Related papers: String diagrams for Strictification and Coherence
Path calculus, or graphical linear algebra, is a string diagram calculus for the category of matrices over a base ring. It is the usual string diagram calculus for a symmetric monoidal category, where the monoidal product is the direct sum…
Twisted diagrams are "diagrams" with components in different categories. Structure maps are defined using auxiliary data which consists of functors relating the various categories to each other. Prime examples of the construction are…
We relate the computational complexity of finite strings to universal representations of their underlying symmetries. First, Boolean functions are classified using the universal covering topologies of the circuits which enumerate them. A…
In previous work, we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases…
We study the monoidal structure of the standard strictification functor $\textrm{st}:\mathbf{Bicat} \rightarrow \mathbf{2Cat}$. In doing so, we construct monoidal structures on the 2-category whose objects are bicategories and on the…
Sequence diagrams are a popular technique for describing interactions between software entities. However, because the OMG group's UML standard is not based on a rigorous mathematical structure, it is impossible to deduce a single…
We extend McCarthy's stabilization construction to exact $\infty$-categories. This is achieved by constructing, for any functor from exact $\infty$-categories to a fixed stable $\infty$-category $\mathcal{A}$, a coherent chain complex in…
We consider a class of conformal models describing closed strings in axially symmetric stationary magnetic flux tube backgrounds. These models are closed string analogs of the Landau model of a particle in a magnetic field or the model of…
Grammar inference deals with determining (preferable simple) models/grammars consistent with a set of observations. There is a large body of research on grammar inference within the theory of formal languages. However, there is surprisingly…
The geometric and algebraic properties of Gray categories with duals are investigated by means of a diagrammatic calculus. The diagrams are three-dimensional stratifications of a cube, with regions, surfaces, lines and vertices labelled by…
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…
Morphisms in a monoidal category are usually interpreted as processes, and graphically depicted as square boxes. In practice, we are faced with the problem of interpreting what non-square boxes ought to represent in terms of the monoidal…
We introduce a notion of compatibility between constraint encoding and compositional structure. Phrased in the language of category theory, it is given by a "composable constraint encoding". We show that every composable constraint encoding…
The goal of this paper is to prove coherence results with respect to relational graphs for monoidal endofunctors, i.e. endofunctors of a monoidal category that preserve the monoidal structure up to a natural transformation that need not be…
Motivated by traces of matrices and Euler characteristics of topological spaces, we expect abstract traces in a symmetric monoidal category to be "additive". When the category is "stable" in some sense, additivity along cofiber sequences is…
Chordal graphs are important in algorithmic graph theory. Chordal digraphs are a digraph analogue of chordal graphs and have been a subject of active studies recently. Unlike chordal graphs, chordal digraphs lack many structural properties…
In a labeling scheme the vertices of a given graph from a particular class are assigned short labels such that adjacency can be algorithmically determined from these labels. A representation of a graph from that class is given by the set of…
We develop a comprehensive string diagrammatic treatment of electrical circuits. Building on previous, limited case studies, we introduce controlled sources and meters as elements, and the impedance calculus, a powerful toolbox for…
I introduce a formalism for representing the syntax of recursively structured graph-like patterns. It does not use production rules, like a conventional graph grammar, but represents the syntactic structure in a more direct and declarative…
We introduce the notion of coherent graphs, and show how those can be used to define dynamic semantics for Multiplicative Linear Logic (MLL) extended with non-determinism. Thanks to the use of a coherence relation rather than mere formal…