David Roberts
weak equivalence

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

$$\begin{array}{ccc}{X}_1& \to & Y_1\\ \downarrow & & \downarrow \\ X_0\times X_0& \to & Y_0\times Y_0\end{array}$$\begin{matrix} X_1 & \to & Y_1 \\ \downarrow && \downarrow\\ X_0\times X_0 & \to & Y_0\times Y_0 \end{matrix}

is a pullback.

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

$$\begin{array}{ccccc}{X}_0{\times }_{Y_0}{Y}_1& \to & Y_1& \stackrel{t}{\to }& Y_0\\ \downarrow & & \downarrow s& & \\ X_0& \underset{f}{\to }& Y_0& & \end{array}$$\begin{matrix} X_0\times_{Y_0} Y_1 & \to & Y_1 & \stackrel{t}{\to} & Y_0\\ \downarrow && \downarrow s &&\\ X_0 & \underset{f}{\to} & Y_0 && \end{matrix}

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…)