No Arabic abstract
The theory of derivators provides a convenient abstract setting for computing with homotopy limits and colimits. In enriched homotopy theory, the analogues of homotopy (co)limits are weighted homotopy (co)limits. In this thesis, we develop a theory of derivators and, more generally, prederivators enriched over a monoidal derivator E. In parallel to the unenriched case, these E-prederivators provide a framework for studying the constructions of enriched homotopy theory, in particular weighted homotopy (co)limits. As a precursor to E-(pre)derivators, we study E-categories, which are categories enriched over a bicategory Prof(E) associated to E. We prove a number of fundamental results about E-categories, which parallel classical results for enriched categories. In particular, we prove an E-category Yoneda lemma, and study representable maps of E-categories. In any E-category, we define notions of weighted homotopy limits and colimits. We define E-derivators to be E-categories with a number of further properties; in particular, they admit all weighted homotopy (co)limits. We show that the closed E-modules studied by Groth, Ponto and Shulman give rise to associated E-derivators, so that the theory of E-(pre)derivators captures these examples. However, by working in the more general context of E-prederivators, we can study weighted homotopy (co)limits in other settings, in particular in settings where not all weighted homotopy (co)limits exist. Using the E-category Yoneda lemma, we prove a representability theorem for E-prederivators. We show that we can use this result to deduce representability theorems for closed E-modules from representability results for their underlying categories.
We develop some aspects of the theory of derivators, pointed derivators, and stable derivators. As a main result, we show that the values of a stable derivator can be canonically endowed with the structure of a triangulated category. Moreover, the functors belonging to the stable derivator can be turned into exact functors with respect to these triangulated structures. Along the way, we give a simplification of the axioms of a pointed derivator and a reformulation of the base change axiom in terms of Grothendieck (op)fibration. Furthermore, we have a new proof that a combinatorial model category has an underlying derivator.
Using the description of enriched $infty$-operads as associative algebras in symmetric sequences, we define algebras for enriched $infty$-operads as certain modules in symmetric sequences. For $mathbf{V}$ a nice symmetric monoidal model category, we prove that strict algebras for $Sigma$-cofibrant operads in $mathbf{V}$ are equivalent to algebras in the associated symmetric monoidal $infty$-category in this sense. We also show that $mathcal{O}$-algebras in $mathcal{V}$ can equivalently be described as morphisms of $infty$-operads from $mathcal{O}$ to endomorphism operads of (families of) objects of $mathcal{V}$.
We prove a rectification theorem for enriched infinity-categories: If V is a nice monoidal model category, we show that the homotopy theory of infinity-categories enriched in V is equivalent to the familiar homotopy theory of categories strictly enriched in V. It follows, for example, that infinity-categories enriched in spectra or chain complexes are equivalent to spectral categories and dg-categories. A similar method gives a comparison result for enriched Segal categories, which implies that the homotopy theories of n-categories and (infinity,n)-categories defined by iterated infinity-categorical enrichment are equivalent to those of more familia
In this paper we present background results in enriched category theory and enriched model category theory necessary for developing model categories of enriched functors suitable for doing functor calculus.
The invertibility hypothesis for a monoidal model category S asks that localizing an S-enriched category with respect to an equivalence results in an weakly equivalent enriched category. This is the most technical among the axioms for S to be an excellent model category in the sense of Lurie, who showed that the category of S-enriched categories then has a model structure with characterizable fibrant objects. We use a universal property of cubical sets, as a monoidal model category, to show that the invertibility hypothesis is consequence of the other axioms.