Cofibrant Replacement Research Articles

Computads for generalised signatures

We introduce a notion of signature whose sorts form a direct category, and study computads for such signatures. Algebras for such a signature are presheaves with an interpretation of every function symbol of the signature, and we describe how computads give rise to signatures. Motivated by work of Batanin, we show that computads with certain generator-preserving morphisms form a presheaf category, and describe a forgetful functor from algebras to computads. Algebras free on a computad turn out to be the cofibrant objects for a certain cofibrantly generated weak factorisation system, and the adjunction above induces the universal cofibrant replacement, in the sense of Garner, for this weak factorisation system. Finally, we conclude by explaining how many-sorted structures, weak ω-categories, and algebraic semi-simplicial Kan complexes are algebras of such signatures, and we propose a notion of weak multiple category.

A monoidal Dold-Kan correspondence for comodules

Cofibrantly generated model categories are generalizations of CW-approximations which provide an inductive cofibrant replacement. We provide examples of inductive fibrant replacements constructed as Postnikov towers for simplicial and differential graded comodules. Our main application is to show that simplicial comodules and connective differential graded comodules are Quillen equivalent and their derived cotensor products correspond. We deduce that the rational A-theory of a simply connected space X is equivalent to the K-theory of perfect chain complexes with a C⁎(X;Q)-comodule structure.

Symmetric homotopy theory for operads

The purpose of this foundational paper is to introduce various notions and constructions in order to develop the homotopy theory for differential graded operads over any ring. The main new idea is to consider the action of the symmetric groups as part of the defining structure of an operad and not as the underlying category. We introduce a new dual category of higher cooperads, a new higher bar-cobar adjunction with the category of operads, and a new higher notion of homotopy operads, for which we establish the relevant homotopy properties. For instance, the higher bar-cobar construction provides us with a cofibrant replacement functor for operads over any ring. All these constructions are produced conceptually by applying the curved Koszul duality for colored operads. This paper is a first step toward a new Koszul duality theory for operads, where the action of the symmetric groups is properly taken into account.

The Tamarkin–Tsygan calculus of an associative algebra<i> à la</i> Stasheff

We show how to compute the Tamarkin-Tsygan calculus of an associative algebra by providing, for a given cofibrant replacement of it, a `small' $\mathsf{Calc}_\infty$-model of its calculus, which we make somewhat explicit at the level of $\mathsf{Calc}$-algebras. To do this, we prove that the operad $\mathsf{Calc}$ is inhomogeneous Koszul; to our best knowledge, this result is new. We illustrate our technique by carrying out some computations for two monomial associative algebra using the cofibrant replacement obtained by the author in 1804.01435.

Delooping derived mapping spaces of bimodules over an operad

To any topological operad O, we introduce a cofibrant replacement in the category of bimodules over itself such that for every map \(\eta :O\rightarrow O'\) of operads, the corresponding model \({\textit{Bimod}}_{O}^{h}(O\,;\,O')\) of derived mapping space of bimodules is an algebra over the one dimensional little cubes operad \(\mathcal {C}_{1}\). We also build an explicit weak equivalence of \(\mathcal {C}_{1}\)-algebras from the loop space \(\Omega {\textit{Operad}}^{h}(O\,;\,O')\) to \({\textit{Bimod}}_{O}^{h}(O\,;\,O')\).

Derived coisotropic structures I: affine case

We define and study coisotropic structures on morphisms of commutative dg algebras in the context of shifted Poisson geometry, i.e. $P_n$-algebras. Roughly speaking, a coisotropic morphism is given by a $P_{n+1}$-algebra acting on a $P_n$-algebra. One of our main results is an identification of the space of such coisotropic structures with the space of Maurer--Cartan elements in a certain dg Lie algebra of relative polyvector fields. To achieve this goal, we construct a cofibrant replacement of the operad controlling coisotropic morphisms by analogy with the Swiss-cheese operad which can be of independent interest. Finally, we show that morphisms of shifted Poisson algebras are identified with coisotropic structures on their graph.

Knuth’s Coherent Presentations of Plactic Monoids of Type A

We construct finite coherent presentations of plactic monoids of type A. Such coherent presentations express a system of generators and relations for the monoid extended in a coherent way to give a family of generators of the relations amongst the relations. Such extended presentations are used for representations of monoids, in particular, it is a way to describe actions of monoids on categories. Moreover, a coherent presentation provides the first step in the computation of a categorical cofibrant replacement of a monoid. Our construction is based on a rewriting method introduced by Squier that computes a coherent presentation from a convergent one. We compute a finite coherent presentation of a plactic monoid from its column presentation and we reduce it to a Tietze equivalent one having Knuth’s generators.

Algebraic weak factorisation systems II: Categories of weak maps

We investigate the categories of weak maps associated to an algebraic weak factorisation system (awfs) in the sense of Grandis–Tholen [14]. For any awfs on a category with an initial object, cofibrant replacement forms a comonad, and the category of (left) weak maps associated to the awfs is by definition the Kleisli category of this comonad. We exhibit categories of weak maps as a kind of “homotopy category”, that freely adjoins a section for every “acyclic fibration” (= right map) of the awfs; and using this characterisation, we give an alternate description of categories of weak maps in terms of spans with left leg an acyclic fibration. We moreover show that the 2-functor sending each awfs on a suitable category to its cofibrant replacement comonad has a fully faithful right adjoint: so exhibiting the theory of comonads, and dually of monads, as incorporated into the theory of awfs. We also describe various applications of the general theory: to the generalised sketches of Kinoshita–Power–Takeyama [22], to the two-dimensional monad theory of Blackwell–Kelly–Power [4], and to the theory of dg-categories [19].

Effective homology for homotopy colimit and cofibrant replacement

We extend the notion of simplicial set with effective homology presented in [22] to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets $X \colon \mathcal{I}\rightarrow \mbox{sSet}$ such that each simplicial set $X(i)$ has effective homology, we present an algorithm computing the homotopy colimit $\mbox{hocolim}\,X$ as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement $X^{\mbox{cof}}$ of $X$ as a diagram with effective homology. This is applied to computing of equivariant cohomology operations.

Transfer of algebras over operads along Quillen adjunctions

Let 𝒱 be a cofibrantly generated monoidal model category and ℳ be a monoidal 𝒱-model category. If C is any set, then we denote by ℳC the product category of copies of ℳ indexed by C. Given a cofibrant C-coloured operad 𝒫 in 𝒱, we give sufficient conditions for the fibrant replacement and cofibrant replacement functors in ℳC to preserve 𝒫-algebra structures. In particular, we show how 𝒫-algebra structures can be transferred along Quillen adjunctions of monoidal 𝒱-model categories, and we apply this result to the Quillen adjunctions defined by enriched Bousfield localizations and colocalizations on ℳ. As an application, we prove that in the category of symmetric spectra the n-connective cover functor preserves A∞ and E∞ module spectra over connective ring spectra, for every n∈ℤ.

On the structure of simplicial categories associated to quasi-categories

AbstractThe homotopy coherent nerve from simplicial categories to simplicial sets and its left adjoint are important to the study of (∞, 1)-categories because they provide a means for comparing two models of their respective homotopy theories, giving a Quillen equivalence between the model structures for quasi-categories and simplicial categories. The functor also gives a cofibrant replacement for ordinary categories, regarded as trivial simplicial categories. However, the hom-spaces of the simplicial category X arising from a quasi-category X are not well understood. We show that when X is a quasi-category, all Λ21 horns in the hom-spaces of its simplicial category can be filled. We prove, unexpectedly, that for any simplicial set X, the hom-spaces of X are 3-coskeletal. We characterize the quasi-categories whose simplicial categories are locally quasi, finding explicit examples of 3-dimensional horns that cannot be filled in all other cases. Finally, we show that when X is the nerve of an ordinary category, X is isomorphic to the simplicial category obtained from the standard free simplicial resolution, showing that the two known cofibrant “simplicial thickenings” of ordinary categories coincide, and furthermore its hom-spaces are 2-coskeletal.

Γ–homology of algebras over an operad

The purpose of this paper is to study generalizations of Gamma-homology in the context of operads. Good homology theories are associated to operads under appropriate cofibrancy hypotheses, but this requirement is not satisfied by usual operads outside the characteristic zero context. In that case, the idea is to pick a cofibrant replacement Q of the given operad P. We can apply to P-algebras the homology theory associated to Q in order to define a suitable homology theory on the category of P-algebras. We make explicit a small complex to compute this homology when the operad P is binary and Koszul. In the case of the commutative operad P=Com, we retrieve the complex introduced by Robinson for the Gamma-homology of commutative algebras.

Homotopy unital [formula omitted]-algebras

It is well known that the differential graded operad of A ∞ -algebras is a cofibrant replacement (a dg-resolution) of the operad of associative differential graded algebras without units. In this article we find a cofibrant replacement of the operad of associative differential graded algebras with units. Algebras over it are called homotopy unital A ∞ -algebras. We prove that the operad bimodule of A ∞ -morphisms is a cofibrant replacement of the operad bimodule of morphisms of dg-algebras without units. Similarly we show that the operad bimodule of homotopy unital A ∞ -morphisms is a cofibrant replacement of the operad bimodule of morphisms of dg-algebras with units.

Cotorsion Pairs of Modules from Left Adjoints

Let R be a ring with 1 and M a right R-module. In this article, we will see that the functor F = M ⊗ R – gives rise to a complete hereditary cotorsion pair where the left class consists of the F-acyclic objects. This cotorsion pair induces a Quillen model structure on Ch(R) which recovers the derived functors . An F-acyclic resolution is as good as a cofibrant replacement in this model structure. So in short, we formalize the fact that can be computed using F-acyclic resolutions.

A homotopy-theoretic universal property of Leinster's operad for weak ω-categories

AbstractWe explain how any cofibrantly generated weak factorisation system on a category may be equipped with a universally and canonically determined choice of cofibrant replacement. We then apply this to the theory of weak ω-categories, showing that the universal and canonical cofibrant replacement of the operad for strict ω-categories is precisely Leinster's operad for weak ω-categories.