Lax monoids, pseudo-operads, and convolution
Brian Day, Ross Street · Contemporary mathematics - American Mathematical Society · 2003
Various weakenings of monoidal category have been in existence almost as long as the notion itself. There are the multicategories of Lambek [Lk], the promonoidal categories of [D1], and the lax monoidal categories involving n-fold tensor products2 with not-necessarily-invertible associativity and unit constraints. There is a diamond Multicategories Monoidal categories Promonoidal categories Lax monoidal categories in which moving down along a side of gradient 1 imposes invertibility on constraints, while moving down along a side of gradient Ð1 imposes representability on the multihoms. A strong form of representability (see Hermida [H]) leads us from the top of the diamond to the bottom in one step. Promonoidal categories were introduced to explain a large variety of convolution monoidal structures on functor categories. What we want to point out in this paper is that convolution formulas are available in weaker settings, but, of course, the resultant functor categories bear weaker monoidal structures too.