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 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 nlab noncommutative noncommutative-geometry number-theory object 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 8th 2011
   • (edited Feb 8th 2011)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexThe statement of HTT Lemma 3.2.2.9 refers to "the map $\Delta^n\times A^n \to M(\Phi)$", but unfortunately, it does not actually define what this map is or what properties we expect it to have (and it is not defined anywhere earlier in the book). Is there some canonical choice of such a map that I'm somehow missing? Got it, $A^n\times \Delta^n$ is the constant mapping simplex of length $n$ at the simplicial set $A^n$. The map is induced by the functoriality of the mapping simplex.

   The statement of HTT Lemma 3.2.2.9 refers to “the map $\Delta^n\times A^n \to M(\Phi)$”, but unfortunately, it does not actually define what this map is or what properties we expect it to have (and it is not defined anywhere earlier in the book). Is there some canonical choice of such a map that I’m somehow missing?

   Got it, $A^n\times \Delta^n$ is the constant mapping simplex of length $n$ at the simplicial set $A^n$. The map is induced by the functoriality of the mapping simplex.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 8th 2011
   • (edited Feb 8th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Is there some canonical choice of such a map Yes. Maybe you want to explicitly write $$ \Delta^n \times A^n \simeq \coprod_{\sigma \in A^n} \Delta^n $$ which in turn you should think of as $$ \cdots = \coprod_{\sigma : \Delta^n \to A} \Delta^n \,. $$ Then go back to the definition of $M(\phi)$ on the top of p. 151: a morphism $\Delta^n \to M(\phi)$ is a pair consisting of a morphism $f : \Delta^n \to \Delta^n$ and a morphism $\sigma : \Delta^n \to A^{max f}$. The first has a canonical choice: the identity. The second choice is parameterized by the index set $A^n$ of the above coproducts.

   Is there some canonical choice of such a map

   Yes. Maybe you want to explicitly write

   $$\Delta^n \times A^n \simeq \coprod_{\sigma \in A^n} \Delta^n$$

   which in turn you should think of as

   $$\cdots = \coprod_{\sigma : \Delta^n \to A} \Delta^n \,.$$

   Then go back to the definition of $M(\phi)$ on the top of p. 151: a morphism $\Delta^n \to M(\phi)$ is a pair consisting of a morphism $f : \Delta^n \to \Delta^n$ and a morphism $\sigma : \Delta^n \to A^{max f}$.

   The first has a canonical choice: the identity. The second choice is parameterized by the index set $A^n$ of the above coproducts.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 8th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexWait a sec, I didn't use the indices as in the book. Just a sec

   Wait a sec, I didn’t use the indices as in the book. Just a sec

4. • CommentRowNumber4.
   • CommentAuthorHarry Gindi
   • CommentTimeFeb 8th 2011
   • (edited Feb 8th 2011)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexUrs, it's even easier than that. Given a simplicial set $A$, let $\c_A:[n]\to sSet$ be the constant functor at $A$. Claim: $M(c_A)\cong A\times \Delta^n$. Giving a simplex of $\Delta^j\to M(c_A)$ gives a map $\Delta^j\to \Delta^n$ and another map $\Delta^j\to \A$ (since $c_A$ is constant). That is the same thing as giving a simplex of the product, by definition, so we're done. Then we obtain the required map (the one I originally wanted) by the functoriality of the mapping simplex functor, since we have an obvious natural transformation $c_{A^n}\to \Phi$ defined component-wise as follows: $\alpha_i:A^n\to A^i$ is given by the composite $A^n \to A^{n-1}\to\dots\to A^i$. We immediately verify that the transformation is natural (by construction).

   Urs, it’s even easier than that.

   Given a simplicial set $A$, let $\c_A:[n]\to sSet$ be the constant functor at $A$.

   Claim: $M(c_A)\cong A\times \Delta^n$.

   Giving a simplex of $\Delta^j\to M(c_A)$ gives a map $\Delta^j\to \Delta^n$ and another map $\Delta^j\to \A$ (since $c_A$ is constant). That is the same thing as giving a simplex of the product, by definition, so we’re done.

   Then we obtain the required map (the one I originally wanted) by the functoriality of the mapping simplex functor, since we have an obvious natural transformation $c_{A^n}\to \Phi$ defined component-wise as follows:

   $\alpha_i:A^n\to A^i$ is given by the composite $A^n \to A^{n-1}\to\dots\to A^i$. We immediately verify that the transformation is natural (by construction).

5. • CommentRowNumber5.
   • CommentAuthorHarry Gindi
   • CommentTimeFeb 9th 2011
   • (edited Feb 9th 2011)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexHmm, a more pressing concern: How can we show that $M(\Phi)\cong M(\Phi')\coprod_{A^n\times \Delta^{n-1}}A^n\times \Delta^n$? It's a pain in the neck because that pushout is defined by a mapping property in, while the pushout has a mapping property in the opposite direction. Here's a [link](http://mathoverflow.net/questions/54866/the-pushout-of-a-mapping-simplex-along-products-is-a-mapping-simplex) to the mathoverflow question.

   Hmm, a more pressing concern:

   How can we show that $M(\Phi)\cong M(\Phi')\coprod_{A^n\times \Delta^{n-1}}A^n\times \Delta^n$?

   It’s a pain in the neck because that pushout is defined by a mapping property in, while the pushout has a mapping property in the opposite direction.

   Here’s a link to the mathoverflow question.