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.

Abstract. This paper contains the construction, examples and properties of a trace and a trace pairing for certain morphisms in a monoidal category with switching isomorphisms. Our construction of the categorical trace is a common generalization of the trace for endomorphisms of dualizable objects in a balanced monoidal category and the trace of nuclear operators on a topological vector space w...

The category of Hopf monoids over an arbitrary symmetric monoidal category as well as its subcategories of commutative and cocommutative objects respectively are studied, where attention is paid in particular to the following questions: (a) When are the canonical forgetful functors of these categories into the categories of monoids and comonoids respectively part of an adjunction? (b) When are ...

Given a horizontal monoid M in a duoidal category F , we examine the relationship between bimonoid structures on M and monoidal structures on the category F ∗M of right M -modules which lift the vertical monoidal structure of F . We obtain our result using a variant of the so-called Tannaka adjunction; that is, an adjunction inducing the equivalence which expresses Tannaka duality. The approach...

For a symmetric monoidal-closed category V and a suitable monad T on the category of sets, we introduce the notion of reflexive and transitive (T,V)-algebra and show that various old and new structures are instances of such algebras. Lawvere’s presentation of a metric space as a V-category is included in our setting, via the Betti-Carboni-Street-Walters interpretation of a V-category as a monad...

A 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we deene a 2-Hilbert space to be an abelian category enriched over Hilb with a-structure, conjugate-linear on the hom-sets, satisfying hf g; hi = hg; f hi = hf; hg i. We also deene monoidal, braided monoidal, and symmetric monoidal versions of 2-Hilbert spaces, which we call 2-H...

LetM be a tensor category with coefficients in a fieldK of characteristic 0, that is, a K-linear pseudo-abelian symmetric monoidal category such that the tensor product ⊗ of M is bilinear. Then symmetric and exterior powers of an object M ∈ M make sense, by using appropriate projectors relative to the action of the symmetric groups on tensor powers of M . One may therefore introduce the zeta fu...