Free skew monoidal categories

Abstract

In the paper Triangulations, orientals, and skew monoidal categories, the free monoidal category Fsk on a single generating object was described. We sharpen this by giving a completely explicit description of Fsk, and so of the free skew monoidal category on any category. As an application we describe adjunctions between the operad for skew monoidal categories and various simpler operads. For a particular such operad L, we identify skew monoidal categories with certain colax L-algebras.

Skew monoidal categories and skew multicategories

“…We are now in a position to prove our first main result, which combines Theorem 5.7 above with Theorem 7.8 of the companion paper [5]. Theorem 6.1.…”

“…In the name LBC-algebra, the "LB" stands for left-bracketed and the "C" for colax; see [5]. We define LBC-Alg to be the full sub-2-category of nColax-L-Alg ℓ consisting of the LBC-algebras.…”

“…See Section 7.3 of the companion paper [5] for further details on the LBCalgebra associated to a skew monoidal category. Accordingly, the corresponding skew multicategory C has C( ; a) = C(i, a)…”

“…Using this, as well as results from our companion paper [5], we describe in Section 6 the perfect correspondence between skew monoidal categories and left representable skew multicategories. In Section 6 we also describe variants of this correspondence, dealing with closed skew monoidal categories and skew closed categories.…”

The Sequent Calculus of Skew Monoidal Categories

“…Szachlányi's paper was immediately noticed by Street, Lack and colleagues who have by now published a whole series of followup works [13,14,6,4,5]. Although the definition of skew monoidal category is simple, it entails some remarkably subtle properties.…”