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 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 sheaves 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.
   • CommentAuthorTim_Porter
   • CommentTimeSep 29th 2011
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexAs an example of working with simplicial automorphisms, I was just wading through a proof that if $K$ is a constant simplicial set, then its automorphism simplicial group must be a constant simplicial group. At several stages what I wrote was looking like sleight of hand. There must be a simple proof using the fact that $K$ is a constant presheaf. Does anyone know of one? (The fact that $K$ is constant bugs me as I have to use confusing notation!!!) This raised the question is something like this true in more generality, say in any presheaf topos, and if so how to prove it simply. Any ideas. And references eith for the simplicial case or for the general? My reason was to have a nice example of working with simplicial automorphism groups so as to look at the automorphism group of a groupoid simplicially (amongst other things). It is for the sake of the exposition.

   As an example of working with simplicial automorphisms, I was just wading through a proof that if $K$ is a constant simplicial set, then its automorphism simplicial group must be a constant simplicial group. At several stages what I wrote was looking like sleight of hand. There must be a simple proof using the fact that $K$ is a constant presheaf. Does anyone know of one? (The fact that $K$ is constant bugs me as I have to use confusing notation!!!)

   This raised the question is something like this true in more generality, say in any presheaf topos, and if so how to prove it simply. Any ideas. And references eith for the simplicial case or for the general?

   My reason was to have a nice example of working with simplicial automorphism groups so as to look at the automorphism group of a groupoid simplicially (amongst other things). It is for the sake of the exposition.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 29th 2011
   • (edited Sep 29th 2011)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI think the fact that the internal endomorphism monoid of a constant object is constant can be boiled down to the fact that the functor $\Delta: Set \to Set^{C^{op}}$ preserves both small coproducts and small products (it has a left adjoint as well as a right adjoint). For the internal hom, we have $$\Delta X^{\Delta X} \cong \Delta X^{\sum_X 1} \cong \prod_X \Delta X \cong \Delta(\prod_X X) \cong \Delta (X^X)$$ That the internal automorphism group is constant follows similar reasoning, since it will involve finite limit constructions also preserved by $\Delta$.

   I think the fact that the internal endomorphism monoid of a constant object is constant can be boiled down to the fact that the functor $\Delta: Set \to Set^{C^{op}}$ preserves both small coproducts and small products (it has a left adjoint as well as a right adjoint). For the internal hom, we have

   $$\Delta X^{\Delta X} \cong \Delta X^{\sum_X 1} \cong \prod_X \Delta X \cong \Delta(\prod_X X) \cong \Delta (X^X)$$

   That the internal automorphism group is constant follows similar reasoning, since it will involve finite limit constructions also preserved by $\Delta$.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeOct 3rd 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexMore abstractly, the constancy of endo/auto-morphism objects of constant presheaves is a consequence of the [[locally connected topos|local connectivity]] of presheaf toposes, which is equivalent to cartesian-closedness of the constant-object functor $\Delta$. Not exactly related, but maybe worth pointing out that automorphism objects of constant sheaves, in a more general context, came up last year in [this discussion](http://golem.ph.utexas.edu/category/2010/11/locally_constant_sheaves.html) about the meaning of locally constancy.

   More abstractly, the constancy of endo/auto-morphism objects of constant presheaves is a consequence of the local connectivity of presheaf toposes, which is equivalent to cartesian-closedness of the constant-object functor $\Delta$.

   Not exactly related, but maybe worth pointing out that automorphism objects of constant sheaves, in a more general context, came up last year in this discussion about the meaning of locally constancy.

4. • CommentRowNumber4.
   • CommentAuthorTim_Porter
   • CommentTimeOct 4th 2011
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexThanks both.

   Thanks both.