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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 19th 2013
   • (edited Jul 19th 2013)
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   Author: Urs
   Format: MarkdownItexFor the sake of illustration I have added to _[[ordinary homology]]_ a section _[In terms of higher linear algebra](http://ncatlab.org/nlab/show/ordinary+homology#InTermsOfHigherLinearAlgebra)_. Currently the main point is to record, after some preliminaries, the standard observation plus detailed proof that for $X$ a topological space, its ordinary chain complex of singular simplices is, up to equivalence, the $\infty$-colimit of the tensor unit local system with coefficients in $H k Mod$. (Its "$Hk$-Thom spectrum".)

   For the sake of illustration I have added to ordinary homology a section In terms of higher linear algebra.

   Currently the main point is to record, after some preliminaries, the standard observation plus detailed proof that for $X$ a topological space, its ordinary chain complex of singular simplices is, up to equivalence, the $\infty$-colimit of the tensor unit local system with coefficients in $H k Mod$. (Its “$Hk$-Thom spectrum”.)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 19th 2013
   • (edited Jul 19th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to _[In terms of higher linear algebra](http://ncatlab.org/nlab/show/ordinary+homology#InTermsOfHigherLinearAlgebra)_ now also the statement that for a Poincaré duality space the $H A$-module of sections of the trivial $H A$-$\infty$-bundle over it is self-dual, up to a degree twist.

   added to In terms of higher linear algebra now also the statement that for a Poincaré duality space the $H A$-module of sections of the trivial $H A$-$\infty$-bundle over it is self-dual, up to a degree twist.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJul 20th 2013
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   Author: Mike Shulman
   Format: MarkdownItexWhy is this colimit functor called $\Gamma$? Usually $\Gamma$ refers to a space of sections, which I would think would be rather the *limit* that computes *cohomology* rather than homology. What is a "flat section"?

   Why is this colimit functor called $\Gamma$? Usually $\Gamma$ refers to a space of sections, which I would think would be rather the limit that computes cohomology rather than homology. What is a “flat section”?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 20th 2013
   • (edited Jul 20th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexA functor $\chi : \Pi(X) \to E Mod$ is a flat $E$-module bundle on $X$. A non-flat one would be a morphism $X \to E\mathbf{Mod}$ where now on the right we have the actual stack of $E$-modules instead of just the $\infty$-category of $E$-modules (over the point). For $\mathbb{I}_X^E$ the tensor unit of $E$-module bundles, a section of $\chi$ is a map $\mathbb{I}_X^E \to \chi$. This is flat in the first case, and not-necessarily flat in the second. By the universal colimit property, a map of $E$-modules $\Gamma(\chi) \to \mathbb{I}_\ast^E$ is equivalently a map of flat $E$-modle bundles $\chi \to \mathbb{I}_X^E$. This is a cosection. Hence $\Gamma(\chi)$ is such that maps out of it are cosections, and hence to the extent that all is dualizable itself it is sections. But maybe another term would be better, I am not sure.

   A functor $\chi : \Pi(X) \to E Mod$ is a flat $E$-module bundle on $X$. A non-flat one would be a morphism $X \to E\mathbf{Mod}$ where now on the right we have the actual stack of $E$-modules instead of just the $\infty$-category of $E$-modules (over the point).

   For $\mathbb{I}_X^E$ the tensor unit of $E$-module bundles, a section of $\chi$ is a map $\mathbb{I}_X^E \to \chi$. This is flat in the first case, and not-necessarily flat in the second.

   By the universal colimit property, a map of $E$-modules $\Gamma(\chi) \to \mathbb{I}_\ast^E$ is equivalently a map of flat $E$-modle bundles $\chi \to \mathbb{I}_X^E$. This is a cosection. Hence $\Gamma(\chi)$ is such that maps out of it are cosections, and hence to the extent that all is dualizable itself it is sections. But maybe another term would be better, I am not sure.