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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.

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up vote 5 down vote accepted

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 :) )

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Perfect! Thanks. – Dan Petersen Feb 2 at 9:21

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