. As already mentioned above, it is enough to show that the left and right action by the same element of T λ,r b on ÿ k`ℓ`|ρ|"b p´1q kφ pkq b1 ℓ,ρ coincide. We obtain commutation with dots and crossings by induction on k, using Lemmas A.11-A. 15. The commutation with a nail also comes from a straightforward induction on k, where the base case is immediate by (25). As explained in [29, §2.3], there is (in general) no right pR, d R q-action on pB i compatible with the left pS, d S q-action. However, there is an induced A 8 -action (defined uniquely up to homotopy), so that the quasi-isomorphism π B : pB » Ý Ñ Ñ B can be upgraded to a map of A 8 -bimodules. Lemma B.3. Let pA, d A q be a dg-algebra, and let U and V be dg-(bi)modules over pA, d A q, with a fixed cofibrant replacement π V : pV Ñ V . Suppose p1 b p V q : U b pA,d A q pV » Ý Ñ U b pA,d A q V is a quasi-isomorphism. If f : Z Ñ U b pA,d A q pV is a map of complexes of graded k-spaces, and f˝p1 b p V q is a map of dg-(bi)modules, then there is an induced map f : Z Ñ U b L V of A 8 -(bi)modules whose degree zero part is f .
Proof. We take f :" p1bp V q´1˝pf˝p1bp V qq, as a composition of maps of A 8 -(bi)modules, since any map of (bi)module can be considered as a map of A 8 -(bi)modules with no higher composition.
Note that the equivalent statement also holds for a cofibrant replacement Uq Ñ U such that pπ U b 1q : B.2. Dg-derived categories. One of the issues with triangulated categories is that the category of functors between triangulated categories is in general not triangulated. To fix this, we work with a dg-enhancement of the derived category. In particular, this allows us to talk about distinguished triangles of dg-functors.
Recall that a dg-category is a category where the hom-spaces are dg-modules over pk, 0q, and compositions are compatible with this structure (see [15, §1.2] for a precise definition). Given such a dg-category C with hom-spaces Hom C pX, Y q " p À hPZ Hom h pX, Y q, d X,Y q, we can consider its underlying category Z 0 p Cq, which is given by the same objects as C and hom-spaces Hom Z 0 p Cq pX, Y q :" ker`d X,Y : Hom 0 pX, Y q Ñ Hom´1pX, Y q˘.
Similarly, the homotopy category H 0 p Cq is given by Hom H 0 p Cq pX, Y q :" H 0 pHom C pX, Y qq.
A dg-enhancement of a category C 0 is a dg-category C such that H 0 p Cq -C 0 .
The dg-derived category D dg pA, d A q of a Z n -graded dg-algebra pA, d A q is the Z n -graded dg-category with objects being cofibrant dg-modules over pA, d A q, and hom-spaces being subspaces of the graded dg-spaces HOM pA,d A q from (16), given by maps that preserve the Z n -grading: Hom D dg pA,d A q pM, Nq :" HOM pA,d A q pM, Nq 0 , for pM, d M q and pN, d N q cofibrant dg-modules. By construction, we have H 0 p D dg pA, d A qq -DpA, d A q. Moreover, D dg pA, d A q is a dg-triangulated category, meaning its homotopy category is canonically triangulated (see [46] for a precise definition, or [34, Appendix A] for a summary oriented toward categorification), and this triangulated structure matches with the usual one on DpA, d A q.
B.2.1. Dg-functors. A dg-functor between dg-categories is a functor commuting with the differentials. Given a dg-functor F : C Ñ C 1 , it induces a functor on the homotopy categories rF s : H 0 p Cq Ñ H 0 p C 1 q. We say that a dg-functor is a quasi-equivalence if it gives quasi-isomorphisms on the hom-spaces, and induces an equivalence on the homotopy categories. We want to consider dg-category up to quasi-equivalence. Let Hqe be the homotopy category of dg-categories up to quasi-equivalence , and we write RHom Hqe for the dg-space of quasi-functors between dg-categories (see [46], [47], or [34, Appendix A]). These quasi-functors induce honest functors on the homotopy categories. Whenever C 1 is dg-triangulated, then RHom Hqe p C, C 1 q is dg-triangulated.
Remark B.4. The space of quasi-functors is equivalent to the space of strictly unital A 8 -functors.
It is in general hard to understand the space of quasi-functors. However, by the results of Toen [46], if k is a field and pA, d A q and pA 1 , d A 1 q are dg-algebras, then it is possible to compute the space of 'coproduct preserving' quasi-functors RHom cop Hqe p D dg pA, d A q, D dg pA 1 , d A 1 qq, in the same way as the category of coproduct preserving functors between categories of modules is equivalent to the category of bimodules. Indeed, we have a quasi-equivalence (64) RHom cop Hqe p D dg pA, d A q, D dg pA 1 , d A 1 qq -D dg ppA 1 , d A 1 q, pA, d A qq, where D dg ppA 1 , d A 1 q, pA, d A qq is the dg-derived category of dg-bimodules. Composition of functors becomes equivalent to derived tensor product. Then, understanding the triangulated structure of RHom cop Hqe p D dg pA, d A q, D dg pA 1 , d A 1 qq becomes as easy as to understand DppA, d A q, pA 1 , d A 1 qq. In particular, a short exact sequence of dg-bimodules gives a distinguished triangle of dg-functors. B.3. Derived hom and tensor dg-functors. Let pR, d R q and pS, d S q be dg-algebras. Let M and N be pR, d R q-module and pS, d S q-module respectively. Let B be a dg-bimodule over pS, d S q-pR, d R q. Then, the derived tensor product is B b L pR,d R q M :" B b pM, and the derived hom space is RHOM pS,d S q pB, Nq :" HOM pS,d S q pB, iNq.
Note that we have quasi-isomorphisms as dg-spaces B b L pR,d R q M -Bq b pR,d R q pM -Bq b pR,d R q M, and RHOM pS,d S q pB, Nq -HOM pS,d S q ppB, iNq -HOM pS,d S q ppB, Nq.
This defines in turns triangulated dg-functors B b L pR,d R q p´q : D dg pR, d R q Ñ D dg pS, d S q, and RHOM pS,d S q pB,´q : D dg pS, d S q Ñ D dg pR, d R q.
B.4.2. C.b.l.f. structures. We fix an arbitrary additive total order ă on Z n . We say that a Z n -graded k-vector space M " À À gPZ n M g is c.b.l.f. (cone bounded, locally finite) dimensional if ‚ dim M g ă 8 for all g P Z n ; ‚ there exists a cone C M Ă R n compatible with ă and e P Z n such that M g " 0 whenever g´e R C M .
Let pA, d A q be a Z n -graded dg-algebra. Suppose that pA, dq is concentrated in nonnegative homological degrees, that is A h g " 0 whenever h ă 0. The c.b.l.f. derived category D cblf pA, d A q of pA, d A q is the triangulated full subcategory of DpA, d A q given by dg-modules having homology being c.b.l.f. dimensional for the Z n -grading. There exists also a dgenhanced version D cblf dg pA, d A q. We write K ∆ 0 pA, dq :" K ∆ 0 p D cblf pA, d A qq.
Definition B.5. We say that pA, dq is a positive c.b.l.f. dg-algebra if (1) A is c.b.l.f. dimensional for the Z n -grading; (2) A is non-negative for the homological grading; (3) A 0 0 is semi-simple; (4) A h 0 " 0 for h ą 0; (5) pA, d A q decomposes a direct sum of shifted copies of modules P i :" Ae i for some idempotent e i P A, such that P i is non-negative for the Z n -grading.
In a Z n -graded triangulated category C, we define the notion of c.b.l.f. direct sum as follows: ‚ take a a finite collection of objects tK 1 , . . . , K m u in C; ‚ consider a direct sum of the form à gPZ n x g pK 1,g '¨¨¨' K m,g q, with K i,g " k i,g à j"1 K i rh i,j,g s, where k i,g P N and h i,j,g P Z such that: ‚ there exists a cone C compatible with ă, and e P Z n such that for all j we have k j,g " 0 whenever g´e R C; ‚ there exists h P Z such that h i,j,g ě h for all i, j, g.
If C admits arbitrary c.b.l.f. direct sums, then K ∆ 0 p Cq has a natural structure of Zppx 1 , . . . , x n qq-module with ÿ gPC a g x e`g rXs :" r à gPC x g`e X 'ag s, where X 'ag " À |ag| ℓ"1 Xrα g s and α g " 0 if a g ě 0 and α g " 1 if a g ă 0. . . , K m u be a finite collection of objects in C, and let tE r u rPN be a family of direct sums of tK 1 , . . . , K m u such that À rPN E r is a c.b.l.f. direct sum of tK 1 , . . . , K m u. Let tM r u rPN be a collection of objects in C with M 0 " 0, such that they fit in distinguished triangles M r fr Ý Ñ M r`1 Ñ E r Ñ Then, we say that an object M P C such that M -T MColim rě0 pf r q in T is a c.b.l.f. iterated extension of tK 1 , . . . , K m u.
Note that under the conditions above, we have rMs " ÿ rě0 rE r s, in the asymptotic Grothendieck group K 0 p Cq.
Definition B.9. Let T be a Z n -graded (dg-)triangulated (dg-)category, and tX j u jPJ be a collection of objects in T. The subcategory of T c.b.l.f. generated by tX j u jPJ is the triangulated full subcategory C Ă T given by all objects Y P T such that there exists a finite subset tX k u kPK such that Y is isomorphic to a c.b.l.f. iterated extension of tX k u kPK in T.