There is a conjecture of Kontsevich which states that Hochschild (co)homology of the Fukaya category of a compact symplectic manifold $X$ is the (co)homology of the manifold. (See …

Many texts which praise the generality of the bar construction associated to a monad, say that Hochschild homology is an example of this. What exactly is in this case the underly …

Background: the Hochschild homology of an associative algebra is the homology of the complex ... --> A (x) A (x) A --> A (x) A --> A where the last two differentials ar …

Let $C$ be an algebra. Let $E = C^{\otimes 2n}$ be the tensor product (over the ground field) of $2n$ copies of $C$. [EDIT: Or better, $E = C\otimes C^{op}\otimes C\otimes C^{op} …

I know that for $X$ a connected space, $THH(\Sigma^\infty \Omega X) = \Sigma^\infty \Lambda X$, the suspension spectrum of the free loop space of $X$. The computation can be carri …

Deligne's conjecture states that the Hochschild cochain complex of an A-infinity algebra is an algebra over the operad of chains on the topological little discs operad. Of course …

As someone whose knowledge of cohomology is patchy and picked up on a need-to-know basis, and whose algebraic geometry is even worse, I wondered if someone could help with this que …

How do you define the Hochschild (co)homology of a dg category or an A-infinity category? I've only seen it defined when the category is equivalent to a category of modules over a …