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.
The 2-category () of categories (groupoids) internal to has a automatic definition of fully faithful internal functor . Namely, that
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 in that satisfy certain properties.
First, we define an internal functor to be essentially -surjective if the composite arrow along the top of
is in (This definition is due to Everaert-Kieboom-van der Linden). If we let be a class of admissible maps, then functors which are fully faithful and essentially -surjective are called -equivalences, or simply weak equivalences when mention of is suppressed.
The easiest example is when is the class of maps admitting local sections for some Grothendieck pretopology.