We show that certain diagrams of $\infty$-logoses are reconstructed in homotopy type theory extended with some lex, accessible modalities, which enables us to use plain homotopy type theory to reason about not only a single $\infty$-logos but also a diagram of $\infty$-logoses. This also provides a higher dimensional version of Sterling's synthetic Tait computability -- a type theory for higher dimensional logical relations.
We define a notion of homotopy Segal cooperad in the category of $ E_\infty $-algebras. This model of Segal cooperad that we define in the paper, which we call homotopy Segal $ E_\infty $-Hopf cooperad, covers examples given by the cochain complex of topological operads and provides a framework for the study of the homotopy of such objects. In a first step, we consider a category of Segal $ E_\infty $-Hopf cooperads, which consists of collections of $ E_\infty $-algebras indexed by trees and equipped with coproduct operators, corresponding to tree morphisms, together with facet operators, corr…
In this paper, we apply an explicit construction of a simplicial powering in dg-categories, due to Holstein (2016) and Arkhipov and Poliakova (2018), as well as our own results on homotopy ends (Arkhipov and Ørsted 2018), to obtain an explicit model for the homotopy limit of a cosimplicial system of dg-categories. We apply this to obtain a model for homotopy descent in terms of $\mathrm{A}_{\infty}$-comodules, proving a conjecture by Block, Holstein, and Wei (2017) in the process.
We compare two models for $\infty$-operads: the complete Segal operads of Barwick and the complete dendroidal Segal spaces of Cisinski and Moerdijk. Combining this with comparison results already in the literature, this implies that all known models for $\infty$-operads are equivalent - for instance, it follows that the homotopy theory of Lurie's $\infty$-operads is equivalent to that of dendroidal sets and that of simplicial operads.
The present article is devoted to the study of transfers for $A_\infty$ structures, their maps and homotopies, as developed in \cite{Markl06}. In particular, we supply the proofs of claims formulated therein and provide their extension by comparing them with the former approach based on the homological perturbation lemma.
We prove the existence of Morita model structures on the categories of small simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of $(\infty,1)$-categories and $\infty$-operads. We give a characterization of the weak equivalences in terms of simplicial presheaves, simplicial algebras and slice categories. In the case of the Morita model structure for simplicial categories and simplicial operads, we also show that each of these model structures can be obtained as an explicit left Bousfield localization of the Bergner model structu…
In previous work, we used an $\infty$-categorical version of ultraproducts to show that, for a fixed height $n$, the symmetric monoidal $\infty$-categories of $E_{n,p}$-local spectra are asymptotically algebraic in the prime $p$. In this paper, we prove the analogous result for the symmetric monoidal $\infty$-categories of $K_{p}(n)$-local spectra, where $K_{p}(n)$ is Morava $K$-theory at height $n$ and the prime $p$. This requires $\infty$-categorical tools suitable for working with compactly generated symmetric monoidal $\infty$-categories with non-compact unit. The equivalences that we prod…
We study extensively the homotopy theory of coalgebras. By coalgebras, we mean the full theory of coalgebras: with counits and not necessarily locally conilpotent. For example $\mathcal E_\infty$-coalgebras, $\mathcal A_\infty$-coalgebras, $\mathcal L_\infty$-coalgebras etc. To do so, we define the category of complete curved algebras -- where the notion of quasi-isomorphims does not make sense -- and endow it with a model category structure, equivalent to that of the category of coalgebras.
A dg-natural transformation between dg-functors is called an objectwise homotopy equivalence if its induced morphism on each object admits a homotopy inverse. In general an objectwise homotopy equivalence does not have a dg-inverse but has an $A_{\infty}$ quasi-inverse. In this note we give a recurrent formula of the $A_{\infty}$ quasi-inverse. This result is useful in studying the compositions of dg-lifts of derived functors of schemes.
We study the left adjoint $\mathbb{D}$ to the forgetful functor from the $\infty$-category of symmetric monoidal $\infty$-categories with duals and finite colimits to the $\infty$-category of symmetric monoidal $\infty$-categories with finite colimits, and related free constructions. The main result is that $\mathbb{D} \mathcal C$ always splits as the product of 3 factors, each characterized by a certain universal property. As an application, we show that, for any compact Lie group $G$, the $\infty$-category of genuine $G$-spectra is obtained from the $\infty$-category of Bredon (\emph{a.k.a} …
Stockholm University（斯德哥尔摩大学）；Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG（巴黎西岱大学和索邦大学，法国国家科学研究中心，IMJ-PRG）；Université d’Angers, LAREMA（昂热大学，LAREMA）
We establish a large class of homotopy coherent Morita-equivalences of Dold-Kan type relating diagrams with values in any weakly idempotent complete additive $\infty$-category; the guiding example is an $\infty$-categorical Dold-Kan correspondence between the $\infty$-categories of simplicial objects and connective coherent chain complexes. Our results generalize many known 1-categorical equivalences such as the classical Dold-Kan correspondence, Pirashvili's Dold-Kan type theorem for abelian $Γ$-groups and, more generally, the combinatorial categorical equivalences of Lack and Street.
Lie $\infty$-groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie $\infty$-groupoids called `Lie $\infty$-groups' by integrating finite type Lie $n$-algebras. In order to study the compatibility between this integration procedure and the homotopy theory of Lie $n$-algebras introduced in the companion paper arXiv:1809.05999, we present a homotopy theory for Lie $\infty$-groupoids. Unlike Kan simplicial sets and the higher geometric groupoids of Behrend and Getzler, Lie $\infty$-group…
Associated to each small category $C$, there is a category of $C$-shaped diagrams of simplicial sets and an $\infty$-category of $NC$-shaped homotopy coherent diagrams of spaces. We present a functor which exhibits the latter as the $\infty$-categorical localisation of the former at the objectwise weak homotopy equivalences. This builds on a Quillen equivalence between the projective and covariant model structures associated to $C$ due to Heuts-Moerdijk, as well as Cisinski's theory of $\infty$-categorical localisations. We use the localisation functor to give simplified proofs that the left (…
For every $\infty$-category $\mathscr{C}$, there is a homotopy $n$-category $\mathrm{h}_n \mathscr{C}$ and a canonical functor $γ_n \colon \mathscr{C} \to \mathrm{h}_n \mathscr{C}$. We study these higher homotopy categories, especially in connection with the existence and preservation of (co)limits, by introducing a higher categorical notion of weak colimit. Using homotopy $n$-categories, we introduce the notion of an $n$-derivator and study the main examples arising from $\infty$-categories. Following the work of Maltsiniotis and Garkusha, we define $K$-theory for $\infty$-derivators and prov…
We axiomatise the theory of $(\infty,n)$-categories. We prove that the space of theories of $(\infty,n)$-categories is a $B(\mathbb{Z}/2)^n$. We prove that Rezk's complete Segal $Θ_n$-spaces, Simpson and Tamsamani's Segal $n$-categories, the first author's $n$-fold complete Segal spaces, Kan and the first author's $n$-relative categories, and complete Segal space objects in any model of $(\infty,n-1)$-categories all satisfy our axioms. Consequently, these theories are all equivalent in a manner that is unique up to the action of $(\mathbb{Z}/2)^n$.
This dissertation comprises three collections of results, all united by a common theme. The theme is the study of categories via algebraic techniques, considering categories themselves as algebraic objects. This algebraic approach to category theory is central to noncommutative algebraic geometry, as realized by recent advances in the study of noncommutative motives. We have success proving algebraic results in the general setting of symmetric monoidal and semiring $\infty$-categories, which categorify abelian groups and rings, respectively. For example, we prove that modules over the semiring…
Many homotopy-coherent algebraic structures can be described by Segal-type limit conditions determined by an "algebraic pattern", bywhich we mean an $\infty$-category equipped with a factorization system and a collection of "elementary" objects. Examples of structures that occur as such "Segal $\mathcal{O}$-spaces" for an algebraic pattern $\mathcal{O}$ include $\infty$-categories, $(\infty,n)$-categories, $\infty$-operads, $\infty$-properads, and algebras for an $\infty$-operad in spaces. In the first part of this paper we set up a general frameworkn for algebraic patterns and their Segal obj…
We prove the conjecture that any Grothendieck $(\infty,1)$-topos can be presented by a Quillen model category that interprets homotopy type theory with strict univalent universes. Thus, homotopy type theory can be used as a formal language for reasoning internally to $(\infty,1)$-toposes, just as higher-order logic is used for 1-toposes. As part of the proof, we give a new, more explicit, characterization of the fibrations in injective model structures on presheaf categories. In particular, we show that they generalize the coflexible algebras of 2-monad theory.
In this paper we prove that various quasi-categories whose objects are $\infty$-categories in a very general sense are complete: admitting limits indexed by all simplicial sets. This result and others of a similar flavor follow from a general theorem in which we characterize the data that is required to define a limit cone in a quasi-category constructed as a homotopy coherent nerve. Since all quasi-categories arise this way up to equivalence, this analysis covers the general case. Namely, we show that quasi-categorical limit cones may be modeled at the point-set level by pseudo homotopy limit…
We prove general adjoint functor theorems for weakly (co)complete $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of (co)complete $\infty$-categories, so these $n$-categories do not admit all small (co)limits in general. We also introduce Brown representability for (homotopy) $n$-categories and prove a Brown representability theorem for localizations of compactly generated $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of presentable $\infty$-categories if $n \geq 2$, and the homotopy $n$-categories of presentable stable $\in…
This is an introduction to the study of abstract homotopy theory by means of model categories and $(\infty,1)$-categories. The only prerequisites are very basic general topology and abstract algebra. None categorical background is needed. The final objective is to show that classical homotopy theory for topological spaces can be more naturally understood in terms of categorical language.
