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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeAug 16th 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexIn response to <a href="http://mathoverflow.net/questions/10246/model-category-structure-on-set-without-axiom-of-choice/10299#10299">this answer</a> to a question at MO I wrote: >This is very late, but I hope it gives some insight. One could cook up anafunctors without pseudoinverses, if instead of all surjections one uses a Grothedieck pretopology J on Set which has as covers surjections from some restricted class. I'm thinking of the example of surjections with finite fibres. If p:X->Y is a surjection with infinite fibres with no projective cover P->Y in J for which p admits a section, the anafunctor Y<-C(X) = C(X) has no pseudoinverse. Here C(X) is the groupoid with object set X and arrow set X\times_Y X, with the obvious structure. Now this is just a application of the contrapositive of theorem 1.83 in [[Fundamental Bigroupoids and 2-Covering Spaces|my thesis]], but it is perhaps a bit contrived. However, I vaguely feel that thinking of surjections with finite fibres as different to surjections with infinite fibres may have arisen elsewhere (if only in some form of finitist ideas). However, the conjunction of denying enough projective covers in $Set$ and only wanting to localise weak equivalences of categories $C \to D$ where $Obj(C) \times_{Obj(D)} Mor(D) \to Obj(D)$ has finite fibres seems very strong. Any thoughts?

   In response to this answer to a question at MO I wrote:

   This is very late, but I hope it gives some insight. One could cook up anafunctors without pseudoinverses, if instead of all surjections one uses a Grothedieck pretopology J on Set which has as covers surjections from some restricted class. I’m thinking of the example of surjections with finite fibres. If p:X->Y is a surjection with infinite fibres with no projective cover P->Y in J for which p admits a section, the anafunctor Y<-C(X) = C(X) has no pseudoinverse. Here C(X) is the groupoid with object set X and arrow set X\times_Y X, with the obvious structure.

   Now this is just a application of the contrapositive of theorem 1.83 in my thesis, but it is perhaps a bit contrived. However, I vaguely feel that thinking of surjections with finite fibres as different to surjections with infinite fibres may have arisen elsewhere (if only in some form of finitist ideas).

   However, the conjunction of denying enough projective covers in $Set$ and only wanting to localise weak equivalences of categories $C \to D$ where $Obj(C) \times_{Obj(D)} Mor(D) \to Obj(D)$ has finite fibres seems very strong.

   Any thoughts?