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 Cat(S) (Gpd(S)) of categories (groupoids) internal to S has a automatic definition of fully faithful internal functor f:XY. Namely, that

X 1 Y 1 X 0×X 0 Y 0×Y 0\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

X 0× Y 0Y 1 Y 1 t Y 0 s X 0 f Y 0 \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…)

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