nLab
finite (infinity,1)-limit
Finite -limits
Context
$(\infty,1)$ -Category theory
(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

Limits and colimits
limits and colimits

1-Categorical
limit and colimit

limits and colimits by example

commutativity of limits and colimits

small limit

filtered colimit

sifted colimit

connected limit , wide pullback

preserved limit , reflected limit , created limit

product , fiber product , base change , coproduct , pullback , pushout , cobase change , equalizer , coequalizer , join , meet , terminal object , initial object , direct product , direct sum

finite limit

Kan extension

weighted limit

end and coend

2-Categorical
(∞,1)-Categorical
Model-categorical
Finite $(\infty,1)$ -limits
Definition
A finite $(\infty,1)$ -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.

Properties
Preservation of finite $(\infty,1)$ -limits
This appears as (Lurie, cor. 4.4.2.5 ).

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

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

This appears as (Lurie, prop. 6.1.5.2 ).

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

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

Related pages
References
Relation to homotopy type theory :

Last revised on April 12, 2021 at 10:03:37.
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