Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: mathematics

Topic: Yoneda ext for complexes

Brendan Murphy (Jun 04 2024 at 19:31):

Does anyone know what was supposed to go in "Higher extensions over general chain complexes" here? Or have a reference for this?

Brendan Murphy (Jun 04 2024 at 19:38):

I would like to deduce the correspondence between yoneda ext and the hom in the derived category from a result about complexes. Let $\mathcal{A}$ be an abelian category and for complexes $X, Y$ with entries in $\mathcal{A}$ define $\operatorname{SES}(X, Y)$ to be the category of short exact sequences of complexes $0 \to X \to Z \to Y \to 0$ (the maps here are ladder diagrams whose left and right vertical maps are isomorphisms; this is a groupoid by the 5 lemma). The connecting homomorphism defines a function $\pi_0 \operatorname{SES}(X, Y) \to \operatorname{Hom}_{D(\mathcal{A})}(Y, \Sigma X)$, well defined by naturality. Is this a bijection? Specifically, how can we show surjectivity?

Brendan Murphy (Jun 04 2024 at 19:39):

We can take a map $Y \to \Sigma X$ and write it as a formal span $Y \leftarrow Y' \rightarrow \Sigma X$, wlog $Y' \to Y$ surjective, but where do we go from there? Presumably some combination of pushing out and pulling back in a clever way?

Brendan Murphy (Jun 04 2024 at 19:41):

My instinct was to form the pullback of $Y' \rightarrow \Sigma X \overset{\partial_X}{\leftarrow} X$ but I don't see how it helps