Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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?