相关论文: Wiring diagrams as normal forms for computing in s…
We construct generalized multicategories associated to an arbitrary operad in Cat that is $\Sigma$-free. The construction generalizes the passage to symmetric multicategories from permutative categories, which is the case when the operad is…
I describe a generalization of the notion of operadic category due to Batanin and Markl. For each such operadic category I describe a skew monoidal category of collections, such that a monoid in this skew monoidal category is precisely an…
We show that every braiding on a monoidal bicategory induces a monoidal structure on its bicategory of monoids, such that if the former is sylleptic or symmetric then the latter is braided or symmetric, respectively. This extends a classic…
In this paper, we extend diagrammatic reasoning in monoidal categories with algebraic operations and equations. We achieve this by considering monoidal categories that are enriched in the category of Eilenberg-Moore algebras for a monad.…
Directed wiring diagrams can be used as a composition pattern for composing input/output systems such as Moore machines. In a Moore machine, the input parametrizes an internal state and the internal state defines the output. Because the…
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 demonstrate how category theory provides specifications that can efficiently be implemented via imperative algorithms and apply this to the field of graph rewriting. By examples, we show how this paradigm of software development makes it…
Actions of monoidal categories on categories, also known as actegories, have been familiar to category theorists for a long time, and yet a comprehensive overview of this topic seems to be missing from the literature. Recently, actegories…
We formalize the simulation paradigm of cryptography in terms of category theory and show that protocols secure against abstract attacks form a symmetric monoidal category, thus giving an abstract model of composable security definitions in…
One goal of applied category theory is to better understand networks appearing throughout science and engineering. Here we introduce "structured cospans" as a way to study networks with inputs and outputs. Given a functor $L \colon…
The unprecedented pace of machine learning research has lead to incredible advances, but also poses hard challenges. At present, the field lacks strong theoretical underpinnings, and many important achievements stem from ad hoc design…
Category and multicategory theory provide abstract frameworks for describing structures and their compositions, with multicategories extending traditional categories to handle multi-input operations. These theories enable modular reasoning…
In this paper we adapt previous work on rewriting string diagrams using hypergraphs to the case where the underlying category has a traced comonoid structure, in which wires can be forked and the outputs of a morphism can be connected to…
We propose a new model for multicategories with symmetries with respect to Zhang's group operads. The fully faithful embedding of the category of group operads into that of crossed interval groups is made use of, and it is shown that every…
A diagram $\mathcal{D} = (G, l)$ over a monoid $M$ is an oriented graph $G = (V, E)$ endowed with a labeling $l\colon E \to M$. A diagram is commutative if and only if for any two oriented paths with the same endpoints, the products in $M$…
Stock and flow diagrams are already an important tool in epidemiology, but category theory lets us go further and treat these diagrams as mathematical entities in their own right. In this chapter we use communicable disease models created…
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…
Monoidal computer is a categorical model of intensional computation, where many different programs correspond to the same input-output behavior. The upshot of yet another model of computation is that a categorical formalism should provide a…
We enhance the calculus of string diagrams for monoidal categories with hierarchical features in order to capture closed monoidal (and cartesian closed) structure. Using this new syntax we formulate an automatic differentiation algorithm…
The development of an IT strategy and ensuring that it is the best possible one for business is a key problem many organizations face. This problem is that of linking business architecture to IT architecture in general and application…