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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?

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    Do you have any guess? – 2017-02-21
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    I know it's supposed to be obvious and maybe I should have thought about it/searched for it more before asking here... My guess is for $f:A\to A'$, we define $Bf$ to be the chain map $(f|_\bar{A}^{\otimes n})_n$ – 2017-02-21
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    And does that work? – 2017-02-21
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    Yes, I believe so. I am however a littles bit confused about what we need the suspension $s$ for, it increases the grades by one but why is this important? – 2017-02-22
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    Never mind i see why the suspension is necessary now, thanks=) – 2017-02-22

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