equivalences in/of $(\infty,1)$-categories
This is the analog of a filtered category in the context of (∞,1)-categories.
The main purpose of considering filtered (∞,1)-categories is to define filtered (∞,1)-colimits, which are the colimits that commute with finite (∞,1)-limits.
Let $\kappa$ be a regular cardinal, and let $C\in \sSet$ be an (∞,1)-category, incarnated as a quasicategory.
$C$ is called $\kappa$-filtered if for all $\kappa$-small $K\in\sSet$ and every morphism $f\colon K\to C$ there is a morphism $\hat p\colon \rcone(K)\to C$ extending $f$, where $\rcone(K)$ denotes the (right) cone of the simplicial set $K$. $C$ is called filtered if it is $\omega$-filtered.
An (∞,1)-category $K$ is filtered precisely if (∞,1)-colimits of shape $K$ in ∞Grpd commute with all finite (∞,1)-limits, hence if
is a left exact (∞,1)-functor.
This is HTT, prop. 5.3.3.3.
A filtered $(\infty,1)$-category is in particular a sifted (∞,1)-category.
This appears as (Lurie, prop. 5.3.1.20). Since sifted (∞,1)-colimits are precisely those that commute with finite products, this is a direct reflection of the fact that finite products are a special kind of finite (∞,1)-limits.
For $C$ a filtered $(\infty,1)$-category, the diagonal (∞,1)-functor $\Delta : C \to C \times C$ if a cofinal (∞,1)-functor.
A filtered $(\infty,1)$-category is weakly contractible, i.e. when incarnated as a quasicategory, it is weakly equivalent to a point in the Kan-Quillen model structure on simplicial sets.
This is (Lurie, Lemma 5.3.1.18).
sifted category, sifted colimit, sifted (∞,1)-category, sifted (∞,1)-colimit
directed set, filtered category, filtered (∞,1)-category
Section 5.3.1 of
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