Elements of ∞-Category Theory

Abstract

The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ∞-categories from first principles in a model-independent fashion using the axiomatic framework of an ∞-cosmos, the universe in which ∞-categories live as objects. An ∞-cosmos is a fertile setting for the formal category theory of ∞-categories, and in this way the foundational proofs in ∞-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.

Topological coHochschild Homology and the Homology of Free Loop Spaces

“…In [26, Section 2.4.3], Lurie defines an ∞-operad C ∐ on an arbitrary ∞-category C and shows this is a symmetric monoidal ∞-category structure on C if and only if C admits finite coproducts. In the case of the under category CAlg(Spec op ) C op / , coproducts are given by pushouts in CAlg(Spec op ), which in turn are pullbacks in CoCAlg(Spec) by [31,Proposition 12.1.7]. Since CoCAlg(Spec) is complete, the necessary pullbacks exist.…”

“…This extends to a split augmented simplicial object ∆ op ⊥ → Set with "extra degeneracies." Here ∆ ⊥ is the category obtained from ∆ by adding an "extra degeneracy" [31,Example B.5.2] and the fact that ∆ 1…”

“…. By [31,Proposition 2.3.15], any cosimplicial object that admits a splitting and coaugmentation has a limit given by the coaugmentation-that is, by evaluation at [−1] ∈ ∆ ⊥ . In this case, by construction the coaugmentation is simply the 1-fold product of C, meaning the limit is C itself.…”

“…Since limits commute, as follows from [31,Lemma 2.4.1], it is equivalent to calculate the pullback of the diagram of totalizations in C:…”

Accessible aspects of 2-category theory

“…In parallel work with Lack and Vokřínek [12] we move in this direction, showing that a variety of simplicially enriched categories of (∞, 1)-categories with structure, and the pseudomaps between them, are accessible with flexible limits. This fits directly into the program of Riehl and Verity [43,44,45] on ∞-cosmoi, which uses classical enriched category theory [20] as a means to develop the theory of ∞-categories. Furthermore in [12] we develop weak adjoint functors theorems and results on the existence of weak colimits in the setting of categories enriched in a monoidal model category, with new applications both in the present 2-categorical and the ∞-categorical settings.…”