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

### Context

#### $\left(\infty ,1\right)$-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 $\left(\infty ,1\right)$-category of $\kappa -$pro-objects in $C$ is the opposite (∞,1)-category of ind-objects in the opposite of $C$:

${\mathrm{Pro}}_{\kappa }\left(C\right):=\left({\mathrm{Ind}}_{\kappa }\left({C}^{\mathrm{op}}\right){\right)}^{\mathrm{op}}\phantom{\rule{thinmathspace}{0ex}}.$Pro_\kappa(C) := (Ind_\kappa(C^{op}))^{op} \,.

For $\kappa =\omega$ we write just $\mathrm{Pro}\left(C\right)$.

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

$\cdots \simeq {\mathrm{Lex}}_{\kappa }\left(C,\infty \mathrm{Grpd}{\right)}^{\mathrm{op}}\subset \mathrm{Func}\left(C,\infty \mathrm{Grpd}{\right)}^{\mathrm{op}}$\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 $\left(\infty ,1\right)$-category with finite limits, we write

$\mathrm{Pro}\left(C\right):=\mathrm{AccLex}\left(C,\infty \mathrm{Grpd}{\right)}^{\mathrm{op}}\phantom{\rule{thinmathspace}{0ex}}.$Pro(C) := AccLex(C,\infty Grpd)^{op} \,.

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

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