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Let $M,B$ be $R$-modules, and suppose we're given an n-extension $E_1\to\dots\to E_n$ of $B$ by $M$, that is, an exact sequence $$0\to M\to E_1\to\dots\to E_n \to B\to 0.$$

A morphism of $n$-extensions of $Y$ by $X$ is defined to be a hammock

$$\begin{matrix} &&A_1&\to&A_2&\to&A_3&\to&\ldots&\to &A_{n-2}&\to &A_{n-1}&\to& A_{n}&&\\ &\nearrow&\downarrow&&\downarrow&&\downarrow&&&&\downarrow&&\downarrow&&\downarrow&\searrow&\\ X&&\downarrow&&\downarrow&&\downarrow&&&&\downarrow&&\downarrow&&\downarrow&&Y\\ &\searrow&\downarrow&&\downarrow&&\downarrow&&&&\downarrow&&\downarrow&&\downarrow&\nearrow&\\ &&B_1&\to&B_2&\to&B_3&\to&\ldots&\to &B_{n-2}&\to &B_{n-1}&\to& B_{n}&&\end{matrix}$$

This detetermines a category $n\operatorname{-ext}(Y,X)$. Further, we're given an almost-monoidal sum on this category given by taking the direct sum of exact sequences, then composing the first and last maps with the diagonal and codiagonal respectively. Taking connected components, we're left with a set $ext^n(Y,X)$, and the sum reduces to an actual sum on the connected components, which turns $ext^n(Y,X)$ into an abelian group (and therefore an $R$-module).

It's well known that these functors, called Yoneda's Ext functors are isomorphic to the Ext functors $\pi_n(\underline{\operatorname{Hom}}(sM,sN))$ where $\underline{\operatorname{Hom}}(sM,sN)$ is the homotopy function complex between the constant simplicial $R$-modules $sM$ and $sN$ (obtained by means of cofibrant replacement of $sM$ in the projective model structure, fibrant replacement of $sN$ in the injective model structure, or by any form of Dwyer-Kan simplicial localization (specifically the hammock localization)).

In a recent answer, Charles Rezk mentioned that we can compute this in the case $n=1$ as $\pi_0(\underline{\operatorname{Hom}}(sM,sN[1]))$, where $sN[1]$ is the simplicial $R$-module with homotopy concentrated in degree $n$ equal to $N$. That is, these are exactly the maps $M\to N[1]$ in the derived category.

It was also mentioned that for the case $n=1$, there exists a universal exact sequence in the derived category: $$0\to N\to C\to N[1]\to 0$$ where $C$ is weakly contractible such that every extension of $N$ by $M$ arises as $$N\to M\times^h_{N[1]} C\to QM,$$ which is $\pi_0$-short exact. (And QM is a cofibrant replacement of $M$, for obvious reasons).

Questions:

Why can we get $\pi_1$ of the function complex by looking at maps into $N[1]$? At least on the face of it, it seems like we would want to look at maps into $N[-1]$, that is, look at maps into the "loop space", not the "suspension" (scare quotes because these are the loop and suspension functors in $sR\operatorname{-Mod}$, not in $sSet$.

What is this this "universal extension" $$0\to N\to C\to N[1]\to 0$$ at the level of simplicial modules?

Is there a similar "universal extension" for $n>1$? If so, what does it (one of its representatives) look like at the level of simplicial modules?

Given an $n$-extension $\sigma$ of $B$ by $M$, how can we produce a morphism in the derived category $B\to M[n]$ that generates an $n$-extension in the same connected component of $n\operatorname{-ext}(B,M)$?

Lastly, since Yoneda's construction of Ext looks suspiciously like Dwyer and Kan's hammock localization, and there is homotopy theory involved in the other construction, I was wondering if there was any connection between the two. That is, I was wondering if there is another construction of Ext using DK-localization directly that shows why Yoneda's construction works.

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1 Answer 1

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For your first question:

Given an $n$-extension $\sigma$ of $B$ by $M$, how can we produce a morphism in the derived category $M\to N[n]$ that generates an n-extension in the same connected component of $n{-}ext(B,M)$?

Do you mean a morphism $B\to M[n]$? In which case, isn't this a the standard construction? Let $E$ be the extension of $B$ by $M$, let $P$ be a projective resolution of $B$, and cover the identity map of $B$ by a map of chain complexes $P\to E$. The resulting map $P_n\to M$ can be thought of as a morphism $B\to M[n]$ in the derived category, and it's the one that represents the corresponding component of $n{-}ext(B,M)$.

As for your second question, there's certainly some connection. What you've described is a category $C=n{-}ext(X,Y)$, whose objects are $n$-extensions of $Y$ by $X$. Let $W=C$, and perform the hammock construction $L_H(C,W)$ of the pair $(C,W)$; then $L_H(C,W)$ is a simplicially eniched category, which represents an $\infty$-groupoid since you inverted everything. The ext group is the set of equivalence classes of objects in $L_H(C,W)$.

Dwyer and Kan showed that if $C=W$, then the simplicial category $L_H(C,W)$ and the simplicial nerve $NC$ of $C$ represent the same $\infty$-groupoid. So $\mathrm{Ext}^n(X,Y)=\pi_0 (NC)$.

What is the homotopy type of $NC$? It is supposed to be equivalent to the space of maps $\mathrm{map}(X,Y[n])$ in the $\infty$-category associated to the derived category. I'm unaware of a reference where this is proved, however.

But take a look at Stefan Schwede's very nice paper on An exact sequence interpretation of the Lie bracket in Hochschild cohomology, which shows that $\pi_1(NC)=\mathrm{Ext}^{n-1}(X,Y)$, and gets some interesting results from that. (He's looking at $A$-bimodule extensions of $A$ by $A$ (i.e., Hochschild cohomology), and he extracts the Gerstenhaber-algebra structure on the ext-groups by means of homotopical constructions involving the $NC$.

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Nifty, and thanks for the reference. –  Harry Gindi Nov 21 '10 at 22:35

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