The classical infinite loopspace machines in fact induce an equivalence of categories between a localization of the category of symmetric monoidal categories and the stable homotopy category of -1-connective spectra. Introduction Since the early seventies it has been known that the classifying spaces of small symmetric monoidal categories are infinite loop spaces, the zeroth space in a spectrum...

We introduce the notion of pseudo-commutative monad together with that of pseudoclosed 2-category, the leading example being given by the 2-monad on Cat whose 2-category of algebras is the 2-category of small symmetric monoidal categories. We prove that for any pseudo-commutative 2-monad on Cat, its 2-category of algebras is pseudo-closed. We also introduce supplementary definitions and results...

Abstract. A compact closed bicategory is a symmetric monoidal bicategory where every object is equipped with a weak dual. The unit and counit satisfy the usual “zig-zag” identities of a compact closed category only up to natural isomorphism, and the isomorphism is subject to a coherence law. We give several examples of compact closed bicategories, then review previous work. In particular, Day a...

Given an algebraic theory which can be described by a (possibly symmetric) operad P , we propose a definition of the weakening (or categorification) of the theory, in which equations that hold strictly for P -algebras hold only up to coherent isomorphism. This generalizes the theories of monoidal categories and symmetric monoidal categories, and several related notions defined in the literature...

Let M be a bicomplete, closed symmetric monoidal category. Let P be an operad in M, i.e., a monoid in the category of symmetric sequences of objects in M, with its composition monoidal structure. Let R be a P-co-ring, i.e., a comonoid in the category of P-bimodules. The co-ring R induces a natural “fattening” of the category of P-(co)algebras, expanding the morphism sets while leaving the objec...

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 introduce the notion of strong concatenable process for Petri nets as the least refinement of non-sequential (concatenable) processes which can be expressed abstractly by means of a functor Q[ ] from the category of Petri nets to an appropriate category of symmetric strict monoidal categories with free non-commutative monoids of objects, in the precise sense that, for each net N , the strong...

We give an axiomatic account of what structure on a category C and an endofunctor H on C yield similar structure on the category H Coalg of H-coalgebras. We give conditions under which completeness, cocompleteness, symmetric monoidal closed structure, local presentability, and subobject classi ers lift. Our proof of the latter uses a general result about the existence of a subobject classi er i...

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...

We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs). These are defined on open-closed cobordisms by which we mean smooth compact oriented 2-manifolds with corners that have a particular global structure in order to model the smooth topology of open and closed string worldsheets. We show that the category of open-closed TQFTs is equivalent to the category...