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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 definitions 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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum 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.
   • CommentAuthorUrs
   • CommentTimeJul 25th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI added the sentence > The factorizing morphism $c \to im(f)$ is sometimes called the **corestriction** of $f$: to [[image]] and made [[corestriction]] redirect to this page.

   I added the sentence

   The factorizing morphism $c \to im(f)$ is sometimes called the corestriction of $f$:

   to image and made corestriction redirect to this page.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeJul 25th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI was convinced that I have once written an entry about corestrictions. The case above is just the most important but special case. Like a morphism can be restricted to any subobject, and a function to any subset of a domain, the function can be also corestricted to any set between the range and the original codomain; similarly a morphism can be corestricted to any subobject of the codomain which contains the image; and the [[corestriction]] to the image is a special case. I am not sure if the best place for all this in full generality is eventually at [[image]] or maybe a separate entry will be needed. There is also a notion of corestriction in representation theory/Hopf algebras.

   I was convinced that I have once written an entry about corestrictions.

   The case above is just the most important but special case. Like a morphism can be restricted to any subobject, and a function to any subset of a domain, the function can be also corestricted to any set between the range and the original codomain; similarly a morphism can be corestricted to any subobject of the codomain which contains the image; and the corestriction to the image is a special case. I am not sure if the best place for all this in full generality is eventually at image or maybe a separate entry will be needed. There is also a notion of corestriction in representation theory/Hopf algebras.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 25th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexGood point. I was wondering about creating a separate entry for it. Maybe we should. But I don't have time for this right now. If you have a minute, please split it off!

   Good point. I was wondering about creating a separate entry for it. Maybe we should. But I don’t have time for this right now. If you have a minute, please split it off!

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeJul 26th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexLet's wait with this for some other day :) Namely, I got for hours absorbed in creating lots of new [[algebraic number theory]] entries. It will get interesting at some point, once I get to the role of fundamental group! Namely an analogue of the fact that the fundamental group which counts homotopy classes of based mappings from circle to a space is isomorphic to the group of deck transformation of the universal covering at the level of the maximal abelian quotient in the case of arithmetic geometry amounts to central facts of [[class field theory]]. In the function field case those are on the other hand analogous to the Takahashi-Ward identities for QFT on the algebraic curve, as shown by [[Lev Tahtajan]]. Setting off to bed now...

   Let’s wait with this for some other day :)

   Namely, I got for hours absorbed in creating lots of new algebraic number theory entries. It will get interesting at some point, once I get to the role of fundamental group! Namely an analogue of the fact that the fundamental group which counts homotopy classes of based mappings from circle to a space is isomorphic to the group of deck transformation of the universal covering at the level of the maximal abelian quotient in the case of arithmetic geometry amounts to central facts of class field theory. In the function field case those are on the other hand analogous to the Takahashi-Ward identities for QFT on the algebraic curve, as shown by Lev Tahtajan.

   Setting off to bed now…