Related papers: Combs, Causality and Contractions in Atomic Markov…
A new, self-contained, proof of a coherence result for categories equipped with two symmetric monoidal structures bridged by a natural transformation is given. It is shown that this coherence result is sufficient for…
A monoidal category has a natural isomorphism $\alpha_{A,B,C}\colon(A\otimes B)\otimes C\to A\times (B\otimes C)$ called the associator. In the case where the objects $(A\otimes B)\otimes C$ and $A\otimes(B\otimes C)$ are equal, it is…
We present a survey of characterisations of cocartesian categories in terms of monoidal categories - and, more generally, magmal categories - satisfying additional properties. In particular, we show that the following are equivalent for a…
A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This…
We consider the category of C*-algebras equipped with actions of a locally compact quantum group. We show that this category admits a monoidal structure satisfying certain natural conditions if and only if the group is quasitriangular. The…
This article studies the categorical setting of Abramsky, Haghverdi, and Scott's untyped linear combinatory algebras, and relates this to more recent work of Abramsky and Heunen on Frobenius algebras in the infinitary setting. The key to…
Markov categories are a recent categorical approach to the mathematical foundations of probability and statistics. Here, this approach is advanced by stating and proving equivalent conditions for second-order stochastic dominance, a widely…
Building on work of Chen, we give a universal property of the Markov category BorelStoch of standard Borel spaces and Markov kernels between them. To do this, we introduce a new notion of *coinflip*, or unbiased binary choice, in a Markov…
In this expository paper, we discuss and compare the notions of braided and coboundary monoidal categories. Coboundary monoidal categories are analogues of braided monoidal categories in which the role of the braid group is replaced by the…
A traced monad is a monad on a traced symmetric monoidal category that lifts the traced symmetric monoidal structure to its Eilenberg-Moore category. A long-standing question has been to provide a characterization of traced monads without…
Categorical probability has recently seen significant advances through the formalism of Markov categories, within which several classical theorems have been proven in entirely abstract categorical terms. Closely related to Markov categories…
In categorical realizability, it is common to construct categories of assemblies and categories of modest sets from applicative structures. These categories have structures corresponding to the structures of applicative structures. In the…
Introduced in the 1990s in the context of the algebraic approach to graph rewriting, gs-monoidal categories are symmetric monoidal categories where each object is equipped with the structure of a commutative comonoid. They arise for example…
The present work develops a construction of a CD category of partial kernels from a particular type of Markov category called a partializable Markov category. These are a generalization of earlier models of categories of partial morphisms…
We show that the category of optics in a monoidal category arises naturally from the free cornering of that category. Further, we show that the free cornering of a monoidal category is a natural setting in which to work with comb diagrams…
Let T be a triangulated category with coproducts, C the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [Adams71]: All contravariant homological functors C --> Ab are the…
Presentations of categories are a well-known algebraic tool to provide descriptions of categories by means of generators, for objects and morphisms, and relations on morphisms. We generalize here this notion, in order to consider situations…
Monoidal closed categories naturally model NMILL, non-commutative multiplicative intuitionistic linear logic: the monoidal unit and tensor interpret the multiplicative verum and conjunction; the internal hom interprets linear implication.…
A premonoidal category is equipped only with a bifunctor and a natural isomorphism for associativity. We introduce a (deformation) natural automorphism representing the deviation from the Pentagon condition. We uncover a binary tree…
We define a traced pseudomonoid as a pseudomonoid in a monoidal bicategory equipped with extra structure, giving a new characterisation of Cauchy complete traced monoidal categories as algebraic structures in $\mathbf{Prof}$, the monoidal…