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.

EE-equivalences of internal categories

The 2-category Cat(S)Cat(S) (Gpd(S)Gpd(S)) of categories (groupoids) internal to SS has a automatic definition of fully faithful internal functor f:XYf:X\to Y. 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 EE in SS that satisfy certain properties.

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

Examples

The easiest example is when EE 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)