Quillen Functor Research Articles

The Simplicial Coalgebra of Chains Under Three Different Notions of Weak Equivalence

Abstract We study the simplicial coalgebra of chains on a simplicial set with respect to three notions of weak equivalence. To this end, we construct three model structures on the category of reduced simplicial sets for any commutative ring $R$. The weak equivalences are given by: (1) an $R$-linearized version of categorical equivalences, (2) maps inducing an isomorphism on fundamental groups and an $R$-homology equivalence between universal covers, and (3) $R$-homology equivalences. Analogously, for any field ${\mathbb{F}}$, we construct three model structures on the category of connected simplicial cocommutative ${\mathbb{F}}$-coalgebras. The weak equivalences in this context are (1′) maps inducing a quasi-isomorphism of dg algebras after applying the cobar functor, (2′) maps inducing a quasi-isomorphism of dg algebras after applying a localized version of the cobar functor, and (3′) quasi-isomorphisms. Building on a previous work of Goerss in the context of (3)–(3′), we prove that, when ${\mathbb{F}}$ is algebraically closed, the simplicial ${\mathbb{F}}$-coalgebra of chains defines a homotopically full and faithful left Quillen functor for each one of these pairs of model categories. More generally, when ${\mathbb{F}}$ is a perfect field, we compare the three pairs of model categories in terms of suitable notions of homotopy fixed points with respect to the absolute Galois group of ${\mathbb{F}}$.

Algebraic models of change of groups functors in (co)free rational equivariant spectra

Free and cofree equivariant spectra are important classes of equivariant spectra which represent equivariant cohomology theories on free equivariant spaces. Greenlees-Shipley [24], [26] and Pol and the author [45] have given an algebraic model for rational (co)free equivariant spectra. In this paper, we extend this framework by proving that the Quillen functors of induction-restriction-coinduction between categories of (co)free rational equivariant spectra correspond to Quillen functors between the algebraic models in the case of connected compact Lie groups. This is achieved using new abstract techniques regarding correspondences of Quillen functors along Quillen equivalences, which we expect to be of use in other applications.

Profinite ∞-operads

We show that a profinite completion functor for (simplicial or topological) operads with good homotopical properties can be constructed as a left Quillen functor from an appropriate model category of ∞-operads to a certain model category of profinite ∞-operads. The construction is based on a notion of lean ∞-operad, and we characterize those ∞-operads weakly equivalent to lean ones in terms of homotopical finiteness properties. Several variants of the construction are also discussed, such as the cases of unital (or closed) ∞-operads and of ∞-categories.

Functorial Factorizations in the Category of Model Categories

We prove that any right Quillen functor between arbitrary model categories admits non trivial functorial factorizations that are similar to those of a model structure. Given a monad, operad or a PROP(erad) $$\mathcal {O}$$ , if we apply one of the factorizations to the forgetful functor $$\textsf {U}: \mathcal {O}{\text {-Alg}}(\textsf {M}) \longrightarrow \textsf {M}$$ , we extend the theory of Quillen–Segal $$\mathcal {O}$$ -algebras initiated in Bacard (Higher Struct 4(1):57–114, 2020), without the hypothesis of $$\textsf {M}$$ being a combinatorial model category.

Comparing the orthogonal and unitary functor calculi

The orthogonal and unitary calculi give a method to study functors from the category of real or complex inner product spaces to the category of based topological spaces. We construct functors between the calculi from the complexification-realification adjunction between real and complex inner product spaces. These allow for movement between the versions of calculi, and comparisons between the Taylor towers produced by both calculi. We show that when the inputted orthogonal functor is weakly polynomial, the Taylor tower of the functor restricted through realification and the restricted Taylor tower of the functor agree up to weak equivalence. We further lift the homotopy level comparison of the towers to a commutative diagram of Quillen functors relating the model categories for orthogonal calculus and the model categories for unitary calculus.

Lie models of simplicial sets and representability of the Quillen functor

Extending the model of the interval, we explicitly define for each n ≥ 0 a free complete differential graded Lie algebra \(\mathfrak{L}_n\) generated by the simplices of Δn, with desuspended degrees, in which the vertices are Maurer-Cartan elements and the differential extends the simplicial chain complex of the standard n-simplex. The family \(\{\mathfrak{L{_\bullet}}\}_{n\geq0}\) is endowed with a cosimplicial differential graded Lie algebra structure which we use to construct two adjoint functors $$Simpset\underset{{\left\langle \cdot \right\rangle }}{\overset{\mathfrak{L}}{\longleftrightarrow}}DGL$$ given by \(\langle{L}\rangle_\bullet={\rm{DGL}}(\mathfrak{L}_\bullet,L)\) and \(\mathfrak{L}(K)=\lim_{\rightarrow K}\mathfrak{L_\bullet}\). This new tool lets us extend the Quillen rational homotopy theory approach to any simplicial set K whose path components are not necessarily simply connected.We prove that \(\mathfrak{L}(K)\) contains a model of each component of K. When K is a 1-connected finite simplicial complex, the Quillen model of K can be extracted from \(\mathfrak{L}(K)\). When K is connected then, for a perturbed differential ϑa, \(H_0(\mathfrak{L}(K),\partial_a)\) is the Malcev Lie completion of π1(K). Analogous results are obtained for the realization 〈L〉 of any complete DGL.

Relative FP-injective and FP-flat complexes and their model structures

The present article studies homological and homotopical aspects of FPn-injective and FPn-flat complexes, and describes homological dimensions associated to them. After establishing a relation between these dimensions via Pontrjagin duality, we construct (pre)covers and (pre)envelopes by complexes with finite FPn-injective and FPn-flat dimensions, thus setting up the bases for an approximation theory relative to complexes of finite type. We also construct on the category of complexes several model structures from modules and complexes with finite FPn-injective and FPn-flat dimensions, and analyze some situations where it is possible to connect these structures via Quillen functors.

Presentably symmetric monoidal ∞–categories are represented by symmetric monoidal model categories

We prove the theorem stated in the title. More precisely, we show the stronger statement that every symmetric monoidal left adjoint functor between presentably symmetric monoidal infinity-categories is represented by a strong symmetric monoidal left Quillen functor between simplicial, combinatorial and left proper symmetric monoidal model categories.

The homotopy theory of simplicial props

The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. In this paper, the second in a series on "higher props," we show that the category of all small colored simplicial props admits a cofibrantly generated model category structure. With this model structure, the forgetful functor from props to operads is a right Quillen functor.

The diagonal of a multicosimplicial object

We show that the functor that takes a multicosimplicial object in a model category to its diagonal cosimplicial object is a right Quillen functor. This implies that the diagonal of a Reedy fibrant multicosimplicial object is a Reedy fibrant cosimplicial object, which has applications to the calculus of functors. We also show that, although the diagonal functor is a Quillen functor, it is not a Quillen equivalence for multicosimplicial spaces. We also discuss total objects and homotopy limits of multicosimplicial objects. We show that the total object of a multicosimplicial object is isomorphic to the total object of the diagonal, and that the diagonal embedding of the cosimplicial indexing category into the multicosimplicial indexing category is homotopy left cofinal, which implies that the homotopy limits are weakly equivalent if the multicosimplicial object is at least objectwise fibrant.

From homotopy operads to infinity-operads

The goal of the present paper is to compare, in a precise way, two approaches of operads up to homotopy which appear in the literature. Namely, we construct a functor from the category of strict unital homotopy colored operads to the category of infinity-operads. The former notion, that we make precise, is the operadic generalization of the notion of A-infinity-categories and the latter notion was defined by Moerdijk–Weiss in order to generalize the simplicial notion of infinity-category of Joyal–Lurie. This functor extends in two directions the simplicial nerve of Faonte–Lurie for A-infinity-categories and the homotopy coherent nerve of Moerdijk–Weiss for differential graded operads; it is also shown to be equivalent to a big nerve à la Lurie for differential graded operads. We prove that it satisfies some homotopy properties with respect to weak equivalences and fibrations; for instance, it is shown to be a right Quillen functor.

Comparing the orthogonal and homotopy functor calculi

Goodwillie's homotopy functor calculus constructs a Taylor tower of approximations to F, often a functor from spaces to spaces. Weiss's orthogonal calculus provides a Taylor tower for functors from vector spaces to spaces. In particular, there is a Weiss tower associated to the functor V↦F(SV), where SV is the one-point compactification of V.In this paper, we give a comparison of these two towers and show that when F is analytic the towers agree up to weak equivalence. We include two main applications, one of which gives as a corollary the convergence of the Weiss Taylor tower of BO. We also lift the homotopy level tower comparison to a commutative diagram of Quillen functors, relating model categories for Goodwillie calculus and model categories for the orthogonal calculus.

Capturing Goodwillie's derivative

Recent work of Biedermann and Röndigs has translated Goodwillie's calculus of functors into the language of model categories. Their work focuses on symmetric multilinear functors and the derivative appears only briefly. In this paper we focus on understanding the derivative as a right Quillen functor to a new model category. This is directly analogous to the behaviour of Weiss's derivative in orthogonal calculus. The immediate advantage of this new category is that we obtain a streamlined and more informative proof that the n-homogeneous functors are classified by spectra with a Σn-action. In a later paper we will use this new model category to give a formal comparison between the orthogonal calculus and Goodwillie's calculus of functors.

Homotopy limits of model categories and more general homotopy theories

Generalizing a definition of homotopy fiber products of model categories, we give a definition of the homotopy limit of a diagram of left Quillen functors between model categories. As has been previously shown for homotopy fiber products, we prove that such a homotopy limit does in fact correspond to the usual homotopy limit, when we work in a more general model for homotopy theories in which they can be regarded as objects of a model category.

Homotopy fiber products of homotopy theories

Given an appropriate diagram of left Quillen functors between model categories, one can define a notion of homotopy fiber product, but one might ask if it is really the correct one. Here, we show that this homotopy pullback is well-behaved with respect to translating it into the setting of more general homotopy theories, given by complete Segal spaces, where we have well-defined homotopy pullbacks.

Calculus of functors and model categories

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure [B. Chorny, W.G. Dwyer, Homotopy theory of small diagrams over large categories, preprint, 2005]. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for a classification of polynomial and homogeneous functors. In the n-homogeneous model structure, the nth derivative is a Quillen functor to the category of spectra with Σ n -action. After taking into account only finitary functors—which may be done in two different ways—the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T.G. Goodwillie [T.G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003) 645–711 (electronic)].

Karoubi-Villamayor and Quillen functors for triangular categories

Karoubi-Villamayor and Quillen functors for triangular categories

Homology of dendroidal sets

We define for every dendroidal set X a chain complex and show that this assignment determines a left Quillen functor. Then we define the homology groups Hn(X) as the homology groups of this chain complex. This generalizes the homology of simplicial sets. Our main result is that the homology of X is isomorphic to the homology of the associated spectrum K(X) as discussed in previous work by the authors. Since these homology groups are sometimes computable we can identify some spectra K(X) which we could not identify before.

Homological properties of non-deterministic branchings of mergings in higher dimensional automata

The branching (resp. merging) space functor of a flow is a left Quillen functor. The associated derived functor allows to define the branching (resp. merging) homology of a flow. It is then proved that this homology theory is a dihomotopy invariant and that higher dimensional branchings (resp. mergings) satisfy a long exact sequence.