nLab finite (infinity,1)-limit

Finite -limits


(,1)(\infty,1)-Category theory

Limits and colimits

Finite (,1)(\infty,1)-limits


A finite (,1)(\infty,1)-limit is an (∞,1)-limit over a finitely presented (∞,1)-category – a finite (∞,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.


Preservation of finite (,1)(\infty,1)-limits


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

This appears as (Lurie, cor.


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

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

This appears as (Lurie, prop.


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

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


Relation to homotopy type theory:

Last revised on July 6, 2023 at 23:16:19. See the history of this page for a list of all contributions to it.