MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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,, and a more classical reference . (Or Chapter 4 in :) )

share|cite|improve this answer
Perfect! Thanks. – Dan Petersen Feb 2 at 9:21

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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