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finite (infinity,1)-limit

Definition

A finite $(\mathrm{\infty },1)$-limit is an (∞,1)-limit over a finitely presented (∞,1)-category.

If we model our (∞,1)-categories by quasicategories, then this can be made precise by saying it is a limit over some simplicial set with finitely many nondegenerate simplices. Note that such a simplicial set is rarely itself a quasicategory; we regard it instead as a finite presentation of a quasicategory.

Properties

Preservation of finite $(\mathrm{\infty },1)$-limits

Proposition

An (∞,1)-functor $F:C\to D$ out of an (∞,1)-category $C$ that has all finite $(\mathrm{\infty },1)$-limits preserves these finite $(\mathrm{\infty },1)$-limits as soon as it preserves (∞,1)-pullbacks and the terminal object.

This appears as (Lurie, cor. 4.4.2.5).

Proposition

Let $C$ be a small (∞,1)-category with finite $(\mathrm{\infty },1)$-limits, and $H$ an (∞,1)-topos. Write $\mathrm{PSh}(C)$ for the (∞,1)-category of (∞,1)-presheaves on $C$.

If a functor $F:{\mathrm{PSh}}_{\mathrm{\infty }}(C)\to H$ preserves (∞,1)-colimits and finite $(\mathrm{\infty },1)$-limits of representables, then it preserves all finite $(\mathrm{\infty },1)$-limits.

This appears as (Lurie, prop. 6.1.5.2).

Examples

• Binary products, pullbacks, and terminal objects are all finite $(\mathrm{\infty },1)$-limits.

• Unlike the case in 1-category theory, the splitting of idempotents is not a finite $(\mathrm{\infty },1)$-limit.

Related pages

• finite limit
• (∞,1)-limit

References

• Jacob Lurie, Higher Topos Theory