A weak equivalence is, generally speaking, a 1-arrow in some n-category that is morally an equivalence, but doesn’t necessarily have a map in the other direction that acts as a weak inverse. The prototypical example is weak homotopy equivalences of topological spaces. These are inverted in the homotopy category, but do not necessarily have even a homotopy inverse.

$E$-equivalences of internal categories

The 2-category $\mathrm{Cat}(S)$ ($\mathrm{Gpd}(S)$) of categories (groupoids) internal to $S$ has a automatic definition of fully faithful internal functor $f:X\to Y$. Namely, that

However, the definition of essential surjectivity is a little more difficult to define well. What is needed is a supplimentary class of arrows $E$ in $S$ that satisfy certain properties.

First, we define an internal functor to be essentially $E$-surjective if the composite arrow along the top of

is in $E$ (This definition is due to Everaert-Kieboom-van der Linden). If we let $E$ be a class of admissible maps, then functors which are fully faithful and essentially $E$-surjective are called $E$-equivalences, or simply weak equivalences when mention of $E$ is suppressed.

Examples

The easiest example is when $E$ is the class of maps admitting local sections for some Grothendieck pretopology.

(More examples…)

Revised on October 1, 2009 12:07:28
by David Roberts
(203.171.199.209)