David Roberts
class of admissible maps

Admissible maps are analogues of surjective functions of sets, used to define ‘essential surjectivity’ in a general ambient category for internal functors

Definition

Let SS be a category with a singleton pretopology JJ (i.e. a site). A class of maps EE is called admissible for JJ if it satisfies the following properties

  1. EE is a singleton pretopology in which JJ is cofinal
  2. EE contains all the split epimorphisms in SS
  3. If the composite xyzx \to y \to z is in EE and xyx\to y is a split epimorphism, then yzy\to z is in EE.

Example The prototype is the class of JJ-epimorphisms for a pretopology JJ on a category with pullbacks. This class is admissible for JJ.

Example If EE is a saturated singleton pretopology, then it is admissible for itself.

If we drop reference to JJ in the above definition, EE is called a class of admissible maps.

Last revised on April 2, 2009 at 01:36:38. See the history of this page for a list of all contributions to it.