Let C be a category with finite colimits, and let (E ,M) be a factorisation system on C with M stable under pushouts. Writing C;M for the symmetric monoidal category with morphisms cospans of the form c → m ←, where c ∈ C and m ∈ M, we give method for constructing a category from a symmetric lax monoidal functor F : (C;M,+) → (Set,×). A morphism in this category, termed a decorated corelation, ...

We show that every braided monoidal category arises as End(I) for a weak unit I in an otherwise completely strict monoidal 2-category. This implies a version of Simpson’s weak-unit conjecture in dimension 3, namely that one-object 3-groupoids that are strict in all respects, except that the object has only weak identity arrows, can model all connected, simply connected homotopy 3-types. The pro...

We show that every braided monoidal category arises as End(I) for a weak unit I in an otherwise completely strict monoidal 2-category. This implies a version of Simpson’s weak-unit conjecture in dimension 3, namely that one-object 3-groupoids that are strict in all respects, except that the object has only weak identity arrows, can model all connected, simply connected homotopy 3-types. The pro...

We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. Also, the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, W -spaces, orth...

We show that the category of Poisson manifolds and Poisson maps, the category of symplectic microgroupoids and lagrangian submicrogroupoids (as morphisms), and the category of monoids and monoid morphisms in the microsymplectic category are equivalent symmetric monoidal categories.

Lyubashenko has described enriched 2–categories as categories enriched over V–Cat, the 2–category of categories enriched over a symmetric monoidal V. This construction is the strict analogue for V–functors in V–Cat of Brian Day’s probicategories for V–modules in V–Mod. Here I generalize the strict version to enriched n–categories for k–fold monoidal V. The latter is defined as by Balteanu, Fied...

Abstract Because an exact pairing between an object and its dual is extraordinarily natural in the object, ideas of the paper [St4] apply to yield a definition of dualization for a pseudomonoid in any autonomous monoidal bicategory as base; this is an improvement on [DS; Definition 11, page 114]. We analyse the dualization notion in depth. An example is the concept of autonomous (which, usually...

In this paper we develops a categorical theory of relations and use this formulation to define the notion of quantization for relations. Categories of relations are defined in the context of symmetric monoidal categories. They are shown to be symmetric monoidal categories in their own right and are found to be isomorphic to certain categories of A−A bicomodules. Properties of relations are defi...

This paper extends the Day Re ection Theorem to skew monoidal categories. We also provide conditions under which a skew monoidal structure can be lifted to the category of Eilenberg-Moore algebras for a comonad.

The notion of proof-net category defined in this paper is closely related to graphs implicit in proof nets for the multiplicative fragment without constant propositions of linear logic. Analogous graphs occur in Kelly’s and Mac Lane’s coherence theorem for symmetric monoidal closed categories. A coherence theorem with respect to these graphs is proved for proof-net categories. Such a coherence ...