Not signed in (Sign In)

Not signed in

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

  • Sign in using OpenID

Site Tag Cloud

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

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 31st 2010

    Added a bit to skeleton about skeletons of internal categories

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeSep 2nd 2010
    • (edited Sep 2nd 2010)

    Entry skeleton says: “If the axiom of choice holds, then every category has a skeleton: simply choose one object in each isomorphism class.”

    When the construction is simple, the proof is a tiny bit more subtle, namely one has to show that the inclusion in:sk(C)Cin:sk(C)\to C is really an equivalence. For the weak inverse one forms a functor :xx-':x\mapsto x' as follows. For every object xx one has chosen already the unique object xx' in sk(X)sk(X) isomorphic to xx, but one also needs to make a choice of isomorphism i x:xxi_x:x\to x' for every xx. This enables to conjugate between C(x,y)C(x,y) and C(x,y)C(x',y') by

    (xfy)(xi x 1xfyi yy). (x\stackrel{f}\to y)\mapsto (x'\stackrel{i_x^{-1}}\to x\stackrel{f}\to y\stackrel{i_y}\to y').

    This correspondence makes -' a functor, that is the rule for morphisms f:=i yfi x 1f' := i_y\circ f\circ i_{x}^{-1} is functorial. Let us show that -' is a weak inverse of inin. In one direction, (in y)=y(in_{y})' = y for ysk(C)y\in sk(C); in another direction notice that i in x 1:in xxi^{-1}_{in_{x'}}:in_{x'}\cong x for xCx\in C is an isomorphism. It suffices to show that these isomorphisms form a natural isomorphism i in 1:in id Ci^{-1}_{in}:in_{-'}\to id_C; the naturality diagram is commutative precisely because of the conjugation formula for the functor -' for morphisms.

    Right ?

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 2nd 2010
    Quite right, Zoran. Well spotted.
    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeSep 2nd 2010
    • (edited Sep 2nd 2010)

    Right, it should say “simply choose one object in each isomorphism class and one isomorphism to that object from each other object in that class” (or something like that).

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeSep 2nd 2010
    OK, I will go to incorporate the remark into the entry, you can feel free to improve the notation etc. Check it.
    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeSep 2nd 2010

    OK, I did.