Related papers: Cartesian Gray-Monoidal Double Categories
We introduce a graphical language for closed symmetric monoidal categories based on an extension of string diagrams with special bracket wires representing internal homs. These bracket wires make the structure of the internal hom functor…
We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives…
We introduce the notion of symplectic microfolds and symplectic micromorphisms between them. They form a monoidal category, which is a version of the "category" of symplectic manifolds and canonical relations obtained by localizing them…
In this chapter we survey some particular topics in category theory in a somewhat unconventional manner. Our main focus will be on monoidal categories, mostly symmetric ones, for which we propose a physical interpretation. These are…
We use a theory of colax Reedy diagrams to show that the category of Segal M-precategories with fixed set of objects has a model structure for a symmetric monoidal model category M = (M,\otimes,I). What is relevant here is when M is…
We present an efficient and user-friendly method for constructing any cofibrantly generated model structure on the category of double categories whose trivial fibrations are the "canonical" ones: the double functors which are surjective on…
Cartesian differential categories come equipped with a differential combinator that formalizes the derivative from multi-variable differential calculus, and also provide the categorical semantics of the differential $\lambda$-calculus. An…
The notion of cartesian bicategory, introduced by Carboni and Walters for locally ordered bicategories, is extended to general bicategories. It is shown that a cartesian bicategory is a symmetric monoidal bicategory.
Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of the desired objects, but also…
In this paper we propose, firstly, a categorification of virtual braid groups and groupoids in terms of "locally" braided objects in a symmetric category (SC), and, secondly, a definition of self-distributive structures (SDS) in an…
Cartesian reverse differential categories (CRDCs) are a recently defined structure which categorically model the reverse differentiation operations used in supervised learning. Here we define a related structure called a monoidal reverse…
Structured and decorated cospans are broadly applicable frameworks for building bicategories or double categories of open systems. We streamline and generalize these frameworks using central concepts of double category theory. We show that,…
The main objective of this paper is to construct a symmetric monoidal closed model category of coherently commutative monoidal quasi-categories. We construct another model category structure whose fibrant objects are (essentially) those…
We construct a compact closed category out of any symmetric monoidal category by freely adding adjoints to its objects. The morphisms of the completion are defined as string diagrams annotated by objects and morphisms from the original…
We define a skew-closed structure for Gray-categories extending the mapping space construction of the author.
Recently, symmetric categorical groups are used for the study of the Brauer groups of symmetric monoidal categories. As a part of these efforts, some algebraic structures of the 2-category of symmetric categorical groups $\mathrm{SCG}$ are…
We describe a class of examples of braided monoidal categories which are built from Hopf algebras in symmetric categories. The construction is motivated by a calculation in two-dimensional conformal field theory and is tailored to contain…
Applied category theory often studies symmetric monoidal categories (SMCs) whose morphisms represent open systems. These structures naturally accommodate complex wiring patterns, leveraging (co)monoidal structures for splitting and merging…
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…
We define a monoidal semantics for algebraic theories. The basis for the definition is provided by the analysis of the structural rules in the term calculus of algebraic languages. Models are described both explicitly, in a form that…