If $J$ is a pretopology on a category with pullbacks, a *$J$-equivalence* $f:X \to Y$ between categories internal to $S$ is a functor that is fully faithful? and essentially $J$-surjective. This last means that the map

$X_0 \times_{f,Y_0,s} Y_1 \to Y_1 \stackrel{t}{\to} Y_0$

admits local sections with respect to $J$.

When no reference to a particular pretopology is mentioned, such maps will be called *weak eqivalences*

Created on March 31, 2009 at 23:28:01. See the history of this page for a list of all contributions to it.