Quantum annular homology and bigger Burnside categories

As part of their construction of the Khovanov spectrum, Lawson, Lipshitz and Sarkar assigned to each cube in the Burnside category of finite sets and finite correspondences, a finite cellular spectrum. In this paper we extend this assignment to cubes in Burnside categories of infinite sets. This is later applied to the work of Akhmechet, Krushkal and Willis on the quantum annular Khovanov spectrum with an action of a finite cyclic group: we obtain a quantum annular Khovanov spectrum with an action of the infinite cyclic group.


Introduction
In 2000, Mikhail Khovanov introduced the first homological link invariant as a categorification of Jones polynomial [6].In its construction, entirely combinatorial, one associates a bigraded chain complex to a given link diagram, such that its homology is a link invariant known as Khovanov homology.
Some years later, Lipshitz and Sarkar introduced a stable homotopy refinement of Khovanov homology [9].More precisely, they provided a method to associate to a given link diagram D a finite suspension spectrum X (D), whose cohomology is the Khovanov homology of the associated link.Together with Lawson [7], they gave an equivalent construction in which they first associate to each link diagram D a functor F (D) : 2 n → B from the cube category to the Burnside category (a cube in the Burnside category), and then define a realization functor from the category of cubes in the Burnside category to the homotopy category of CW-spectra.The spectrum X (D) is obtained as the realization of the cube F (D).
The goal of the present paper is to extend the realization functor to cubes in several enlargements of the Burnside 2-category: the locally finite Burnside 2-category, the bilocally finite Burnside 2-category and the Burnside 2-category with an action of a group G (see Section 2.2 for definitions).This is achieved in Sections 3.2, 3.3 and 3.4, where we respectively prove the following theorems: Theorem 1.1.Every cube in the locally finite Burnside category yields a CWspectrum, well-defined up to homotopy equivalence.Moreover, any natural transformation between such cubes induces a map between CW-spectra well-defined up to homotopy.Theorem 1.2.Every cube in the bilocally finite Burnside category yields a locally compact CW-spectrum, well-defined up to proper homotopy equivalence.Moreover, any natural transformation between such cubes induces a map between CW-spectra well-defined up to proper homotopy.
A locally compact CW-spectrum has two associated cohomologies: its usual cohomology and the compactly supported cohomology, Theorem 1.3.Every cube in the G-equivariant Burnside category yields a free G-CW-spectrum, well-defined up to G-homotopy equivalence.Moreover, any natural transformation between such cubes induces a G-equivariant map between CWspectra well-defined up to G-homotopy .
A G-CW-spectrum E is G-finite if its quotient is a finite spectrum.In that case E is also a locally compact CW-spectrum, and thus has an associated compactly supported cohomology.
Application.In [2] Asaeda, Przytycki and Sikora introduced annular Khovanov homology, a triply-graded homological invariant of annular links (i.e., links in the thickened annulus A × I) which categorifies the annular Jones polynomial from [5].Some years later, Beliakova, Putyra and Wehrli defined a deformation of this homology theory, known as quantum annular Khovanov homology [4].This arises as the homology of a chain complex CKh Aq (D) with two extra gradings and with a free action of the infinite cyclic group Z.In particular, it is not finitely generated.Quotienting by the subgroup rZ ⊂ Z one obtains a finitely generated chain complex CKh r Aq (D) with an action of the finite cyclic group Z r of order r.Recently, Akhmechet, Krushkal and Willis [1] associated to each link diagram, a cube in the Burnside category with a free action of Z r whose realization is a finite CW-spectrum X r (D) endowed with a Z r -action, whose cellular chain complex is CKh r Aq (D).An application of our Theorem 1.3 allows to extend their construction as follows: Theorem 1.4.Every link diagram in the annulus yields a cube in the Z-equivariant Burnside category, whose realization is a Z-finite free Z-CW-spectrum well-defined up to stable equivariant homotopy equivalence.This equivariant stable homotopy type does not depend on the chosen diagram.Moreover, the compactly supported cohomology of this locally compact spectrum recovers the quantum annular Khovanov homology of the link.
The structure of the paper is as follows.In Section 2 we introduce the categories we deal with in the paper; in particular, we introduce Burnside category, its generalizations and the cubes in these categories.Section 3 is devoted to extend the realization of Lawson, Lipshitz and Sarkar to the setting of (bi)locally finite and G-equivariant Burnside categories and to provide the proofs of Theorems 1.1-1.3.Section 4 compares the constructions in the previous section to the constructions of cubes in abelian groups used to define the various Khovanov homologies.Finally, in Section 5 we apply our results to the setting of quantum annular Khovanov homology and provide a proof of Theorem 1.4.The Appendix summarizes some definitions and results about locally compact spectra.

The cube and the Burnside category
2.1.The cube.The (n-dimensional) cube 2 n is the partially ordered set whose elements are n-tuples u = (u 1 , . . ., u n ) ∈ {0, 1} n and u ≥ v if u i ≥ v i for every i.The elements of this poset are called vertices.It has an initial element 1 = (1, 1, . . ., 1) and a terminal element 0 = (0, 0, . . ., 0).The degree of a vertex u is |u| = n i=1 u i .We may regard the poset 2 n as a category whose objects are the vertices of the poset and the morphism set Hom(u, v) has a single element ϕ u,v if u ≥ v and is empty otherwise.We say that the morphism ϕ u,v is an edge between vertices u and v.

The Burnside categories.
Let G be a group.A G-set is a set with an action of G.A G-set is G-finite if it has finitely many orbits.
A correspondence from a G-set X to a G-set Y is a triple (A, s, t), where A is a G-set and s : A → X and t : A → Y are equivariant maps, called the source and target map, respectively.A correspondence is • free if the action of G is free on X, Y and A; • finite if the sets X, Y and A are finite; • G-finite if the sets X, Y and A are G-finite; • locally finite if the source map s has finite preimages (i.e., |s −1 (x)| < ∞, for every x ∈ X); • bilocally finite if both the source and target maps have finite preimages (i.e., |s The category of correspondences from X to Y has as objects the correspondences from X to Y and the following morphisms: a morphism from a correspondence (A, s A , t A ) to a correspondence (B, s B , t B ) is an equivariant fibrewise bijection from A to B, that is, an equivariant bijection f : A → B making the following diagram commute: These bijections are composed as usual, and the identity bijection plays the role of the identity morphism.
The horizontal composition and the action on C is inherited from the diagonal action, that is, g(b, a) = (gb, ga), for every g ∈ G.
The horizontal composition of two fibrewise bijections is defined as their fibrewise product too.The identity correspondence of a G-set X is the correspondence Definition 2.1.The G-equivariant Burnside category B G is the 2-category whose objects are free G-finite G-sets and whose category of morphisms from a G-set X to a G-set Y is the category of G-finite correspondences from X to Y .The horizontal composition in B G is the horizontal composition of correspondences and fibrewise bijections.Therefore, 1-morphisms are correspondences and 2-morphisms are fibrewise bijections.
For the next definition, we take G to be the trivial group.Definition 2.2.The big Burnside category B b is the 2-category whose objects are sets and whose category of morphisms from a set X to a set Y is the category of correspondences from X to Y .As in the previous definition, 2-morphisms are given by fibrewise bijections.We will consider the following subcategories: • The locally finite Burnside category is the subcategory B lf of B b whose correspondences are required to be locally finite.• The bilocally finite Burnside category is the subcategory B bf of B b whose correspondences are required to be bilocally finite.• The Burnside category is the subcategory B of B b whose correspondences are required to be finite.Let R be a commutative ring.The linearization functor whose value on a generator x ∈ X is: t(a).
Precomposing this functor with the forgetful functor B G → B bf ⊂ B lf , we obtain a functor • For each u ∈ {0, 1} n , a set F (u).
• For every u > v, a correspondence F (u) ) such that the following diagram commutes for every u > v > w > z: Fv,w,z ×Id Fu,w,z Fu,v,z We may replace the big Burnside category by any of the subcategories introduced in Definition 2.2, or by the G-equivariant Burnside category B G .Let B 2 n b be the functor category, whose objects are strictly unitary lax 2-functors from 2 n to B b and morphisms are natural transformations between these functors.
Let F : 2 n → B lf be a strictly unitary lax 2-functor.
Definition 2.5.The totalization of F is the chain complex of R-modules with underlying graded module with homological degree of A(F (u)) equal to |u| and differential This chain complex can alternatively be constructed as the following homotopy colimit: Define the extended cube category of dimension n, 2 n + , as the result of adding to the cube category an extra object * that receives an arrow ϕ u, * from every vertex u in 2 n \ 0. Then the cube of R-modules A •F can be extended to a functor K : 2 n + → R-Mod by setting F ( * ) to be the trivial module.As a consequence we have [10, Section 2.1]: Lemma 2.6.The totalization of F is homotopy equivalent to the homotopy colimit of K.
This rule is functorial and defines the totalization functor to the category of chain complexes of R-modules

Cubes in Burnside categories and spectra
In [8] Lawson, Lipshitz and Sarkar defined a functor from the category of cubes in the Burnside category to the homotopy category of CW-spectra.In this section we explain how to promote this construction to cubes in the locally finite Burnside category.Then, we show that for cubes in the bilocally finite Burnside category, the CW-spectra obtained are locally compact (see the Appendix for the definition of locally compact CW-spectrum).Finally, we explain that the realization of certain cubes in a G-equivariant Burnside category yield locally compact G-CW-spectra.
When defining the geometric realization of the cubes above, we need to take the homotopy colimit of some functors satisfying certain properties, called homotopy coherent diagrams; we recall these notions: Given a homotopy coherent diagram F : C → Top * , its homotopy colimit can be computed as the following quotient (1) hocolim We refer to this model of the homotopy colimit as the Vogt homotopy colimit [14].
Observe that in the particular case when all isomorphisms in the category C correspond to identity maps, one can assume that none of the maps f i considered when n > 0 are equal to the identity, and take the equivalence relation generated by the last four conditions when constructing the homotopy colimit of F (see [8,Observation 4.12]).
3.1.The realization of a cube in the Burnside category.Given a set X, denote by X + the disjoint union of X together with a basepoint, and let S k be the pointed sphere of dimension k.Recall from [8] (1) g is the composition of a continuous map f : The set of all box maps E(s) refining a given correspondence X s ← A t → Y is in bijection with a subset of the space of all embeddings of [0, 1] k × A into [0, 1] k × X, and this endows E(s) with a topology.The inclusion of the set of box maps E(s) into the set of continuous maps ( When the families of box maps F (f n , . . ., f 1 ) are only defined for n ≤ ℓ, we say F is an ℓ-partial k-spatial refinement.(1) If k ≥ n then there is a k-spatial refinement of F .
(2) If k ≥ n + 1 then any two k-spatial refinements of F are homotopic (as homotopy coherent diagrams).(3) If Fk is a k-spatial refinement of F then the result of suspending each Fk (u) and Fk (f 1 , . . ., f n )( t) gives a (k + 1)-spatial refinement of F .
The proof of the above proposition implicitly uses a recursive application of the following lemma.
An inclusion ι : X ′ ⊂ X gives rise to a correspondence X ′ id ← − X ′ ι − → X that we denote by ι too.Write f | X ′ for the restriction of f to X ′ , i.e., for the composition of correspondences f • ι.
The following result is an extension of Proposition 3.3 to the setting of the locally finite Burnside category.It will be crucial in the realization of the cubes in this category.
Proposition 3.7.Consider C a small category such that every sequence of composable nonidentity morphisms has length at most n, and let F : C → B lf .
(1) If k ≥ n then there is a k-spatial refinement of F .
(2) If k ≥ n + 1 then any two k-spatial refinements of F are homotopic (as homotopy coherent diagrams).(3) If Fk is a k-spatial refinement of F then the result of suspending each Fk (u) and Fk (f 1 , . . ., f n )( t) gives a (k + 1)-spatial refinement of F .
Proof.On objects, we set F (u) = F (u) + ∧ S k .Then we extend the spatial refinement recursively on the length of the chain of composable morphisms using Lemma 3.6.This completes the proof of (1).The proof of ( 2) and ( 3) is analogous to that in [8, Proposition 5.22].
Given a cube in the locally finite Burnside category F : 2 n → B lf , define its realization |F | in the same way as described in Section 3.1 for the case of the Burnside category.Proposition 3.8.Let F : 2 n → B lf be a strictly unitary lax 2-functor and k > n.
(1) If F1 and F2 are two k-spatial refinements of F , then F1 ≃ F2 . ( (3) If η : F → F ′ is a natural transformation between two cubes F, F ′ : 2 n → B lf , and F and F ′ are k-spatial refinements of F and F ′ , respectively, then η induces a map η : F → F ′ well-defined up to homotopy. Proof.
(1) and ( 2) are consequences of Proposition 3.7 (2) and (3), respectively.To prove (3), observe that a natural transformation from F to F ′ is a (strictly unitary lax) 2-functor η : 2 n+1 → B lf that restricts to F and F ′ on 2 n × {1} and 2 n × {0}, respectively.By Lemma 3.6, the spatial refinements F and F ′ can be extended to a spatial refinement of η.This spatial refinement yields a map from the coherent diagram F to the coherent diagram F ′ that commutes up to homotopy, which in turn yields a map between the CW-complexes F and F ′ .
Corollary 3.9.The realization |F | of a cube in the locally finite Burnside category B lf is a CW-spectrum well-defined up to homotopy.Moreover, any natural transformation between cubes in the locally finite Burnside category induces a welldefined map up to homotopy.
This completes the proof of Theorem 1.1.

3.3.
The realization of a cube in the bilocally finite Burnside category.
We will prove that the realization of cubes in the bilocally finite Burnside category leads to locally compact CW-spectra.To do this, we first prove a result in the setting of functors over locally finite categories.
A proper homotopy coherent diagram is a homotopy coherent diagram in the category of r-locally compact CW-complexes and proper maps CW lc (see Appendix for the definition of this category).A proper homotopy coherent natural transformation η : F → F ′ between proper homotopy coherent diagrams F, F ′ : D → CW lc is a proper homotopy coherent diagram η : D × 2 1 → CW lc .A proper homotopy H : F × [0, 1] → F ′ between two proper homotopy natural transformations η, η ′ is a proper homotopy coherent natural transformation H : D × 2 1 → CW lc between F × [0, 1] and F ′ such that the restriction to F × {1} is η and the restriction to F × {0} is η ′ .Lemma 3.10.Let C be a small category such that for each object u of C the comma categories C ↓ u and u ↓ C have finitely many sequences of composable non-identity morphisms.The Vogt homotopy colimit of a proper homotopy coherent diagram F over C is a locally compact space.If η : F → F ′ is a level-wise proper natural transformation of proper homotopy coherent diagrams, then η induces a proper map between Vogt homotopy colimits.If η and η ′ are two naturally properly homotopic natural transformations, then the induced maps between Vogt homotopy colimits are properly homotopic.
Proof.The Vogt homotopy colimit of such a diagram is the quotient of a disjoint union of a basepoint * and spaces of the form [0, 1] m × F (u 0 ), indexed by sequences composable non-identity morphisms.Observe first that this is a disjoint union of compact spaces, and therefore is locally compact.To show that the quotient is locally compact, notice first that every equivalence class has a unique representative g = (g ℓ , . . ., g 1 ; s 1 , . . ., s ℓ ; q) such that (s 1 , . . ., s ℓ ) ∈ (0, 1) ℓ and q ∈ F (v 0 ), where v 0 g1 − → . . .
Any other element in this equivalence class is of the form f = (f m , . . ., f 1 ; t 1 , . . ., t m ; p) with m > ℓ and t ji = s i with j 1 < . . .< j ℓ and the remaining t j are equal to either 0 or 1.
Let A be the set of sequences of transformations of type (a) that can be applied to g.For each a ∈ A let g a be the representative defined by a and let B a be the set of sequences of transformations of type (b) that can be applied to g a .Finally, for each b ∈ B a , let g a,b be the representative obtained by applying b to g a and let C a,b be the set of all representatives of g that are obtained by applying transformations of type (c) to g a,b .
Since the comma category v ℓ ↓ C has finitely many sequences of composable non-identity morphisms, A is finite.The same condition over C ↓ v 0 implies that B a is finite.Lastly, since u ↓ C (or C ↓ u) has finitely many sequences of composable non-identity morphisms, C a,b is finite.
Since A, B a , C a,b are all finite, there are finitely many representatives in each class and therefore the quotient is locally compact.
If η : F → F ′ is a level-wise proper homotopy coherent natural transformation, we can view it as a proper homotopy coherent diagram η : C × 2 1 → Top * such that it restricts to F and F ′ on C × {1} and C × {0}, respectively.There is a proper homotopy equivalence hocolim η ≃ hocolim F ′ and a proper inclusion hocolim F → hocolim η.The composition of these two maps is the map between homotopy colimits.
Given a cube in the bilocally finite Burnside category F : 2 n → B bf , consider its realization |F | in the same way as described for category in Section 3.1 using the Vogt homotopy colimit (1).
Proposition 3.11.The realization |F | of a cube in the bilocally finite Burnside category B bf is a locally compact CW-spectrum well-defined up to proper homotopy.Moreover, any natural transformation between cubes in the bilocally finite Burnside category induces a well-defined map up to proper homotopy.
Proof.A spatial refinement of a cube in B bf maps each vertex to a wedge of spheres, which is a r-locally compact CW-complex.An edge is sent to the realization of the spatial refinement of a bilocally finite correspondence f , which is a proper map f : X → Y , where X and Y are wedges of spheres.Finally, the homotopies that arise from isotopies of box maps are also proper: if H : X × I → Y is the homotopy constructed between the spatial refinements f , g : X → Y of two bilocally finite correspondences f and g, and e is an open cell of Y , then H −1 (e) only meets product cells e ′ × I ⊂ X × I with e ′ a cell that meets f −1 (e).Since there are finitely many of the latter, H is proper.As a consequence, the spatial refinement built in Proposition 3.7 is a proper homotopy coherent diagram.Then, by Lemma 3.10, F is a locally compact CW-complex, hence locally finite, and from Remark A.9 it follows that |F | is a locally finite CW-spectrum.
To prove that the proper homotopy type of |F | is well-defined, suppose that we are given two spatial refinements F1 and F2 and let I = 2 1 be the poset (1 → 0).By Lemma 3.6, we can extend F1 and F2 to a spatial refinement Fa : 2 n × I → Top * that restricts to F1 on 2 n × {1} and to F2 on 2 n × {0}, and is the identity on {u} × I.Then, Fa induces a proper map Fa : F1 → F2 .
Similarly, we can produce a spatial refinement Fb : 2 n × I → Top * that restricts to F2 on 2 n × {1} and to F1 on 2 n × {0}, and is the identity on {u} × I.It induces a proper map Fb : F2 → F1 .
The compositions Fb • Fa and Fa • Fb are induced by the diagrams that result from gluing Fa and Fb , and Fb and Fa , respectively.Using once more Lemma 3.6, define another spatial refinement F : 2 n ×I ×I → Top * that restricts to Fa • Fb on 2 n × I × {1} and to the identity on 2 n × I × {0}, with arrows {u} × {i} × I being identities.
Then F induces a proper homotopy between Fb • Fa and the identity.In a similar way we obtain a proper homotopy between Fa • Fb and the identity, so F1 and F2 are proper homotopy equivalent.By Remark A.9, |F | is well-defined up to proper homotopy type.
Consider now a natural transformation η : F → F ′ between cubes in the bilocally finite category F, F ′ : 2 n → B bf , and let F and F ′ be two spatial refinements of F and F ′ , respectively.Then η induces a map between the spatial refinements η : F → F ′ , as well as between the CW-complexes, η : F → F ′ .
On the other hand, we can see η as a (n + 1)-dimensional cube η : 2 n × I → B bf that restricts to F on 2 n × {1} and to F ′ on 2 n × {0}, and η as a spatial refinement extending F and F ′ given by Lemma 3.6.By the second part of Lemma 3.10, η is proper.By Remark A.9, η induces a well-defined map up to proper homotopy between the realizations |F | and |F ′ |.
This completes the proof of Theorem 1.2.

3.4.
The realization of a cube in the equivariant Burnside category.Given a cube in the G-equivariant Burnside category F : 2 n → B G , we can construct a spatial refinement FG of the quotient F/G, and then define an equivariant spatial refinement F of F by setting F (u) = F (u) + ∧ S k and then defining the maps F (u) → F (v) as the only refinements that make the following diagram commute: Proceeding in the same manner with higher dimensional faces, we get a G-equivariant spatial refinement F of F that is free away from the basepoint.Now, following Section 3.1, the Vogt homotopy colimit F of the homotopy coherent diagram F * is then a G-CW-complex and its associated spectrum |F | = Σ −k Σ ∞ F is a free G-CW-spectrum well-defined up to G-homotopy.Therefore we have: Proposition 3.12.Let F : 2 n → B G be a cube.Then the realization |F | is a locally compact CW-spectrum with a free action of G well-defined up to G-homotopy.
Moreover, any natural transformation between cubes in the G-Burnside category induces a map between CW-spectra well-defined up to G-homotopy.
Proof.The spatial refinement FG constructed above is also a refinement of the composition 2 n → B G → B bf , and therefore yields a locally compact spectrum.
If η : F → F ′ is a natural transformation between cubes F, F ′ : 2 n → B G , it is a (n + 1)-dimensional cube in B G that restricts to F on 2 n × {1} and F ′ on 2 n × {0}.Let FG and F ′ G be spatial refinements of the quotients F/G and F ′ /G.By Lemma 3.6, we can extend them to a spatial refinement ηG of η/G, that is, a spatial refinement of η as a cube in B bf .We can lift ηG to a spatial refinement of η by the same arguments used in the discussion preceding this proposition.Then, η induces a map η : F → F ′ which in turn yields a map |η| : |F | → |F ′ | well-defined up to G-homotopy.
For the last assertion, notice that the quotient of the G-equivariant spatial refinement FG by the subgroup N is a G/N -equivariant spatial refinement of F N .Since the action of G on FG is free, taking quotient by N commutes with the homotopy colimit used to build the realization of F , and therefore |F N | ≃ |F |/N .

Duality for cubes in the Burnside category
The Burnside category is equivalent to its opposite: reversing the role of the maps in a correspondence defines a functor D : B → B op .This duality disappears in the locally finite Burnside category, but still exists in the bilocally finite Burnside category, since the locally finiteness assumption is imposed in both maps of each correspondence.
The same happens with the category R-Mod f b of finite free R-modules with a basis: Taking duals defines a functor D : R-Mod f b → R-Mod op f b which is not the identity on objects, but the existence of a basis yields a canonical isomorphism between the finite free R-module M and D(M ).The linearization functor A : B → R-Mod f b commutes with the duality functors in the sense that there is a natural isomorphism between D • A and A • D.
The dual of a non-finite free R-module is not isomorphic to the original Rmodule.Nonetheless, this can be solved by changing the category: Define the category R-Mod b whose objects are modules endowed with a prescribed basis and morphisms f : M → N are homomorphisms such that for each basis element v of N there are finitely many basis elements e i of M such that (v ∨ )(f (e i )) = 0. Define the finitely supported dual of a module M ∈ R-Mod b as the subspace D lf (M ) ⊂ D(M ) of those functions f : M → R such that f (e i ) = 0 for all basis elements except for finitely many of them (alternatively, the subspace generated by the e ∨ i ).Again, there is a canonical isomorphism between D lf (M ) and M , and the linearization functor A : B bf → R-Mod b commutes with the finitely supported duality functor in the sense that there is a natural isomorphism between D lf • A and A • D. Additionally, the category R-Mod b has finite coproducts and the functor D lf commutes with them.Define Ch b (R) as the category of chain complexes in R-Mod b , and CW-Sp lc as the category of r-locally compact CW-spectra and proper maps.
Applying this discussion levelwise to cubes in the Burnside category, we obtain the following commutative diagram: The arrows in the lower left-hand square are compositions with the indicated functors, where we identify functors (2 n ) op → B op bf with functors 2 n → B bf , and the same with R-Mod b .The bottom right map is defined so that the right square commutes.
The commutativity of the upper square follows by the same reasoning as in [8].Let D be the category 2 n + .Then Tot(F ) ≃ hocolim D A •F , and we have Here is an application of this discussion to Khovanov homology that only uses the small Burnside category: Khovanov homology is the totalization of a contravariant cube F of R-modules (middle bottom of the diagram).The work of Lawson, Lipshitz and Sarkar produces a (strictly unitary, lax) covariant cube F in the Burnside category (left middle of the diagram) such that the cube of R-modules A •F (center of the diagram) is dual to the cube of Khovanov.As a consequence, the dual of the cellular chain complex of |F | is isomorphic to the Khovanov chain complex.
When working equivariantly we have the following counterpart of the previous diagram:

Quantum annular homology and Burnside functor
In [2] Asaeda, Przytycki and Sikora generalized Khovanov homology to the more general setting of links in thickened surfaces, that is, [0, 1]-bundles over surfaces.In the particular case when the surface is the annulus A = S 1 × [0, 1], the homology is called annular Khovanov homology.The idea to adapt Viro's approach in [13] to the setting of annular links (i.e., links embedded in the thickened annulus A × [0, 1]) consists of considering diagrams as projections over the first factor A and noticing that after smoothing crossings to obtain Kauffman states, there are two different types of closed curves in A: the ones bounding disks (trivial), and the ones parallel to the core of the annulus (essential).Authors manage to extend Viro's differential to the case when essential curves are involved in the mergins and splittings.As in the classical case, annular Khovanov homology is functorial and its construction can be summarized in the annular Khovanov functor F A : (2 n ) op → Z-Mod.The totalization of the functor F A associated to a diagram D is the annular Khovanov chain complex CKh A (D).Some years later, Beliakova, Putyra and Wehrli [4] introduced quantum annular Khovanov homology, a triply graded homology theory carrying an action which behaves well with the action of cobordisms.This construction can be summarized in a functor F Aq : (2 n ) op → k-Mod, a deformation of the annular Khovanov homology functor, where k = Z[q, q −1 ] is a ring and q ∈ k a distinguished unit.As in the previous case, the totalization of the functor F Aq associated to a diagram D is the quantum annular Khovanov chain complex CKh Aq (D).
5.1.The quantum annular Burnside functor.In [1] the authors consider the finite cyclic group G r = q | q r = 1 with r ≥ 1, and construct the quantum annular Burnside functor F r q : 2 n → B Gr .The quotient F 1 q = F r q /G r is precisely the Equivalently, a pointed CW-complex is r-locally compact if any open cell different from the basepoint intersects only finitely many cells of any given dimension.A CW-complex is reduced if it has a single 0-cell.
Example A.2.The reduced suspension of a locally compact pointed CW-complex is a r-locally compact CW-complex.
Example A.3.The quotient X/T of a T -complex by its tree T [3,12] is a r-locally compact CW-complex.
Definition A.4.A cellular map between r-locally compact CW-complexes is proper if it is proper relative to the basepoint.
Equivalently, a cellular map f is proper if the preimage of each open cell different from the basepoint intersects only finitely many cells of each dimension.
Example A.5.The reduced suspension of a proper map between locally compact CW-complexes is a proper map between r-locally compact CW-complexes.
Definition A.6.The compactly supported cochain complex of a r-locally compact CW-complex X is the subcomplex C * c (X, * ) ⊂ C * (X, * ) of compactly supported cochains relative to the basepoint.The compactly supported cohomology of X is the cohomology H * (X, * ) of this cochain complex.
The relative chain complex C * (X, * ) of a r-locally compact CW-complex is a locally finite chain complex, and its finitely supported dual is precisely C * c (X, * ) ⊂ C * (X, * ).Proper maps induce homomorphisms between compactly supported cohomology groups.Properly homotopic proper maps induce the same homomorphism between compactly supported cohomology groups.
Let G be a countable group.A free pointed G-CW-complex is a pointed CWcomplex X with a cellular action of G that fixes the basepoint and is free away from it.The quotient X/G is a pointed CW-complex, and we say that X is Gcompact if the quotient is compact.If that is the case, then X is a r-locally compact CW-complex.An equivariant map f : X → Y between G-finite free pointed CWcomplexes is always proper.An equivariant map f : X → Y is a G-homotopy equivalence if there is an equivariant map g : Y → X such that f • g and g • f are equivariantly homotopic to the identity.In that case, f is also a proper homotopy equivalence.The following proposition is a consequence of Theorem 5.3 in [11].
Proposition A.7.Let f : X → Y be an equivariant map between finite dimensional free pointed G-CW-complexes.Then f is a G-homotopy equivalence if and only if f is a homotopy equivalence.
A.2. Locally compact spectra.Recall that a CW-spectrum is a sequence of pointed CW-complexes {E n } n≥0 together with structural maps ε n : ΣE n → E n+1 that include ΣE n as a subcomplex of E n+1 .
Definition A.8.A finite dimensional CW-spectrum E is locally compact if every CW-complex E n is r-locally compact and the structural maps ε n are proper and eventually homeomorphisms.
Proper functions and proper maps of finite dimensional locally compact CWspectra are defined as expected.
Remark A.9.The suspension spectrum of a finite dimensional r-locally compact CW-complex is locally finite.If f : X → Y is a proper map of finite dimensional locally finite CW-complexes, then Σ ∞ f is a proper function of finite dimensional locally finite CW-spectra.
The structural maps of a CW-spectrum E = {E n , ε n } induce chain homomorphisms (ε n ) * : C * (E n , * ) → Σ −1 C * (E n+1 , * ).The chain complex of the CWspectrum E = {E n } n≥0 is the colimit The chain complex of a locally compact CW-spectrum is locally finite, so it has a well-defined finitely-supported dual, that will be called the compactly cochain complex of E. The compactly supported cohomology H * c (E; R) of a spectrum E is the cohomology of the cochain complex C * c (E).A proper map of locally compact CW-spectra induces a map between their compactly supported cochain complexes.A proper homotopy between two maps of locally compact CW-spectra induces a homotopy between their induced maps on compactly supported cochain complexes.
A free G-CW-spectrum is a CW-spectrum E = {E n , ε n } such that every CWcomplex E n is a free pointed G-CW-complex and the structure maps ε n are equivariant.The quotient E/G is a CW-spectrum, and we say that E is G-finite if the quotient is a finite CW-spectrum.If that is the case and E is finite-dimensional, then E is a locally compact CW-spectrum.An equivariant map between finite dimensional free G-finite CW-spectra is always proper.An equivariant map f : E → E ′ is a G-homotopy equivalence if there is an equivariant map g : E ′ → E such that f • g and g • f are equivariantly homotopic to the identity.The following is the stable version of Proposition A.7: Proposition A.10.An equivariant map f : E → E ′ of free G-CW-spectra is a G-homotopy equivalence if and only if it is a homotopy equivalence.
s B (b)} with source and target maps given by s(b, a) = s A (a) and t(b, a) = t B (b),

Lemma 2 . 3 .
The forgetful functor B G → B b factors through the bilocally finite Burnside category B bf .Proof.If f : A → B is an equivariant map to a free G-set B and b ∈ B, then f −1 (b) contains at most one element from each orbit.If there are finitely many orbits, then f −1 (b) is finite.Applying this to the source and target maps in a G-finite correspondence X s ← − A t → Y yields the result.

Definition 3 . 1 .− → u 1 f2−
A homotopy coherent diagram from a small category C to the category of based topological spaces Top * , F : C → Top * , consists of: • For each u ∈ ob(C ) a space F (u) ∈ ob(Top * ).• For each n ≥ 1 and each sequence u 0 f1 → . . .fn −→ u n of composable morphisms in C , a continuous map

Proposition 3 . 3 .
[8, Prop.5.22] Consider a small category C such that every sequence of composable nonidentity morphisms has length at most n, and let F : C → B.

Lemma 3 . 4 . 3 . 2 .
If k ≥ ℓ and F is an (ℓ − 1)-partial k-spatial refinement of F , then there is an ℓ-partial k-spatial refinement of F extending F .We are ready now to describe the realization |F | of a cube F : 2 n → B following[8]: Consider the extended cube category 2 n + and choose a k-spatial refinement F of F for some k > n (this can be done by Proposition 3.3(1)).The next step is to extend F to a homotopy coherent diagram F * indexed by 2 n + by setting F ( * ) to be the basepoint and F (ϕ u, * ) the constant map.If we denote by F the homotopy colimit of the homotopy coherent diagram F * , the geometric realization of F is defined as|F | = Σ −k Σ ∞ F ,the k-fold desuspension of the suspension spectrum of F .Theorem 3.5.[8, Section 5.3] The realization |F | of a cube in the Burnside category is a CW-spectrum well-defined up to homotopy.Moreover, any natural transformation between cubes in the Burnside category induces a map between CWspectra well-defined up to homotopy.The realization of a cube in the locally finite Burnside category.Let f : X → Y be a correspondence given as X s ← A t → Y and x ∈ X. Define the cosupport of x under f as the subset cosupp

f1 → u 1 f2→
refinement of a functor F : C → B lf is defined as in Definition 3.2, replacing the Burnside category by the locally finite Burnside category.Lemma 3.6.Let F : C → B lf be a strictly unitary lax 2-functor.If k ≥ ℓ and F is an (ℓ − 1)-partial k-spatial refinement of F , then there is an ℓ-partial k-spatial refinement of F extending F .Proof.Let u 0 . ..f ℓ → u ℓ be a sequence of ℓ composable morphisms in C. module.