nLab pro-object in an (infinity,1)-category

Context

$(\infty,1)$-Category theory

(∞,1)-category theory

Models

Limits and colimits

limits and colimits

Contents

Definition

For small (∞,1)-categories

For $C$ a small (∞,1)-category and $\kappa$ a regular cardinal, the $(\infty,1)$-category of $\kappa-$pro-objects in $C$ is the opposite (∞,1)-category of ind-objects in the opposite of $C$:

$Pro_\kappa(C) := (Ind_\kappa(C^{op}))^{op} \,.$

For $\kappa = \omega$ we write just $Pro(C)$.

By the properties listed there, if $C$ has all $\kappa$-small (∞,1)-limits then this is equivalent to

$\cdots \simeq Lex_\kappa(C, \infty Grpd)^{op} \subset Func(C,\infty Grpd)^{op}$

the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors on those that preserve these limits.

For large (∞,1)-categories

Generalizing this definition, if $C$ is a non-small $(\infty,1)$-category with finite limits, we write

$Pro(C) := AccLex(C,\infty Grpd)^{op} \,.$

for the category of left exact functors $C\to \infty Gprd$ which are moreover accessible. In other words, when $C$ is large, $Pro(C)$ consists only of those left-exact functors which are “small cofiltered limits of representables”.

• pro-object / pro-object in an $(\infty,1)$-category

References

The large version is mentioned around def. 7.1.6.1 of

Revised on December 16, 2012 15:46:32 by Urs Schreiber (71.195.68.239)