Enriched Koszul Duality

We show that the category of non-counital conilpotent dg-coalgebras and the category of non-unital dg-algebras carry model structures compatible with their closed non-unital monoidal and closed non-unital module category structures respectively. Furthermore, we show that the Quillen equivalence between these two categories extends to a non-unital module category Quillen equivalence, i.e. providing an enriched form of Koszul duality.


Introduction
Koszul duality appears in several areas of algebra, topology and geometry where its origin can be traced back to the inception of rational homotopy theory as developed by Quillen in [Qui69].We will mainly be working in the context of associative dg Koszul duality which, in its modern formulation, can be expressed as Quillen equivalence between the category of non-unital differential graded algebras, DGA 0 , and the category of conilpotent noncounital differential graded coalgebras, coDGA conil , over some field k.This case was initially shown in [Lef03] and further developed in [Pos11].A similar result in what we will refer to as com-Lie dg Koszul duality, previously achieved in [Hin01], can be expressed as a Quillen equivalence between the category of differential graded Lie algebras, DGLA, and the category of conilpotent non-counital cocommutative coalgebras, coCDGA conil , over some field k of characteristic zero.For a general survey of these results as well as further reading on Koszul duality we refer the reader to [Pos22].
On a somewhat different track, it has been shown in [AJ13] that the category DGA 0 is enriched, tensored, and cotensored over the closed symmetric monoidal category of non-counital differential graded coalgebras (coDGA 0 , ⊗) equipped with the ordinary tensor product.This was further extended to the operadic setting in [LG19].Our main aim with this paper is to provide a strengthening of Koszul duality that respects this enrichment of algebras over coalgebras.However in the case of associative Koszul duality we are dealing with the category of conilpotent coalgebras, coDGA conil , which does not have a monoidal unit under the ordinary tensor product making it into a semi-monoidal category.
Motivated by the lack of a unit, we introduce the notion of semi-module categories over a semi-monoidal category.Taking this further in the homotopical direction, we introduce semi-monoidal model categories and semimodule model categories analogous to their unital counterparts.Remembering that a closed module category is precisely the same thing as a tensored and cotensored enriched category, over the same monoidal category, this also allows us to speak about what we will refer to as a semi-enrichment over a semi-monoidal category.While a priori a weaker concept than enriched category theory, it nevertheless puts structure limitations on the categories in question.Furthermore, while our main interest as well as initial motivation is that of Koszul duality, one quickly finds that semi-monoidal model categories that are not monoidal do appear naturally in homotopical algebra.
Having established the framework of semi-monoidal-and semi-modulecategories as well as their model categorical analogues, we proceed to show that DGA 0 can be given a closed semi-module category structure over the semi-monoidal category (coDGA conil , ⊗) using the same procedure as in [AJ13].Furthermore, we show that their semi-module structures are compatible with their respective model category structures as well as with the Quillen equivalence that is associative Koszul duality.Proceeding similarly in the case of com-Lie Koszul duality, one obtains the corresponding result in that setting.Specifically, our main result in the associative setting is Theorem 1.1.The category (coDGA conil , ⊗, coDGA conil ) is a semi-monoidal model category and (DGA 0 , ⊲, DGA 0 , {−, −}) is a semi-module model category over coDGA conil .Furthermore, the Quillen equivalence upgrades to a coDGA conil -module Quillen equivalence.
Here we used the notation coDGA conil for the internal hom of coDGA conil , while ⊲, DGA 0 , and {−, −} corresponds to the tensoring, enrichment, and cotensoring functors of the closed semi-module structure of DGA 0 respectively.In particular the cotensoring {−, −} is the convolution algebra functor.
The corresponding result in the com-Lie context is the following.Here we used the notation coCDGA conil for the internal hom functor of coCDGA conil , while ⊲, DGLA, and {−, −} corresponds to the tensoring, enrichment, and cotensoring functors of the closed semi-module structure of DGLA respectively.In particular the cotensoring {−, −} is the convolution algebra functor.
Finally, let us note that similar results in the context of dg-categories have been achieved in [HL22], where they provide a homotopical enrichment of dg-categories over pointed coalgebras.Note that the category of pointed algebras is monoidal, as opposed to just semi-monoidal, and as such they provide the category of dg-categories with an enriched category structure.

Conilpotent coalgebras
We will here recall some facts about dg-coalgebras and conilpotent dg-coalgebras in particular.We will be working with the category of non-counital coassociative dg-coalgebras, coDGA 0 , over some field k.Explicitly, this means that we require that the comultiplication ∆ : and that the differential d : However we will make no demands on the existence of a counit morphism.
x is the braiding morphism of dg-vector spaces, we say that C is cocommutative.As all our coalgebras will be differential graded, we will from now take coalgebra to implicitly mean dg-coalgebra.
Definition 1.3.Let (C, ∆) be a non-counital coalgebra.We say that an element x ∈ C is conilpotent if there exists some n such that ∆ n (x) = 0.If all elements of C are conilpotent we say that C is conilpotent.
We will denote the category of conilpotent dg-coalgebras by coDGA conil .
Definition 1.4.Let (C, ∆, d) be a non-counital coalgebra.We say that an element c ∈ C is an atom if ∆(c) = c ⊗ c and dc = 0.
The set of atoms of a non-counital coalgebra C is in one-to-one correspondence with coalgebra morphisms k → C from the monoidal unit.We note that a conilpotent coalgebra C has exactly one atom 0 ∈ C. In particular, if C is conilpotent, the zero morphism is the only morphism k → C.
where we have not done any reordering of the terms as it only affects the sign.
An admissible filtration of C is an increasing filtration F starting at F 1 compatible with the differential and comultiplication, meaning that Example 1.7.For any conilpotent dg-coalgebra (C, ∆) there is a canonical admissible filtration, known as the coradical filtration, given by F n = ker ∆ n .
Definition 1.8.We say that a morphism f : C → D in coDGA conil is a filtered quasi-isomorphism if there exists admissible filtrations F C and F D of C and D respectively such that the induced morphism of the associated graded complexes gr f : gr(F C ) → gr(F D ) is a quasi-isomorphism in each degree.
Several of our proofs rely on the existence of cofree objects in the categories of coDGA 0 and coDGA conil .The cofree functor Ť0 is defined as the right adjoint to the forgetful functor to the category of differential graded vector spaces, DGVec, i.e.
The existence of this adjunction was shown in [BL85].We will say that a coalgebra of the form Ť0 V for some V ∈ DGVec is cofree.Analogously the conilpotent cofree functor T co is defined to satisfy the adjunction and we say that a conilpotent coalgebra of the form T co V for some V ∈ DGVec is conilpotent cofree.Conilpotent cofree coalgebras are also known as tensor coalgebras as they are constructed analogously to the tensor algebra but instead given the cut comultiplication.
There is also an adjunction, between the inclusion functor ι : coDGA conil → coDGA 0 and the conilpotent radical functor R co : coDGA 0 → coDGA conil .The latter is defined by taking a coalgebra C to the subcoalgebra consisting of conilpotent elements of C.
The adjunction follows from that the image of a conilpotent element under a coalgebra morphisms is also conilpotent.As a consequence of this adjunction we have that T co ∼ = R co Ť0 .When working with the category of cocommutative conilpotent noncounital dg-coalgebras coCDGA conil we will also make use of the adjunction Here S co denotes the cocommutative conilpotent cofree functor, which takes a dg-vector space V to the coabelianization of the conilpotent cofree algebra over V .We say that a cocommutative conilpotent coalgebra of the form S co V for V ∈ DGVec is cocommutative conilpotent cofree.

Maurer-Cartan elements and Koszul duality
Definition 1.9.Let (A, d A ) be a non-unital dg-algebra.We say an element a ∈ A of degree −1 is a Maurer-Cartan element if it satisfies the Maurer-Cartan equation We denote the set of Maurer-Cartan elements of A by MC(A).
We define the universal Maurer-Cartan algebra mc as the non-unital free algebra T 0 x , where x denotes the vector space generated by the variable x of degree −1, and given the differential induced from d : x → −x 2 .Noting that any Maurer-Cartan element a ∈ A corresponds to the unique morphism in DGA(mc, A) generated by x → a we get the following proposition.
Proposition 1.10.The functor MC : DGA 0 → Set taking a dg-algebra to its set of Maurer Cartan elements is representable by the universal Maurer-Cartan element mc.
Similarly in the case of dg-Lie algebras we have the following definition.
Definition 1.11.Let (g, [−, −], d) be a dg-Lie algebra.We say an element x ∈ g of degree −1 is a Maurer-Cartan element if it satisfies the Maurer-Cartan equation We denote the set of Maurer-Cartan elements of g by MC Lie (g).
Similar to the associative case, we define the universal Maurer-Cartan Lie algebra mc Lie as the free Lie algebra T Lie x , where x denotes the vector space generated by the variable x of degree −1, and given the differential induced from d : Noting that any Maurer-Cartan element a ∈ g corresponds to the unique morphism in DGLA(mc Lie , g) generated by x → a we get the following proposition.
Proposition 1.12.The functor MC Lie : DGLA → Set taking a dg-Lie algebra to its set of Maurer-Cartan elements is representable by the universal Maurer-Cartan element mc Lie .
Next we remind ourselves of the convolution coalgebra construction.Definition 1.13.Let (C, ∆ C , d C ) be a dg-coalgebra and (A, µ A , d A ) a dgalgebra.Then the internal hom of dg-vector space, DGVec(C, A) has the structure of a dg-algebra with multiplication defined as the convolution product We will refer to this construction as the convolution algebra of C into A and denote it by {C, A}.
We caution the reader that our choice of notation for the convolution algebra conflicts with the notation used in [AJ13].They use the notation [−, −] for the convolution algebra while {−, −} instead denotes the measuring coalgebra construction.
Proposition 1.14.Let C be a cocommutative non-unital coassociative dgcoalgebra and g a dg-Lie algebra.Then the convolution algebra {C, g}, defined as above, takes the form of a dg-Lie algebra.
Proof.We check that {C, g} is a Lie algebra.As C is cocommutative, i.e. satisfying that ∆(c) = c (1) ⊗c (1) , antisymmetry is satisfied by [f, f ]•∆ C (c) = 0 for all f ∈ {C, g}.We also have the Jacobi identity as for all f, g, h ∈ {C, A} we have that We will from now on consider the convolution algebra functor restricted to the category of conilpotent coalgebras Let us now briefly recall Koszul duality and the bar and cobar constructions.We refer the reader to [Pos11] for the proofs and further background.We will adopt the convention of using homological grading throughout.For an algebra (A, m, d A ) ∈ DGA 0 , we define the bar construction BA as a graded coalgebra to be the conilpotent cofree coalgebra Conversely, given a coalgebra (C, ∆, d C ) we define the cobar construction ΩC as a graded algebra to be the free algebra T ΣC with differential induced from d C + ∆.
The bar and cobar functors can be shown to be adjoint by Furthermore when considering categories DGA 0 and coDGA conil with their standard model structures the above adjunction upgrades to a Quillen equivalence.Explicitly the model structure on DGA 0 is given as follows.We say a morphism in DGA 0 is a iii) cofibration if it has the left lifting property with respect to all acyclic fibrations.
The category of coDGA conil admits the left transferred model structure over the above adjunction.We say that a morphism in coDGA conil is a i) weak equivalence if it belongs to the minimal class of morphisms generated by filtered quasi-isomorphism under the 2 out of 3 property, ii) cofibration if it is injective, iii) fibration if it has the right lifting property with respect to all acyclic cofibrations.
For a proof of the existence of these model structures as well as the Quillen equivalence we refer the reader to [Pos11].
The story in the com-Lie case of Koszul duality is very similar but we will in this case require that the ground field k has characteristic zero.This is needed for the existence of the model structure on DGLA and coCDGA conil as shown in [Hin01].As in the associative case we say that a morphism in DGLA is a iii) cofibration if it has the left lifting property with respect to all acyclic fibrations.
Similarly we say that a morphism in coCDGA conil is a i) weak equivalence if it belongs to the minimal class of morphisms generated by filtered quasi-isomorphism under the 2 out of 3 property, ii) cofibration if it is injective, iii) fibration if it has the right lifting property with respect to all acyclic cofibrations.
For a Lie-algebra (g, [−, −], d g ) ∈ DGLA we define the bar construction Bg to be S co (Σ −1 g) with differential induced from d g + [−, −].In the other direction for a cocommutative coalgebra (C, ∆, d C ) ∈ coCDGA conil we define the cobar construction ΩC as T Lie (ΣC) with differential induced by d C + ∆.These functors are Quillen equivalent by the adjunction DGLA(ΩC, g) ∼ = MC Lie ({C, g}) ∼ = coCDGA conil (C, Bg), as shown in [Hin01].Note in particular that {C, A} has the structure of a dg Lie-algebra by Proposition 1.14.

Semi-monoidal categories, semi-module categories, and semi-enrichments
Categories that are monoidal except missing a unit, known as semi-monoidal, semi-groupal or non-unital monoidal in the literature have previously been studied in e.g.[Koc08], [Abu13], and [LYH19].We will introduce the notion of semi-module categories over a semi-monoidal category, fully analogous to the definition in the unital case.Working in the model category setting this will lead us to the definition of semi-monoidal model categories and semi-module model categories.
We will take the definition of (symmetric) semi-monoidal categories, semi-monoidal functors, and semi-monoidal natural transformations to be fully analogous to the monoidal ones by dropping the unit and unit axioms at every step.Similarly, we take take the definitions of semi-modules, semimodule functors and semi-module natural transformations to be those found in appendix B of [AJ13] or chapter 4 in [Hov99] by dropping the unit axioms.When working over a semi-monoidal category V we will commonly use the terminology V-module to mean a semi-module over V etc. Definition 2.1.A semi-monoidal category (V, ⊗, a) consists of a category V together with a functor and a natural isomorphism commute for all X, Y, Z ∈ V.
Definition 2.3.We say that a semi-monoidal category (V, ⊗) is (left) closed if there exists a functor The functor V is known as the internal hom functor.
Definition 2.4.Let (V, ⊗, a) and (V, ⊗ ′ , a ′ ) be two semi-monoidal categories.Then a (strong) semi-monoidal functor (F, m) from V to V ′ consists of a functor F : V → V ′ and a natural isomorphism for all X, Y, Z ∈ V.
Definition 2.5.Let (F, m) and (F ′ , m ′ ) be two semi-monoidal functors from We could at this point further proceed with the theory of semi-monoidal categories, by defining semi-monoidal adjunctions and semi-monoidal equivalences etc.As we will not explicitly need them, we instead proceed with the definition of semi-module categories.
Definition 2.6.Let (V, ⊗, a) be a symmetric semi-monoidal category.A (left) V-module is a category C together with a functor commutes for all X, Y, Z ∈ V and A ∈ C.
A module over the opposite symmetric semi-monoidal category (V op , ⊗) is known as a V-opmodule.We should at this point note that Mac Lane's coherence theorem holds in the case of semi-monoidal categories and the corresponding version for module categories similarly holds for semi-module categories.
Definition 2.7.We say that a V-module C is right closed if there exists a functor [−, −] : Proposition 2.9.Let (C, ⊲) be a V-module.Then for any Y ∈ V the tensoring functor Y ⊲ − is itself a V-module functor.
Proof.We take the natural isomorphism, to be the composition Definition 2.10.Let V be a symmetric semi-monoidal category and (F, m) and (F ′ , m ′ ) be two V-module functors from (C, ⊲) to (D, ⊲ ′ ).A V-module natural transformation is a natural transformation η : commutes for all X ∈ V and A ∈ C.
We define the concept of semi-module adjunctions and equivalences as follows.
Definition 2.12.A V-module equivalence (F, U, φ, m) is an equivalence of categories (F, U, φ) such that (F, m) is a V-module functor.
Note that these are the notion of adjunction and equivalence in the 2category of V-module categories and lax V-module functors, i.e.where the natural transformation is not required to be an isomorphism.
Finally we would like to mention that we can define the concept of what we call semi-enriched categories by extracting the properties of the enrichment functor in a closed semi-module category.Proceeding similarly as in the monoidal case we get the definition of such a structure.Definition 2.13.Let (V, ⊗, a) be a symmetric semi-monoidal category.A semi-enriched V-category is a category C together with a functor Note this structure is different from that of semi-categories, i.e. categories without a unit morphism.Most notably, the underlying category is an essential part of the structure.This is necessary as if we had taken a definition as for semi-categories, i.e. enriched categories without units, as has been studied in [MBB02] and [Stu05] we would have no way of recovering an underlying category.In particular, going to back to our definition, the enriched hom objects don't necessarily provide information about the hom set of the underlying category.

Semi-monoidal model categories and semi-module model categories
We take the definitions of semi-monoidal model categories and module model categories to be analogous to the corresponding definitions with units as found in [Hov99].
If furthermore V is a model category and the tensor functor, Example 3.5 (Reduced simplicial sets).The category of reduced simplicial sets sSet 0 , under the smash product ∧, gives an example of a semi-monoidal model category that is not monoidal.To see this, consider the coreflective adjunction where ι is the inclusion into the category of pointed simplicial sets, sSet * / , and it's adjoint R takes a pointed simplicial set to the subsimplicial set whose n-cells are those who have the marked point as 0-cells.The category of reduced simplicial sets admits the left transferred model structure, from the classical model structure on sSet * / by the above adjunction, as shown by Proposition 6.2 in [GJ09].As the wedge product of pointed simplicial sets restricts to a functor ∧ : sSet 0 × sSet 0 → sSet 0 , we see that sSet 0 is closed semi-monoidal with internal hom functor sSet 0 (−, −) := RsSet * / (−, −).
Note that the unit, * ⊔ * , of sSet * / is not reduced so sSet 0 is not monoidal.
As a consequence of sSet 0 being semi-monoidal and the coreflective Quillen adjunction to sSet * / we have get the following.
Corollary 3.6.Any pointed simplicial model category M is canonically also a sSet 0 -module model category.
Corollary 3.7.Let M be a pointed model category.Then its homotopy category can be given the structure of a closed Ho(sSet 0 )-module.
We will not further develop the theory here but only note that the standard proofs for monoidal-and module model categories also applies to the non-unital setting.In particular from Section 4.3 in [Hov99] we obtain the following statements.
• Let (V, ⊗, V) be a symmetric semi-monoidal model category.Then its homotopy category Ho(V) has the structure of a semimonoidal category (Ho(V), ⊗ L , RV) induced from V. 4 The semi-module structure of DGA 0 over coDGA conil From [AJ13] we know that DGA 0 admits a closed coDGA 0 -module structure.
To show that DGA 0 also admits a closed coDGA conil -semi-module structure we will make use of the same procedure.Indeed, applying the conilpotent radical functor R co to the internal hom functor and the enrichment functor in the coDGA 0 case would provide the results in the conilpotent case.Nevertheless, for the convenience of the reader we provide categorical proofs of the needed results.

The semi-monoidal structure of conilpotent coalgebras
It is well known from e.g.[AJ13] that the category of non-counital dg coalgebras (coDGA 0 , ⊗) is symmetric monoidal.We consider the full subcategory of conilpotent non-counital dg coalgebras coDGA conil .By Proposition 1.5 we know that the tensor product functor of conilpotent coalgebras restrict to a functor thus providing a semi-monoidal structure on coDGA conil .However coDGA conil does not quite form a monoidal category under the tensor product as the monoidal unit k of coDGA 0 is not conilpotent.
Proof.We need to construct an internal hom functor For the case of a conilpotent cofree coalgebra T co V we have the natural isomorphisms Hence we can define In the case of an arbitrary conilpotent coalgebra D we can write D as an equaliser We then define the internal hom coDGA conil (C, D) as the equaliser of Since the coDGA conil (C 0 , −) functor preserves limits this indeed gives the desired natural bijection Note that the internal hom of coDGA conil is substantially different from the internal hom of coDGA 0 .The latter can be constructed analogously using the non-unital cofree functor T 0 in place of the conilpotent cofree functor T co .In particular we have that coDGA conil (C, D) ∼ = R co coDGA 0 (C, D) for all conilpotent coalgebras C and D. Remark 4.2.As we have no unit in a semi-monoidal category we have no way of recovering the hom sets from the internal hom.Indeed in our case that information is lost as the only atom of coDGA conil (C, D) is 0, corresponding to the zero morphism.This is also the reason we cannot consider some larger subcategory of coDGA 0 containing k as, even if such a category admits an internal hom, it will never be conilpotent.

The semi-module structure of non-unital DG-algebras
In addition to the internal hom adjunction for conilpotent coalgebras established in the previous section we will need to establish tensoring and cotensoring adjunctions.
Our starting point is to consider the convolution algebra functor restricted to conilpotent coalgebras We will show that this will be the cotensoring functor of the closed semimodule structure of DGA 0 .We construct the enriched hom functor as the left adjoint to the opposite convolution algebra functor.
Proof.For a free algebra T 0 V we have natural isomorphisms )), so we can define the enriched hom as DGA 0 (T 0 V, B) := T co 0 DGVec(V, B).Given an arbitrary algebra A we can write it as a coequaliser and define DGA 0 (A, B) as the equaliser of Since coDGA conil (C, −) preserves limits we get the desired natural bijection We construct what will be the tensoring functor similarly.
Proposition 4.4.There exists a functor such that C ⊲(−) is left adjoint to the convolution algebra functor {C, −} for each coalgebra C ∈ coDGA conil .We will refer to ⊲ as the tensoring functor.
Proof.For a free algebra T 0 V we have the natural isomorphisms Hence we can define Given an arbitrary algebra A we can write it as a coequaliser and define C ⊲ A as the the coequaliser of Since DGA 0 (−, B) takes colimits to limits we get the desired natural bijection DGA 0 (A, {C, B}) ∼ = DGA 0 (C ⊲ A, B).
By combining Proposition 4.3 and Proposition 4.4 we also get a third adjunction between the tensoring and the enrichement functor.These are analogous to those shown in [AJ13] for the non-conilpotent case.
In particular the tensoring and cotensoring functors are the same as in the non-conilpotent case while the enrichment functor differs.

Measurings and coherence
To give DGA 0 the structure of a module category we also need it to satisfy the coherence axiom.To show this we will use the concept of measurings developed in [Swe69].We will briefly repeat the definition of measurings and some properties we will make use of, while referring the reader to [AJ13] for a more extensive coverage.
Definition 4.6.Let A, B be non-unital dg-algebras and C a non-counital dg-coalgebra.We say that a dg-linear morphism f : is a morphism of non-unital dg-algebras.
As an immediate consequence of the definition we have the following.
Proposition 4.7.Let v : C ⊗ A → B be a measuring, f : A ′ → A and g : B → B ′ be algebra maps and h : is also a measuring.
Proposition 4.8.Let C be a non-counital dg-coalgebra, A a non-unital dg-algebra and V a dg-vector space.Then a dg-linear map extends uniquely to a measuring Proof.By the tensor-hom adjunction for dg-vector spaces, the free forgetful adjunction for dg-algebras, and the definition of measurings we have natural equivalences We denote by M(C, A, B) the set of measurings from C ⊗ A → B. By Proposition 4.7 this assignment extends to a functor which we will refer to as the measurement functor.We will consider the measurement functor restricted to conilpotent coalgebras, which we will refer to as the restricted measurement functor.The measurement functor is representable in each variable, which is shown in Section 4.1 of [AJ13].The argument to show that it is also representable in the restricted case is similar which we briefly repeat here.Remark 4.10.Explicitly we have that the restricted measuring functor is represented in the third variable by where u is the measuring induced from the inclusion map by Proposition 4.8.Similarly, the restricted measuring functor is represented in the first variable by (DGA 0 (A, B), ev : DGA 0 (A, B) ⊗ A → B).
Here the evaluation map ev is the composition with ǫ ⊲ denoting the counit of the tensoring-enrichment functor adjunction of Corollary 4.5.
We can now proceed with showing the semi-module structure of (DGA 0 , ⊲).
Lemma 4.11.There exists a unique natural isomorphism Proof.This follows from the universal property of u i.e. for every measuring v : C ⊗ A → B there exists a unique algebra morphism f : We now have sufficient background to prove the main result of this section.The reader should however note that the following result also follows as a straightforward corollary from Theorem 4.1.18in [AJ13].
Proof.It remains to show that the coherence axiom is satisfied for the associator a constructed in the lemma 4.11.Consider the diagram where the vertical arrows consist of the universal element u applied as demanded by the diagram.We conclude that every vertical face commutes by Lemma 4.11 and that the top face commutes by the monoidal structure of the tensor product ⊗ on vector spaces.Thus after precomposition with the morphism u : For showing that coDGA conil is a semi-monoidal model category, we will first establish that the tensor product functor of coDGA conil preserves (acyclic) cofibrations in each variable separately.
Lemma 5.1.The tensor product functor preserves cofibrations and weak equivalences in each variable separately.
Proof.We first note that the forgetful functor U : coDGA conil → DGVec commutes with the tensor product and preserves cofibrations.The preservation of cofibrations under the tensor product of coDGA conil then follows from the DGVec case.It remains to show the preservation of weak equivalences.Since the class of weak equivalences is the closure of the class of filtered quasi-isomorphism under the 2 out of 3 property it suffices to show the preservation of filtered quasi-isomorphisms.Thus let f : C → D be a filtered quasi-isomorphism and let E be a conilpotent coalgebra.By assumption there exist admissible filtrations F C and F D , of C and D respectively, such that gr i f : gr i C F C → gr i D F D is a quasi-isomorphism in each degree.We define filtrations F C⊗E := F C ⊗E and F D⊗E := F D ⊗ E and note that they are admissible.Further noting that gr C⊗E ∼ = gr C ⊗ E we get an induced quasi-isomorphism, using that the tensor product of dg-vector spaces preserves quasi-isomorphism.Thus f ⊗ E is a filtered quasi-isomorphism.
Proof.We have to show that ⊗ is a Quillen bifunctor.Let i : C → C ′ be a cofibration and j : D → D ′ an (acyclic) cofibration in coDGA conil .The relevant pushout diagram is where we use that the (acyclic) cofibrations are closed under pullback, and Lemma 5.1.We see that in the acyclic case we get that the pushout map is a weak equivalence by the 2 out of 3 property.That the pushout map is injective follows from the dg-vector space case as colimits and cofibrations are preserved by the forgetful functor to DGVec, which furthermore commutes with the tensor product.

Homotopical enrichment of DGA 0
To show that DGA 0 is a model coDGA conil category we need to show that the tensoring functor ⊲ is a Quillen bifunctor.However since cofibrations in coDGA conil and fibrations in DGA 0 are particularly easy to work with, we will make use of the equivalent condition for the cotensoring functor {−, −}.That is we will show that for every cofibration i : C → C ′ in coDGA conil and every fibration j : B → B ′ in DGA 0 the induced map is a cofibration, which furthermore is acyclic if either i or j is.Proof.That we get fibrations in either case is immediate.For the second part we show the stronger statement that the convolution algebra functor preserves quasi-isomorphisms.This is sufficient as every weak equivalence in coDGA conil by necessity is also a quasi-isomorphism.Next note that the forgetful functor to DGVec preserves quasi-isomorphisms and commutes with the cotensoring functor.That is the convolution algebra functor is taken to the internal hom of dg-vector spaces.As the internal hom of dgvector spaces is exact the preservation of quasi-isomorphisms follows.
Proof.Let i : C ֒→ C ′ be a cofibration in coDGA conil and j : A ։ A ′ a (acyclic) fibration in DGA 0 .The relevant pullback diagram is Proof.For a free Lie algebra T Lie V we have the natural isomorphism

Hence we define
Given an arbitrary Lie algebra g, we write it as a coequaliser and define C ⊲ g the coequaliser of Since DGLA(−, h) takes colimits to limits we get the desired natural bijection DGLA(g, {C, h}) ∼ = DGLA(C ⊲ g, h).
By combining Proposition 7.2 and Proposition 7.3 we also get a third adjunction between the tensoring and the enrichement functor.As in the associative case we make use of the concept of measurings to show the coherence axiom for the module structure of DGLA.Adapted to the Lie algebra case the definition becomes the following.

Semi-enriched Koszul duality
Having established the the semi-module category of structures of DGA 0 and DGLA we will now return our attention to Koszul duality.Our main aim is to establish Theorem 1.1 and Theorem 1.2.That is we will show that the Quillen equivalences in both the associative and the com-Lie case of Koszul duality upgrades to semi-module Quillen equivalences.
As pointed out in [AJ13] the bar and cobar constructions are directly related to the constructed enrichment functor and the tensoring functor respectively.Specifically we have the following result.As a consequence of Proposition 8.1 we see that the bar-cobar adjunction, in the associative case, upgrades to a coDGA conil -module Quillen equivalence.Similarly Proposition 8.2 implies that the bar-cobar adjunction, in the com-Lie case, upgrades to a coCDGA conil -module Quillen equivalence.As a consequence, we have established Theorem 1.1 and Theorem 1.2.Remark 8.3.Note that it also follows that the bar construction B, in both the associative and the com-Lie case, is a quasi-strong semi-module functor.Explicitly the weak equivalence is given by here in the notation of the associative case.
Remark 8.4.It may seem that our results should generalise to the operadic context of Koszul duality.This is however not the case as can be seen from considering a third case of Koszul duality between the category of conilpotent dg-Lie algebra, coDGLA conil , and the category of commutative non-unital dg-algebras cDGA 0 .This case was established in [LM15] and shows that there is a Quillen equivalence coDGLA conil cDGA 0 .
The problem here, in regards to extending our results, is that coDGLA conil does not have the notion of a tensor product.Instead one could consider the monoidal structure given by the direct product, which indeed gives a closed monoidal structure on coDGLA conil albeit with a quite different internal hom functor from the other cases.However one quickly runs into trouble with defining a coDGLA conil -module structure for cDGA 0 as we don't have the concept of a convolution algebra or the Sweedler theory adjunctions to rely on.

Proposition 1. 5 .
Let (C, ∆ C ) be a conilpotent coalgebra and (D, ∆ D ) an arbitrary non-unital coalgebra.Then (C ⊗ D, ∆ C⊗D ) is also conilpotent.Proof.Let c ⊗ d be a pure tensor in C ⊗ D. By assumption there exists some n > 0 such that ∆ n C (c) = 0. Then we have right adjoint to X ⊲ −.Similarly we say that C is left closed if there exists a functor C(−, −) : C op × C → V such that C(A, −) is right adjoint to − ⊲ A. If C is both left-and right closed we say it is closed.Particularly in the closed case, we will refer to the [−, −] functor as the cotensoring functor and C(−, −) as the enrichment functor.Note that the cotensoring functor [−, −] makes C into a V-opmodule.The motivation for referring to C(−, −) as an enrichment is that in the monoidal case the notion of a closed V-module is equivalent to a tensored and cotensored C-category.Definition 2.8.Let (V, ⊗) be a symmetric semi-monoidal category and (C, ⊲, α) and (D, is a (left) Quillen bifunctor, then we say that V is a semi-monoidal model category.Definition 3.2.Let (V, ⊗) be a semi-monoidal model category and (C, ⊲) a closed V-module.If furthermore C is a model category and the tensoring functor, ⊲ : V × C → C, is a (left) Quillen bifunctor, then we say that C is a V-module model category.Definition 3.3.Let C and D be semi-monoidal model categories.Then a semi-monoidal Quillen adjunction (F, U, φ, m) is a Quillen adjunction (F, U, φ) such that (F, m) is a semi-monoidal functor.Definition 3.4.Let V be a semi-monoidal model category and C and D be V-module model categories.A V-module Quillen adjunction (F, U, φ, µ) is a Quillen adjunction such that (F, µ) is a V-module functor.

Proposition 4. 9 .
The restricted measurement functor M is represented in each variable.Proof.By definition of measuring the restricted measurement functor is represented in the second variable by ({C, B}, ǫ : C ⊗ {C, B} → B) , where ǫ is the counit of the tensor-hom adjunction for dg-vector spaces.The representability in the remaining variables now follows from the tensored and cotensored adjunctions constructed in Proposition 4.3 and Proposition 4.4.That is we have isomorphisms, M(C, A, B) ∼ = DGA 0 (A, {C, B}) ∼ = DGA 0 (C ⊲ A, B) ∼ = coDGA conil (C, DGA 0 (A, B)), natural in each variable.
the bottom face commutes.But by the universal property of u we have that u is rightcancellative on algebra morphisms.Hence the bottom face commutes.5Semi-monoidal model structure on coDGA conil