We define a notion of symmetric monoidal closed (smc) theory, consisting of a smc signature augmented with equations, and describe the classifying categories of such theories in terms of proof nets.

We show that any pivotal Hopf monoid $H$ in a symmetric monoidal category $\mathcal{C}$ gives rise to actions of mapping class groups oriented surfaces genus $g \geq 1$ with $n boundary components. These group are given by homomorphisms into the automorphisms certain Yetter-Drinfeld modules over $H$. They associated edge slides embedded ribbon graphs generalise chord diagrams. give concrete des...

The suspension loop construction is used to de ne a process in a symmetric monoidal category The algebra of such processes is that of symmetric monoidal bicategories Processes in categories with prod ucts and in categories with sums are studied in detail and in both cases the resulting bicategories of processes are equipped with opera tions called feedback Appropriate versions of traced monoida...

Bicat is the tricategory of bicategories, homomorphisms, pseudonatural transformations, and modifications. Gray is the subtricategory of 2-categories, 2functors, pseudonatural transformations, and modifications. We show that these two tricategories are not triequivalent. 1. Background. Weakening the notion of 2-category by replacing all equations between 1-cells by suitably coherent isomorphism...

The concatenable processes of a Petri net N can be characterized abstractly as the arrows of a symmetric monoidal category ~(N). However, this is only a partial axiomatization, since it is based on a concrete, ad hoe chosen, category of symmetries Sym N. In this paper we give a completely abstract characterization of the category of concatenable processes of N, thus yielding an axiomatic theory...

We define higher Frobenius-Schur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a k-linear semisimple rigid monoidal category, which we call the Frobenius-Schur endomorphisms. For a k-linear semisimple pivotal mono...

Main contribution of this paper is an investigation of expressive power of the database category DB. An object in this category is a database-instance (set of n-ary relations). Morphisms are not functions but have complex tree structures based on a set of complex query computations. They express the semantics of view-based mappings between databases. The higher (logical) level scheme mappings b...

we give the concept of ‘braiding’ for 2-groupoids, and we show that this structure is equivalent tobraided regular, crossed modules.

Since the early 1990's there have been several useful symmetric monoidal model structures on the underlying category of a point set category of spectra. Topological Hochschild homology (THH) is constructed from smash powers of ring-spectra, that is, monoids in the category of spectra. Already in the 1980's such a construction was made. Building on ideas of Goodwillie and Waldhausen, Bökstedt in...