nLab infinity-cosmos

Idea

An $\infty$-cosmos is a “good place in which to do higher category theory,” for 1-categories, (∞,1)-categories , (∞,n)-categories, or fibered category-versions of any of the above.

The word is chosen by analogy with the notion of cosmos in enriched category theory, which is similarly “a good place to do (ordinary) category theory.” The notion is more similar to Street’s “fibrational cosmoi” than to Bénabou’s cosmoi.

A source of examples

Roughly speaking, an $\infty$-cosmos is a simplicially enriched category of fibrations and fibrant objects.

Let $\mathcal{M}$ be a model category that is enriched over the Joyal model structure on simplicial sets. Then the full subcategory of fibrant objects defines an $\infty$-cosmos. If, as is frequently the case in practice, every fibrant object is cofibrant, then this subcategory defines an $\infty$-cosmos with all objects cofibrant, which admits a simpler axiomatization.

Definition

For simplicity, we define only an $\infty$-cosmos with all objects cofibrant. See the references below for the general definition.

An $\infty$-cosmos (with all objects cofibrant) is a simplicially enriched category $\mathcal{K}$ whose homs $Fun(A,B)$ are quasi-categories equipped with a specified class of isofibrations so that

1. As a simplicially enriched category, $\mathcal{K}$ admits products, cotensors with simplicial sets, pullbacks of isofibrations, splittings of idempotents, and limits of towers of isofibrations.

2. The class of isofibrations is closed under product, pullback, retract, limits of towers, and Leibniz cotensors with monomorphisms of simplicial sets. Furthermore, if $A \twoheadrightarrow B$ is a fibration in $\mathcal{K}$, then for any $X$, $\Fun(X,A) \twoheadrightarrow \Fun(X,B)$ is an isofibration of quasi-categories.

A map $A \to B$ is an equivalence if for any $X$, $\Fun(X,A) \rightarrow \Fun(X,B)$ is an equivalence of quasi-categories, and a trivial fibration if it is both an isofibration and an equivalence.

1. For any trivial fibration $p\colon E \twoheadrightarrow B$ and any map $f \colon A \to B$ there exists a lift of $f$ along $p$.

The prototypical example

The category of quasi-categories and isofibrations, inner fibrations that lift also against the inclusion of either endpoint into the nerve of the walking isomorphism, defines an $\infty$-cosmos.

Related concepts

• homotopy 2-category of (∞,1)-categories

References

For an overview:

• Emily Riehl, Dominic Verity, $\infty$-Category theory from scratch, (arXiv:1608.05314)

For the most general notion of $\infty$-cosmos and constructions of many examples:

• Emily Riehl, Dominic Verity, Fibrations and Yoneda’s lemma in an $\infty$-comos (arXiv:1506.05500)

For a simplified notion with “all objects cofibrant”

• Emily Riehl, Dominic Verity, Kan extensions and the calculus of modules for $\infty$-categories (arXiv:1507.01460)

Textbook account:

• Emily Riehl, Dominic Verity, Elements of ∞-Category Theory, Cambridge studies in advanced mathematics 194, Cambridge University Press (2022) $[$doi:10.1017/9781108936880, ISBN:978-1-108-83798-9, pdf$]$

For an exploration of atypical examples of ∞-cosmoi whose objects are 2-categories or bicategories

• Emily Riehl, Mira Wattal, On ∞-cosmoi of bicategories (arXiv:2108.11786)