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Given a field $k$, an associative $k$-algebra $A$, and an $A$-bimodule $M$, one can define as the Hochschild homology and cohomology as the homology of the complexes $$M\otimes A^{\otimes n}$$ and $$\ …

Let $G$ be a (discrete) group, $X$ a topological space that works as a classifying space for $G$, and $\mathcal{L}$ a local system on $X$ with stalk $L$. It is a fairly standard result that $$ H^i(G, …

Suppose we have a non symmetric operad $\mathcal{O}$, a collection of sets $\{P(n)\}_{n\geq 0}$ and maps $$P(n)\otimes \mathcal{O}(k_1)\otimes\cdots \otimes \mathcal{O}(k_n)\to P(k_1+\cdots + k_n)$$ s …

We assume $A$ and $M$ are finite dimensional, and denote by $A*$ and $M*$ their respective duals. Denote by $HH_n$ and $CH_n$ the Hochschild homology of an algebra and a coalgebra respectively, and si …

It is a standard fact in the theory of coalgebras and comodules that, given a coalgebra $C$ and a comodule $M$ over $C$, the coradical filtration $$M_n := \Delta^{-1}(M\otimes C_0 + M_{n-1}\otimes C)$ …