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1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeSep 29th 2011
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   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)
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   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
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   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
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   Author: Tim_Porter
   Format: MarkdownItexThanks both.

   Thanks both.