I have just read (in Algebraic Operads, Loday & Vallette) that the bar construction is a functor $B$ from the category of augmented graded algebras to the category of conilpotent differential graded coalgebras. I have seen what this functor does on objects: If $A$ is an augmented graded algebra, then $BA=(T^c(s\bar{A}),d_2)$ is a conilpotent differential graded coalgebra. My question is what does this functor do with morphisms of augmented graded algebras?
2026-03-26 03:27:10.1774495630
How is the bar construction on augmented graded algebras a functor?
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