Modules and Morita theorem for operads (Q2762637)

In the article the authors investigate generalisations of classical Morita theory in the context of linear operads. For a \(k\)-linear operad \(\mathcal P\) matrix operads \(\text{Mat}(n,{\mathcal P})\) have been be constructed, and it has been shown that their categories of representations are equivalent.NEWLINENEWLINENEWLINEA \(k\)-linear operad can be defined as a monoid in the category of \({\mathcal S}\)-spaces, \(({{\mathcal S}}\)-\({\mathcal V}ect,\circ)\), where ``\(\circ\)'' is the so-called plethysm tensor product and \({\mathcal S}\) is the category of standard finite sets and their bijections. The plethysm product, however, is not symmetric. In particular the notions of left and right modules over an operad get asymmetric. Besides the plethysm product two other important tensor products can be defined on \({{\mathcal S}}\)-\({\mathcal V}ect\). And one of these, call it \(\otimes\), corresponds to composition of formal power series, whereas the plethysm is an analogue of multiplication of power series.NEWLINENEWLINENEWLINEIn order to construct a Morita theory for operads the authors work with right \(\mathcal P\)-modules and replace left modules by \(\mathcal P\)-algebras as follows. The relative plethysm product for modules over operads will be defined in a way similar to the usual tensor product of a right and a left module over an algebra or over a monoid in a monoidal category. Furthermore for two operads \(\mathcal P\) and \(\mathcal Q\) a \(({\mathcal Q},{\mathcal P})\)-bimodule can be defined. Then a canonical associativity isomorphism can be obtained for the relative plethysm product NEWLINE\[NEWLINE(L\circ_{\mathcal Q} M)\circ_{\mathcal P} N \simeq L\circ_{\mathcal Q}(M\circ_{\mathcal P} N)NEWLINE\]NEWLINE and NEWLINE\[NEWLINE(L\circ_{\mathcal Q} M)\circ_{\mathcal P}\langle A\rangle \simeq L\circ_{\mathcal Q}\langle M\circ_{\mathcal P} A\rangleNEWLINE\]NEWLINE for \(L\) a left \(\mathcal Q\)-module, \(N\) a right \(\mathcal P\)-module, \(M\) a \(({\mathcal Q},{\mathcal P})\)-bimodule, and \(A\) a \(\mathcal P\)-algebra. Moreover, each such bimodule \(M\) defines a functor \(f_M: {\mathcal P}\)-\(\text{Alg}\to {\mathcal Q}\)-\(\text{Alg}\) by \(f_M(A)=M\circ_{\mathcal P}\langle A\rangle\).NEWLINENEWLINENEWLINEFor a right \(\mathcal P\)-module \(M\) the endomorphism operad \(\text{OpEnd}_{\mathcal P}(M)(n) := \text{Hom}_{\text{Mod}_{\mathcal P}}(M^{\otimes_n},M)\) will be defined. The following result can then be shown. It is an analogue of the classical Morita equivalence. Given a \(\mathcal P\)-module \(M\), and put \(Q=\text{OpEnd}_{\mathcal P}(M)\). Then \(M\) is a \(({\mathcal Q},{\mathcal P})\)-module which yields the functor \(f_M: {\mathcal P}\)-\(\text{Alg}\to {\mathcal Q}\)-\(\text{Alg}\). If \(M\) is a free \(\mathcal P\)-module, i.e., is isomorphic to \({\mathcal P}^{\oplus_d}\) for some \(d\), then \(f_M\) is an equivalence of categories.NEWLINENEWLINENEWLINEA super-version of this theorem can be shown as well, since most of the constructions and results can be obtained in any \(k\)-linear abelian symmetric monoidal category instead of \({\mathcal V}ect\).NEWLINENEWLINENEWLINEIt is hoped that the fuller statement providing the necessary conditions for this equivalence can be obtained in a more extended context.