Let $B$ be the symmetric monoidal category of finite sets and bijections with disjoint union. Let $C$ be a symmetric monoidal category. Is there a standard name for a lax monoidal functor $F:B \to C$? In other words, we are considering a sequence $F(n)$ of $S_n$-representations in $C$ (i.e. a species in $C$) together with $S_n\times S_m$-equivariant maps $F(n) \otimes F(m) \to F(n+m)$ satisfying certain associativity conditions.

I could invent a name for this notion (like "a $B^\otimes$-module in $C$") but if there is already an established terminology I'd prefer to use that.


1 Answer 1


Depending on whether you want it to agree with the symmetric structure or only with monoidal structure, this would be usually referred to, respectively, as twisted commutative algebras or twisted associative algebras. See, for example, http://arxiv.org/pdf/0710.3392.pdf, and a more classical reference http://www.sciencedirect.com/science/article/pii/0022404993901064 . (Or Chapter 4 in http://www.maths.tcd.ie/~vdots/AlgebraicOperadsAnAlgorithmicCompanion.pdf :) )


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.