Homotopy units in \(A\)-infinity algebras (Q2787968)

The purpose of this paper is to compare associative operads and unital associative operads in a fairly general class of model categories. The starting point is the categorical result that, in any close symmetric monoidal category \(\mathcal V\) with an initial object, the morphism from the associative operad to the unital associative operad is an epimorphism in the category of operads in \(\mathcal V\). To generalize to the model category setting, one first needs for \(\mathcal V\) to be a sufficiently nice closed symmetric monoidal model category with some appropriate axioms for the unit. Then the appropriate analogue for the theorem is that the map from the associative operad to the unital associative operad in \(\mathcal V\) to be a homotopy epimorphism.NEWLINENEWLINEThe paper begins with a treatment of homotopy epimorphisms, which do not seem to be as well-known as the dually-defined homotopy monomorphisms. It continues with a lengthy section on operads, which in particular gives an extended description of free operads on trees. These two topics are likely to be of independent interest; the treatment of trees is perhaps more familiar in the context of dendroidal sets, but is used in an elegant way here. Later in the operad section are a number of technical results used to prove the main theorem which make use of weaker \(A_\infty\) operads and intermediate versions which the author calls \(u\)-\textit{infinity (unital) associative operads}.NEWLINENEWLINEThe author first proves his main result for the examples where \(\mathcal V\) is the category of groupoids or \(\mathcal V\) is the category of chain complexes, using the intermediate operads defined here. From there, he proves a result which allows the conclusion of the desired map being a homotopy epimorphism to be transferred across a Quillen adjunction. This result is used first to prove the result for the examples of non-negatively graded chain complexes and simplicial sets, and from there to any closed symmetric monoidal category which has a simplicial or complicial (here meaning compatibly enriched in chain complexes) structure.