equivalences in/of $(\infty,1)$-categories
An (∞,1)-category is sifted if a quasi-category $K \in sSet$ modelling it has the property that
it is not empty, $K \neq \emptyset$
The diagonal $K \to K \times K$ (in sSet) models a cofinal (∞,1)-functor.
Let $C$ be an (∞,1)-category such that products preserve sifted (∞,1)-colimits (for instance an (∞,1)-topos, see universal colimits).
Then sifted (∞,1)-colimits preserve finite products.
This is (Lurie, lemma 5.5.8.11).
The opposite category $\Delta^{op}$ of the simplex category is a sifted $(\infty,1)$-category.
Every filtered (∞,1)-category is sifted.
This appears as (Lurie, prop. 5.3.1.20).
In a category of commutative monoids in a symmetric monoidal $(\infty,1)$-category, sifted colimits are computed as sifted colimits in the underlying $(\infty,1)$-category.
See commutative monoid in a symmetric monoidal (∞,1)-category for details.
sifted category, sifted $(\infty,1)$-category
Section 5.5.8 of
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