nLab regular (infinity,1)-category

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Idea

A regular $(\infty,1)$ -category is the analog of a regular category for (∞,1)-category theory.

Let $\mathcal{C}$ be an (∞,1)-category. This is called an exact $(\infty,1)$ -category if

$\mathcal{C}$ has a terminal object and homotopy fiber products;
$\mathcal{C}$ admits a factorization system $(S_L,S_R)$ , where $S_L$ is the collection of regular n-connected morphisms and $S_R$ is the collection of n-truncated morphisms.

An (∞,1)-category $C$ is regular if it admits finite (∞,1)-limits, every morphism in $C$ has an image, i.e., can be written as a composition of an (∞,1)-quotient morphism and an (∞,1)-monomorphism, and the collection of (∞,1)-quotient morphisms is closed under base changes.

Last revised on December 14, 2023 at 20:46:17. See the history of this page for a list of all contributions to it.