# Idea

The IPC-property (“inductive limit product commutation property”) is a technical condition on a category $A$ which ensures that presheaves with values in $A$ have a good notion of sheafification.

# Definition

A category $A$ is said to satisfy the IPC-property if

• it admits small products and filtered colimits;

• for

• every family $\left\{{I}_{s}{\right\}}_{s\in S}$ of small filtered categories

• and for any family $\left\{{\alpha }_{s}:{I}_{s}\to A\right\}$ of functors

• indexed by a small set $S$

the canonical morphism

$\mathrm{colim}\left(\prod _{s}{I}_{s}\stackrel{k↦\prod _{s}{\alpha }_{s}\left({\pi }_{s}\left(j\right)\right)}{\to }A\right)\to \prod _{s}\mathrm{colim}{\alpha }_{s}$colim \left( \prod_s I_s \stackrel{k \mapsto \prod_s \alpha_s(\pi_s(j))}{\to} A \right) \to \prod_s colim \alpha_s

is an isomorphism.

# References

The IPC property is definition 3.1.10 in

It is invoked for sheafification in section 17.4 there.

Created on March 31, 2009 19:06:54 by Urs Schreiber (134.100.222.156)