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1. • CommentRowNumber1.
   • CommentAuthorYaron
   • CommentTimeMay 18th 2012
   • (edited May 19th 2012)
   • PermaLink
   Author: Yaron
   Format: MarkdownItexI got stuck on something that should probably be obvious in Section VII.6 in CWM (the bar construction). If $A$ is an $Ab$-category and $\langle L,\varepsilon\colon L\Rightarrow I_A, \nu\colon L\Rightarrow L^2\rangle$ is a comonad in $A$ (that is, a comonoid in the strict monoidal category $A^A$), then because the opposite (augmented) simplicial category $\Delta^{\mathrm{op}}$ has the universal comonoid, there is a unique strict monoidal functor $\Delta^{\mathrm{op}}\to A^A$ with $1\mapsto L$, $(\delta_{0}^{0})^{\mathrm{op}}\mapsto \varepsilon$ and $(\sigma_0^1)^{\mathrm{op}}\mapsto \nu$ (using the indexing convention of CWM for face maps $\delta_{?}^{?}$ and degeneracy maps $\sigma_{?}^{?}$ in $\Delta$). Composing with the ``evaluate at $a$'' functor $E_a\colon A^A\to A$, we get an augmented simplical object $Y_a\colon\Delta^{\mathrm{op}}\to A$ with $Y_a(n)=L^n(a)$. As usual, this defines a chain complex $$ (Y_a(0)=a)\leftarrow (Y_a(1)=La)\leftarrow (Y_a(2)=L^2a)\leftarrow\cdots $$ in $A$ (where the boundary morphisms are the appropriate sums of face maps with alternating sums). Now, at the bottom of p. 181 (second edition), it is written the this complex is a resolution. Why is this true in this general setting? [For the special case of the bar resolution appearing in any homology book (that is, when $A={\Pi}$-$\mathbf{Mod}$ for some group ${\Pi}$, $L=\mathbb{Z}({\Pi})\otimes-$, $a=\mathbb{Z}$, etc.), an explicit contracting homotopy is specified (by elements). The problem is that I don't understand how to deal with the general case.]

   I got stuck on something that should probably be obvious in Section VII.6 in CWM (the bar construction).

   If is an -category and is a comonad in (that is, a comonoid in the strict monoidal category ), then because the opposite (augmented) simplicial category has the universal comonoid, there is a unique strict monoidal functor with , and (using the indexing convention of CWM for face maps and degeneracy maps in ). Composing with the “evaluate at ” functor , we get an augmented simplical object with .

   As usual, this defines a chain complex

   in (where the boundary morphisms are the appropriate sums of face maps with alternating sums).

   Now, at the bottom of p. 181 (second edition), it is written the this complex is a resolution. Why is this true in this general setting?

   [For the special case of the bar resolution appearing in any homology book (that is, when - for some group , , , etc.), an explicit contracting homotopy is specified (by elements). The problem is that I don’t understand how to deal with the general case.]

2. • CommentRowNumber2.
   • CommentAuthorFinnLawler
   • CommentTimeMay 18th 2012
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   Author: FinnLawler
   Format: MarkdownItex[This nCafé post](http://golem.ph.utexas.edu/category/2007/05/on_the_bar_construction.html) by Todd might help you.

   This nCafé post by Todd might help you.

3. • CommentRowNumber3.
   • CommentAuthorYaron
   • CommentTimeMay 18th 2012
   • PermaLink
   Author: Yaron
   Format: MarkdownItexThank you very much, Finn. I will surely look into this (although from a first glance it seems a little above my current knowledge). Is there some simple straightforward explanation (something like: "just take ... as the contracting homotopy")?

   Thank you very much, Finn. I will surely look into this (although from a first glance it seems a little above my current knowledge). Is there some simple straightforward explanation (something like: “just take … as the contracting homotopy”)?

4. • CommentRowNumber4.
   • CommentAuthorFinnLawler
   • CommentTimeMay 19th 2012
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   Author: FinnLawler
   Format: MarkdownItexI don't really know complexes, but for simplicial objects the idea is that if you have a monad $T$ on a category $C$, then the bar construction turns each $T$-algebra into a simplicial $T$-algebra, using the comonad $F^T U^T$ on $C^T$ arising from $T$. If you apply $U^T$ to this simplicial algebra you get a simplicial object of $C$, and it turns out that the unit of $T$ supplies a contracting homotopy for this. (Compare the example at [[split coequalizer]].) Mac Lane says that the complex $L^* a$ is a resolution, even though the associated simplicial object of $A$ won't be contractible in general. Maybe it's exact for some other reason, but you'll have to ask someone who knows more homological algebra than I do (i.e. pretty much anyone).

   I don’t really know complexes, but for simplicial objects the idea is that if you have a monad on a category , then the bar construction turns each -algebra into a simplicial -algebra, using the comonad on arising from . If you apply to this simplicial algebra you get a simplicial object of , and it turns out that the unit of supplies a contracting homotopy for this. (Compare the example at split coequalizer.)

   Mac Lane says that the complex is a resolution, even though the associated simplicial object of won’t be contractible in general. Maybe it’s exact for some other reason, but you’ll have to ask someone who knows more homological algebra than I do (i.e. pretty much anyone).

5. • CommentRowNumber5.
   • CommentAuthorYaron
   • CommentTimeMay 19th 2012
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   Author: Yaron
   Format: MarkdownItexFinn, thanks! I will try to work out the details of your explanation - this now looks as a manageable task. > pretty much anyone ...certainly not me :)

   Finn, thanks! I will try to work out the details of your explanation - this now looks as a manageable task.

   pretty much anyone

   …certainly not me :)