A Quillen model category structure on some categories of comonoids
aa r X i v : . [ m a t h . C T ] J a n A QUILLEN MODEL CATEGORY STRUCTURE ON SOMECATEGORIES OF COMONOIDS
ALEXANDRU E. STANCULESCU
Abstract.
We prove that for certain monoidal (Quillen) model categories,the category of comonoids therein also admits a model structure. Introduction A monoidal model category is a closed symmetric monoidal category which ad-mits a Quillen model category structure compatible in a certain sense with themonoidal product [6],[9]. The majority of the natural occurring examples of modelcategories are monoidal model categories. In [9], the authors gave a sufficient con-dition which ensured that the category of monoids in a monoidal model categoryadmits a model structure, extended in an appropriate sense from the base category.This condition was called the monoid axiom , and it is satisfied in many examples.Dually, one can consider comonoids in a monoidal category which has a modelstructure and ask for a model structure for comonoids, somehow inherited from thebase category. We were not able to find in the literature a general result along theselines. The situation turns out to be more complicated than with monoids. In thisnote we give a (very) partial answer to this problem. We prove Theorem 1.1.
Let E be a symmetric monoidal category with unit I and let Comon ( E ) be the category of (coassociative and counital) comonoids in E . We assume that ( i ) E is locally presentable, abelian and the monoidal product preserves colimitsand finite limits in each variable; ( ii ) E has two classes of maps W and Cof such that
Cof and the class ofmonomorphisms of E are the cofibrations of two model structures on E with thesame class W of weak equivalences; furthermore, either of the two model structuresis cofibrantly generated; ( iii ) the pushout-product axiom between the two model structures holds: if i : K → L belongs to Cof and i ′ : X → Y is a monomorphism, then the canonical map K ⊗ Y [ K ⊗ X L ⊗ X −→ L ⊗ Y is a monomorphism, which is a weak equivalence if either one of i or i ′ is; ( iv ) I is Cof -cofibrant and E has a coalgebra interval, by which we mean a fac-torisation of the codiagonal I ⊔ I ▽ / / i ⊔ i $ $ IIIIIIIII
ICyl ( I ) p < < yyyyyyyyy such that i ⊔ i belongs to Cof , p is a weak equivalence and the whole diagram livesin Comon ( E ) . Then
Comon ( E ) admits a model category structure in which a map is a weakequivalence (resp. cofibration) if and only if the underlying map is a weak equiva-lence (resp. monomorphism) in E . An analogue of 1.1 for the category of comodules over a comonoid in E is pre-sented in section 3.One of the motivations for writing this note was the paper [5]. In [2], the authorsextended the main result of [5] to the category of cooperads, or F -comonoids, inthe category of non-negatively graded chain complexes of vector spaces. We do notknow whether the technique used in this paper would provide a model structure onthe category of cooperads in E .2. Proof of theorem 1.1
In order to prove theorem 1.1 we shall use two results of J.H. Smith, recalledbelow.
Theorem 2.1. ( [3] , Thm. 1.7) Let E be a locally presentable category, W a fullaccessible subcategory of Mor( E ) , and I a set of morphisms of E . Suppose theysatisfy: c : W has the three-for-two property. c : inj( I ) ⊆ W . c : The class cof( I ) ∩ W is closed under transfinite composition and underpushout.Then setting weak equivalences:= W , cofibrations:= cof( I ) and fibrations:= inj(cof( I ) ∩ W ), one obtains a cofibrantly generated model structure on E . Theorem 2.2.
The class of weak equivalences of a combinatorial model categoryis accessible.
Proofs of the preceding theorem has been given in [7] and [8]. By general argu-ments the forgetful functor U : Comon ( E ) → E has a right adjoint and the category Comon ( E ) is locally presentable, see e.g. ([1], Remark below Lemma 2.76 and thedual of Corollary 2.75). We shall define a set I which will generate the class ofcofibrations and then check condition c1 of 2.1.Let C ∈ Comon ( E ). We say that ( D, i ) ∈ Comon ( E ) /C is an E - subobject of C if U ( i ) : U ( D ) → U ( C ) is a monomorphism. As pointed out to us by Steve Lack, the E -subobjects are precisely the strong subobjects in Comon ( E ). This can be seenusing the left exactness of the monoidal product.For example, if f : C → D is a map of comonoids, then the subobject m : Im ( f ) U ( D ) is an E -subobject of D and the canonical epi e : U ( C ) → Im ( f )is a map of comonoids. To see this, one uses the fact that Coim ( f ) ∼ = Im ( f ) andagain the left exactness of the monoidal product. For C and D comonoids we write C (cid:22) D if C is an E -subobject of D . Lemma 2.3.
There is a regular cardinal κ such that every comonoid C is a κ -filtered colimit C = colim C i , with C i (cid:22) C .Proof. The functor U preserves and reflects epimorphisms. Let λ be a regularcardinal such that Comon ( E ) is locally λ -presentable and let C be a comonoid.Write C = colim D i , with canonical arrows ϕ i : D i → C and with D i λ -presentable.Factor U ( ϕ i ) as U ( D i ) e i → C i m i → U ( C ), with m i mono and e i epi. By the above, C i (cid:22) D and one clearly has C = colim C i . Since Comon ( E ) is co-well-powered,there is a set (up to isomorphism) Q of all quotients of all λ -presentable objects.Therefore there is a regular cardinal κ such that Q is contained in the set of all κ -presentable objects of Comon ( E ). (cid:3) QUILLEN MODEL CATEGORY STRUCTURE ON SOME CATEGORIES OF COMONOIDS 3
We define I to be the set of all isomorphism classes of cofibrations A → B with B κ -presentable.
Lemma 2.4.
A map has the right lifting property with respect to the cofibrationsiff it has the right lifting property with respect to the maps in I . To prove this lemma we need the following general result.
Lemma 2.5.
Let E be an abelian and monoidal category with monoidal product ⊗ which is left exact in each variable. If A X and B Y are subobjects, then A ⊗ B = ( A ⊗ Y ) ∩ ( X ⊗ B ) . As a consequence, if i : D → C and j : E → C aremaps of comonoids in E such that U ( i ) and U ( j ) are monomorphisms, then D ∩ E is a comonoid in E .Proof. For the first part, start with the short exact sequences 0 → A → X → X/A → → B → Y → Y /B →
0. By tensoring them one produces a3 × E D ∩ E / / % % KKKKKKKKKK (cid:15) (cid:15) E j % % KKKKKKKKKKK (cid:15) (cid:15) D i / / (cid:15) (cid:15) C (cid:15) (cid:15) P / / % % JJJJJJJJJJ E ⊗ E j ⊗ j % % JJJJJJJJJ D ⊗ D i ⊗ i / / C ⊗ C in which the top and bottom faces are pullbacks. The bottom face can be calculatedas an iterated pullback P / / (cid:15) (cid:15) ( D ∩ E ) ⊗ E (cid:15) (cid:15) / / E ⊗ E j ⊗ E (cid:15) (cid:15) D ⊗ ( D ∩ E ) / / (cid:15) (cid:15) D ⊗ E / / (cid:15) (cid:15) C ⊗ E C ⊗ j (cid:15) (cid:15) D ⊗ D D ⊗ i / / D ⊗ C i ⊗ C / / C ⊗ C, therefore P is ( D ∩ E ) ⊗ ( D ∩ E ) by the first part. This provides D ∩ E a comulti-plication. The counit of D ∩ E is the counit of C restricted to D ∩ E . (cid:3) Proof. (of lemma 2.4) The proof is standard. Let C f / / i (cid:15) (cid:15) X p (cid:15) (cid:15) D / / Y be a commutative diagram with i a cofibration and p having the right lifting prop-erty with respect to the maps in I . Let S be the set consisting of pairs ( E, l ), where
ALEXANDRU E. STANCULESCU C (cid:22) E (cid:22) D and l : E → X is a morphism making the diagram C / / (cid:15) (cid:15) X p (cid:15) (cid:15) E / / l nnnnnnnnnnnnnn D / / Y commutative. We order S by ( E, l ) ( E ′ , l ′ ) iff E (cid:22) E ′ and l ′ is an extension of l . Then S is nonempty, as it contains ( C, f ). Let C be any chain in S and let κ ′ be a regular cardinal such that both E and Comon ( E ) are locally κ ′ -presentable.Then C is κ ′ -directed and therefore colim C is defined in Comon ( E ), and U (colim C )is the colimit of the U ( F ), ( F, m ) ∈ C . Hence colim C → D is a cofibration. Also,we have a unique l : colim C → X extending each m , and clearly (colim C , l ) is anelement of S . This shows that Zorn’s lemma is applicable, therefore the set S hasa maximal element ( E, l ). We are going to show that E ∼ = D by showing thatfor each κ -presentable comonoid B (cid:22) D , one has B (cid:22) E . This suffices since D ,being the κ -filtered colimit of all of its E -subobjects, is the least upper bound of its κ -presentable E -subobjects.Take B (cid:22) D with B κ -presentable. Using lemma 2.5 and the hypothesis we havea diagonal filler d in the commutative diagram E ∩ B / / (cid:15) (cid:15) E l / / X p (cid:15) (cid:15) B / / D / / Y. Therefore in the diagram E ∩ B / / (cid:15) (cid:15) B (cid:15) (cid:15) d (cid:25) (cid:25) E / / l ) ) TTTTTTTTTTTTTTTTTTTT E ∪ B l ′ GGGGGGGGG X in which the square is a pushout, there is a map l ′ : E ∪ B → X extending l , andso ( E ∪ B, l ′ ) ∈ S . This shows that ( E ∪ B, l ′ ) ( E, l ) since (
E, l ) was maximal.It follows that B (cid:22) E . (cid:3) By performing the small object argument it follows from lemma 2.4 and a re-tract argument that the class of cofibrations is the class
Cof ( I ). It remains to checkcondition c1 of 2.1. For this we shall use Let E be a model categoryand let F : C ⇄ E : G be an adjoint pair ( F : C → E is the left adjoint). We define a map f of C to be a cofibration ( resp. weak equivalence) if F ( f ) is such in E . Suppose that C is finitelycocomplete, it has a cofibrant replacement functor and a functorial cylinder objectfor cofibrant objects. Then a map of C that has the right lifting property with respectto all cofibrations is a weak equivalence. QUILLEN MODEL CATEGORY STRUCTURE ON SOME CATEGORIES OF COMONOIDS 5
Proof.
We recall its proof for the sake of completeness. Let f : X → Y be map of C which has the right lifting property with respect to all cofibrations. Letˆ CX i X / / ˆ C ( f ) (cid:15) (cid:15) X f (cid:15) (cid:15) ˆ CY i Y / / Y be the cofibrant replacement of f . Then the diagram ∅ / / (cid:15) (cid:15) ˆ CX i X / / X f (cid:15) (cid:15) ˆ CY ˆ CY i Y / / Y has a diagonal filler d . Let ˆ CX ⊔ ˆ CX i ⊔ i −→ Cyl ( ˆ CX ) p → ˆ CX be the cylinder objectfor ˆ CX . Consider the commutative diagramˆ CX ⊔ ˆ CX ( d ˆ C ( f ) ,i X ) / / i ⊔ i (cid:15) (cid:15) X f (cid:15) (cid:15) Cyl ( ˆ CX ) fi X p / / Y. By hypothesis it has a diagonal filler H , and so d ˆ C ( f ) is a weak equivalence. Sincethe weak equivalences of E satisfy the two out of six property, it follows that d is aweak equivalence. (cid:3) We return to the proof of 1.1. By 2.6 it suffices to show that there is a functorialcylinder object for comonoids. This is guaranteed by hypotheses ( iii ) and ( iv ). Theproof of theorem 1.1 is complete. Remark 2.7.
Let E be as in the statement of theorem 1.1. If, moreover,the cylinder object Cyl ( I ) for I is a cocommutative comonoid, then the category CComon ( E ) of cocommutative comonoids in E admits a model category structurein which a map is a weak equivalence (resp. cofibration) if and only if the under-lying map is a weak equivalence (resp. monomorphism) in E . Examples. ( a ) Let R be a commutative von Neumann regular ring and let Ch ( R ) be the category of unbounded chain complexes of R -modules. We consideron Ch ( R ) the projective and injective model structures [6]. Ch ( R ) has a well-knowncoalgebra interval given by ... → → Re ∂ → Ra ⊕ Rb → → ..., where ∂ ( e ) = b − a and Ra ⊕ Rb is in degree 0. The maps i and i are the inclusionsand the map p is a, b
1, see e.g. ([9], section 5). The last part of ( i ) is shown in([10], Proof of Prop. 3.3 for Ch ).( b ) The above considerations apply to the category of non-negatively gradedchain complexes as well. ALEXANDRU E. STANCULESCU Comodules
Let E be a monoidal category with monoidal product ⊗ . Given a (coassociativeand counital) comonoid C in E , we denote by M od C the category of right C -comodules in E . There is a forgetful-cofree adjunction U : M od C ⇄ E : − ⊗ C (1) Theorem 3.1.
Let E be a cofibrantly generated monoidal model category and let C be a (coassociative and counital) comonoid in E . Suppose that ( i ) the cofibrations of the model structure are precisely the monomorphisms; ( ii ) E is locally presentable, abelian, and for each object X of E the functor −⊗ X is left exact, where ⊗ denotes the monoidal product of E .Then M od C admits a cofibrantly generated model structure in which a map f isa weak equivalence (resp. cofibration) if and only if the underlying map is a weakequivalence (resp. monomorphism) in E . The proof of the above theorem follows the same steps as the proof of 1.1, exceptthat condition c1 of 2.1 will be a consequence of lemma 3.2 below.We say that a map of C -comodules is a fibration if it has the right lifting prop-erty with respect to the maps which are both cofibrations and weak equivalences.We say that a map of C -comodules is a trivial fibration if it is both a fibrationand a weak equivalence. Lemma 3.2.
The category
M od C has a weak factorisation system (cofibrations,trivial fibrations).Proof. We follow an idea from [5]. Let f : M → N be a map of C -comodules.We factor the map U ( M ) → U ( M ) i → X →
0. Then f factors as M j → N × ( X ⊗ C ) p → N where j = ( i ∗ , f ), i ∗ is the adjoint transpose of i and p : N × ( X ⊗ C ) → N is theprojection. The map p is a weak equivalence since it is the map N ⊕ ( X ⊗ C ) → N ⊕ (0 ⊗ C ) ∼ = N , which is a weak equivalence. We show that the underlying mapof j is a monomorphism. One has i = ǫ X U ( p j ), where ǫ X is the counit of theadjunction (1) and p : N × ( X ⊗ C ) → ( X ⊗ C ) is the projection. Therefore j is a cofibration. Next we show that p : N × ( X ⊗ C ) → N has the right liftingproperty with respect to all cofibrations. Let M ′ / / k (cid:15) (cid:15) N × ( X ⊗ C ) p (cid:15) (cid:15) N ′ / / N be a commutative diagram with k a cofibration. This diagram has a diagonal fillerif and only if the diagram U ( M ′ ) / / U ( k ) (cid:15) (cid:15) X (cid:15) (cid:15) U ( N ′ ) / / X . Therefore p is a fibration. Letnow f : M → N be a trivial fibration. Factor it as above M j → N × ( X ⊗ C ) p → N . QUILLEN MODEL CATEGORY STRUCTURE ON SOME CATEGORIES OF COMONOIDS 7
Since j is a weak equivalence, there is a diagonal filler in the diagram M j (cid:15) (cid:15) M f (cid:15) (cid:15) N × ( X ⊗ C ) / / N hence f is a (domain) retract of a map which has the right lifting property withrespect to all cofibrations, therefore f has the right lifting property with respectto all cofibrations. Conversely, let f : M → N have the right lifting property withrespect to all cofibrations. The same argument shows that f is a (domain) retractof a trivial fibration, hence f is a trivial fibration. (cid:3) Acknowledgements.
We are indebted to Professor Michael Makkai for manyuseful discussions and suggestions. We thank the referee for useful comments.
References [1] J. Ad´amek, J. Rosick´y,
Locally presentable and accessible categories , London MathematicalSociety Lecture Note Series, 189. Cambridge University Press, Cambridge, 1994. xiv+316 pp.[2] M. Aubry, D. Chataur,
Cooperads and coalgebras as closed model categories , J. Pure Appl.Algebra 180 (2003), no. 1-2, 1–23.[3] T. Beke,
Sheafifiable homotopy model categories
Math. Proc. Cambridge Philos. Soc. 129(2000), no. 3, 447–475.[4] F. Borceux,
Handbook of categorical algebra. 2. Categories and structures , Encyclopediaof Mathematics and its Applications, 51. Cambridge University Press, Cambridge, 1994.xviii+443 pp.[5] E. Getzler, P. G. Goerss,
A model category structure for differential graded coalgebras
Model categories , Mathematical Surveys and Monographs, 63. American Mathe-matical Society, Providence, RI, 1999. xii+209 pp.[7] J. Lurie,
Higher topos theory , Preprint arXiv:math/0608040, March 2007.[8] J. Rosick´y,
On combinatorial model categories , Preprint arXiv:math/0708.2185, August 2007.[9] S. Schwede, B. E. Shipley,
Algebras and modules in monoidal model categories , Proc. LondonMath. Soc. (3) 80 (2000), no. 2, 491–511.[10] B. Shipley, H Z -algebra spectra are differential graded algebras , Amer. J. Math. 129 (2007),no. 2, 351–379. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str.West, Montr´eal, Qu´ebec, Canada, H3A 2K6
E-mail address ::