The category of all binary relations between arbitrary sets turns out to be a certain symmetric monoidal category Rel with an additional structure characterized by a family d = (dA : A → A⊗A | A ∈ |Rel|) of diagonal morphisms, a family t = (tA : A → I | A ∈ |Rel|) of terminal morphisms, and a family∇ = (∇A : A⊗A → A | A ∈ |Rel|) of diagonal inversions having certain properties. Using this prope...

One fundamental aspect of linear logic is that its conjunction behaves in the same way as a tensor product in linear algebra. Guided by this intuition, we investigate the algebraic status of disjunction – the dual of conjunction – in the presence of linear continuations. We start from the observation that every monoidal category equipped with a tensorial negation inherits a lax monoidal structu...

String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to vertices at both ends, and these unconnected ends can be interpreted as the inputs and outputs of a diagram. In this paper, we give a concrete construction for str...

It is proved that MacLane’s coherence results for monoidal and symmetric monoidal categories can be extended to some other categories with multiplication; namely, to relevant, affine and cartesian categories. All results are formulated in terms of natural transformations equipped with “graphs” (g-natural transformations) and corresponding morphism theorems are given as consequences. Using these...

This paper centres around two dagger compact categories and the relationship between them. The first is a category whose morphisms are electrical circuits comprising resistors, inductors, and capacitors, with marked input and output terminals. To understand this category, we begin by recalling Ohm’s law and Kirchhoff’s laws, and show that we may also understand these circuits through a formal m...

A category with biproducts is enriched over (commutative) additive monoids. A category with tensor products is enriched over scalar multiplication actions. A symmetric monoidal category with biproducts is enriched over semimodules. We show that these extensions of enrichment (e.g. from hom-sets to homsemimodules) are functorial, and use them to make precise the intuition that “compact objects a...

We discuss an example of a triangulated Hopf category related to SL(2). It is an equivariant derived category equipped with multiplication and comultiplication functors and structure isomorphisms. We prove some coherence equations for structure isomorphisms. In particular, the Hopf category is monoidal.

For an arbitrary symmetric monoidal∞-category V, we define the factorization homology of V-enriched∞-categories over (possibly stratified) 1-manifolds and study its basic properties. In the case that V is cartesian symmetric monoidal, by considering the circle and its self-covering maps we obtain a notion of unstable topological cyclic homology, which we endow with an unstable cyclotomic trace ...

We discuss an example of a triangulated Hopf category related to SL(2). It is an equivariant derived category equipped with multiplication and comultiplication functors and structure isomorphisms. We prove some coherence equations for structure isomorphisms. In particular, the Hopf category is monoidal.

This paper is an extended version of my talk given in Zürich during the Conference “Quantization and Geometry”, March 2-6, 2009. The main results are the following. 1. We construct a 2-fold monoidal structure [BFSV] on the category Tetra(A) of tetramodules (also known as Hopf bimodules) over an associative bialgebraA. According to an earlier result of R.Taillefer [Tai1,2], Ext q Tetra(A)(A,A) i...