6
votes
Bar construction in commutative algebras is calculated by pushout
A way to see this which doesn't dive into the specifics of the simplicial diagram "$C\otimes D^{\otimes n}\otimes E$" is to apply 3.2.4.7 to the symmetric monoidal $\infty$-category $Mod_D(\...
6
votes
Bar construction in commutative algebras is calculated by pushout
Welcome to MathOverflow! First, let me point out that what you're asking is already true at the 1-categorical level. The pushout in the category of commutative rings is computed by the tensor product. ...
3
votes
Accepted
Bar construction in commutative algebras is calculated by pushout
This is just to be explicit about the role of the bar construction in David's answer.
If $I$ is the category $a \leftarrow b \rightarrow c$ parametrizing pushout diagrams, then there is a functor $f: ...
1
vote
Strictifying strong monoidal functors
Here is another example that shows the answer to the second part (where both $C_i$ are strict) is negative.
Let $C_1$ be the discrete strict monoidal category whose monoid of objects is the Booleans. ...
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