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.
   • CommentAuthorHarry Gindi
   • CommentTimeFeb 9th 2011
   • (edited Feb 9th 2011)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexReposted from [MO](http://mathoverflow.net/questions/54866/the-pushout-of-a-mapping-simplex-along-products-is-a-mapping-simplex): Background ---- ---- The mapping simplex: We define a functor $M^n:Fun([n],sSet)\to sSet$ sending the functor $\phi:[n]\to sSet$ classifying the sequence $$A^0\leftarrow A^1\leftarrow\dots\leftarrow A^n$$ (notice that $A^k=\phi(n-k)$.) to the simplicial set $M^n(\phi)$ defined as a representable functor as follows: $$Hom_{sSet}(\Delta^k,M^n(\phi))=\coprod_{\{g:[k]\to[n]\}}Hom_{sSet}(\Delta^k,A^{g(k)})$$ The identity map $M^n(\phi)\to M^n(\phi)$ then determines a unique map $M^n(\phi)\to \Delta^n$ by the fact that simplicial sets are colimits of their simplices. Also, given a map $\Delta^m\to \Delta^n$ corresponding to a functor $F:[m]\to [n]$ (by full and faithfulness of the Yoneda embedding), it's not hard to see that $$M^n(\phi)\times_{\Delta^n} \Delta^m\cong M^m(F^*\phi)$$ Let $\phi:[n]\to sSet$ be a functor classifying a sequence of composable maps $$A^0\leftarrow A^1\leftarrow\dots\leftarrow A^n$$ Then let $c_{A^n}$ be the constant functor $[n]\to sSet$ at $A^n$. Then by the functoriality of $M^n$, we have a map $M^n(c_{A^n})\to M^n(\phi)$ induced by the obvious natural transformation $\alpha:c_{A^n}\to \phi$, which is defined componentwise as the composite map $\alpha_i:A^n\to A^i$ for each $i$ (the naturality of this map is immediate). Notice that $M^n(c_A)$ for any simplicial set $A$ is canonically isomorphic to the product $A\times \Delta^n$ since $$Hom(\Delta^m,A\times \Delta^n)\cong Hom(\Delta^m,A)\times Hom(\Delta^m,\Delta^n)=\coprod_{\{[m]\to [n]\}}Hom(\Delta^m,A).$$ Problem ----- ----- Let $\iota:[n-1]\to [n]$ be the obvious inclusion on the last $n-1$ objects of $[n]$, and let $\phi:[n]\to sSet$ be a functor parameterizing a sequence of composable maps $$A^0\leftarrow A^1\leftarrow\dots\leftarrow A^n.$$ Let $\phi'=\iota^*\phi$, which parameterizes the sequence: $$A^0\leftarrow A^1\leftarrow\dots\leftarrow A^{n-1}$$ (yes, the indexing is annoying, since $A^k=\phi(n-k)$). Also, let $c_{A^n}:[n]\to sSet$ be the constant functor at $A^n$, and let $c'_{A^n}=\iota^*c_{A^n}$. We also define a third sequence $d_{A^n}:[n]\to sSet$ to be the sequence $$A^n\leftarrow A^n\leftarrow\dots\leftarrow A^n\leftarrow \emptyset$$ which extends $c'_{A^n}$ by the empty object on the right. It is immediate from the definition that we have a canonical isomorphism $M^{n-1}(c'_{A^n})\cong M^n(d_{A^n})$. This gives us a natural inclusion $M^{n-1}(c'_{A^n})\cong M^n(d_{A^n})\hookrightarrow M^n(c_{A^n})$ Then we apparently have the following pushout square: $$\begin{matrix}M^{n-1}(c'_{A^n})&\hookrightarrow &M^n(c_{A^n})\\ \downarrow&&\downarrow\\ M^{n-1}(\phi')&\hookrightarrow& P\end{matrix}$$ Claim: $P\cong M^n(\phi)$ Then why does the claim hold?

   Reposted from MO:

   Background

   ---

   The mapping simplex: We define a functor $M^n:Fun([n],sSet)\to sSet$ sending the functor $\phi:[n]\to sSet$ classifying the sequence

   $$A^0\leftarrow A^1\leftarrow\dots\leftarrow A^n$$

   (notice that $A^k=\phi(n-k)$.) to the simplicial set $M^n(\phi)$ defined as a representable functor as follows:

   $$Hom_{sSet}(\Delta^k,M^n(\phi))=\coprod_{\{g:[k]\to[n]\}}Hom_{sSet}(\Delta^k,A^{g(k)})$$

   The identity map $M^n(\phi)\to M^n(\phi)$ then determines a unique map $M^n(\phi)\to \Delta^n$ by the fact that simplicial sets are colimits of their simplices.

   Also, given a map $\Delta^m\to \Delta^n$ corresponding to a functor $F:[m]\to [n]$ (by full and faithfulness of the Yoneda embedding), it’s not hard to see that

   $$M^n(\phi)\times_{\Delta^n} \Delta^m\cong M^m(F^*\phi)$$

   Let $\phi:[n]\to sSet$ be a functor classifying a sequence of composable maps

   $$A^0\leftarrow A^1\leftarrow\dots\leftarrow A^n$$

   Then let $c_{A^n}$ be the constant functor $[n]\to sSet$ at $A^n$. Then by the functoriality of $M^n$, we have a map $M^n(c_{A^n})\to M^n(\phi)$ induced by the obvious natural transformation $\alpha:c_{A^n}\to \phi$, which is defined componentwise as the composite map $\alpha_i:A^n\to A^i$ for each $i$ (the naturality of this map is immediate).

   Notice that $M^n(c_A)$ for any simplicial set $A$ is canonically isomorphic to the product $A\times \Delta^n$ since

   $$Hom(\Delta^m,A\times \Delta^n)\cong Hom(\Delta^m,A)\times Hom(\Delta^m,\Delta^n)=\coprod_{\{[m]\to [n]\}}Hom(\Delta^m,A).$$

   Problem

   ---

   Let $\iota:[n-1]\to [n]$ be the obvious inclusion on the last $n-1$ objects of $[n]$, and let $\phi:[n]\to sSet$ be a functor parameterizing a sequence of composable maps

   $$A^0\leftarrow A^1\leftarrow\dots\leftarrow A^n.$$

   Let $\phi'=\iota^*\phi$, which parameterizes the sequence:

   $$A^0\leftarrow A^1\leftarrow\dots\leftarrow A^{n-1}$$

   (yes, the indexing is annoying, since $A^k=\phi(n-k)$).

   Also, let $c_{A^n}:[n]\to sSet$ be the constant functor at $A^n$, and let $c'_{A^n}=\iota^*c_{A^n}$. We also define a third sequence $d_{A^n}:[n]\to sSet$ to be the sequence

   $$A^n\leftarrow A^n\leftarrow\dots\leftarrow A^n\leftarrow \emptyset$$

   which extends $c'_{A^n}$ by the empty object on the right. It is immediate from the definition that we have a canonical isomorphism $M^{n-1}(c'_{A^n})\cong M^n(d_{A^n})$. This gives us a natural inclusion $M^{n-1}(c'_{A^n})\cong M^n(d_{A^n})\hookrightarrow M^n(c_{A^n})$

   Then we apparently have the following pushout square:

   $$\begin{matrix}M^{n-1}(c'_{A^n})&\hookrightarrow &M^n(c_{A^n})\\ \downarrow&&\downarrow\\ M^{n-1}(\phi')&\hookrightarrow& P\end{matrix}$$

   Claim: $P\cong M^n(\phi)$

   Then why does the claim hold?

2. • CommentRowNumber2.
   • CommentAuthorHarry Gindi
   • CommentTimeFeb 10th 2011
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexDoes anyone know how to show this? It's geometrically obvious, but I'm having a rough time showing it to be true. The mapping properties are in exactly the wrong directions =(.

   Does anyone know how to show this? It’s geometrically obvious, but I’m having a rough time showing it to be true. The mapping properties are in exactly the wrong directions =(.

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeFeb 11th 2011
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexAlright, I think I answered my own question on MO. Any chance anyone can tell me if the proof works?

   Alright, I think I answered my own question on MO. Any chance anyone can tell me if the proof works?