5
votes
2answers
303 views

Homotopy factorization of morphisms of chain complexes

This is a sort of follow up to this MO question. Let $R$ be a ring (eventually with good properties) and $\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). ...
1
vote
1answer
167 views

Could we extend the Atiyah class to the sheaf of poly-vector fields to get a Poisson bracket?

Let $X$ be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
9
votes
4answers
578 views

What is the relation between the Lie bracket on $TX$ as commutator and that coming from the Atiyah class?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
7
votes
1answer
241 views

Analogue of cyclic homology for e_n-algebras?

Cyclic homology may be defined as the primitive part (with respect to a natural product) of the homology of the Lie algebra associated with the "stabilization" of an associative algebra $A$. Here the ...
7
votes
3answers
581 views

A 2-category of chain complexes, chain maps, and chain homotopies?

First-time here... I hope my question isn't silly or anything... anyway... Consider the category of chain complexes and chain maps. We can also define chain homotopies between chain maps. Does this ...
0
votes
0answers
244 views

cokernel for $L_\infty$-algebra morphisms

As I have asked a wrong question previously, I edited a bit. It is obvious that the cokernel construction does not work well for the category of Lie algebras. The cokernel exists only for a normal ...
18
votes
3answers
2k views

Homology theory constructed in a homotopy-invariant way

Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological ...
8
votes
1answer
391 views

Is there a “derived” Free $P$-algebra functor for an operad $P$?

Recall that an operad (in vector spaces, say) $P$ consists of a collection of vector spaces $P(n)$ for $n\geq 0$, such that $P(n)$ is equipped with an action by the symmetric group $S_n$, with maps ...
4
votes
3answers
481 views

Chain Complexes and Linear Infinity-Categories

A statement I heard recently is that "chain complexes are the same thing as strict linear $\infty$-categories". Can someone explain how to see this?