David Roberts anafunctor

This is my own notational preference for how to describe anafunctors. For others see the nLab.

Constructions

Let XX be a category internal to a site (S,J)(S,J) where JJ is a subcanonical singleton pretopology. Let UX 0U \to X_0 be a cover, and X[U]X[U] the category with objects UU and arrows U×U× X 0×X 0X 1U\times U \times_{X_0 \times X_0} X_1. Call this the induced category. There is a canonical functor

X[U]X X[U] \to X

which is a JJ-equivalence.

Definition

An anafunctor in (S,J)(S,J) from XX to YY, both categories internal to SS, is a cover UX 0U\to X_0 and a functor f:X[U]Yf:X[U] \to Y. Denote it (U,f):XY(U,f):X \to Y.

Ordinary functors g:XYg:X \to Y can be considered as anafunctors, with the identity map of X 0X_0 as the cover. Denote these by (X 0,g)(X_0,g). There is a composition of anafunctors, which is composition of the spans that they define. This requires a little lemma to say that the pullback category so defined is (isomorphic to one) of the required form

There are also transformations between anafunctors, which are defined in a manner entirely analogous to coboundaries between Čech cocycles (which are, of course, examples of said transformations).

Definition

..of transformation goes here.


Here is a result that helps to show that anafunctors compute the localisation of Gpd(S)(S) at the JJ-equivalences.

Let (S,J)(S,J) be a site with a subcanonical singleton pretopology, and f:XYf:X \to Y a JJ-equivalence. Then there is an anafunctor (U,f¯):YX(U,\bar{f}):Y \to X and isotransformations

(U,f¯)(X 0,f)id X(X 0,f)(U,f¯)id Y. (U,\bar{f})\circ(X_0,f) \Rightarrow id_X \quad (X_0,f) \circ (U,\bar{f}) \Rightarrow id_Y.

That is, ff has an anafunctor pseudoinverse.

Created on February 27, 2009 at 02:59:45. See the history of this page for a list of all contributions to it.