Tags: ? Refs: simplicial sets

bar construction

The cosimplicial cobar construction keeps track, in a very explicit sense, of all the ways that a given object \(F(i)\) of the diagram is mapped into. It remembers every finite sequence of arrows whose terminus is \(F(i)\). The cobar construction then looks at ways of coherently mapping a “fat point” into this enormous arrangement.

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Cyclic Bar Construction

See geometric realization, trace, factorization homology: Pasted image 20210630211527

Simplicial bar construction

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See classifying space

Reduced bar construction

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THH

Let \(A \in \mathsf{Alg} _{/R}\) such that \(A\) is projective as an \(R{\hbox{-}}\)module and \(M\in ({A}, {A}){\hbox{-}}\mathsf{biMod}\). Then define the Hochschild complex as the bar construction \begin{align*} C_n(A, M) := M \otimes A^{\otimes n} \end{align*}
DGA

bar construction
A_infty

cobar construction