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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…
We show that every action operad gives rise to a notion of monoidal category via the categorical version of the Borel construction, embedding action operads into the category of 2-monads on $\mathbf{Cat}$. We characterize those 2-monads in…
Most existing pre-trained language models for source code focus on learning the static code text, typically augmented with static code structures (abstract syntax tree, dependency graphs, etc.). However, program semantics will not be fully…
The basic data for a skew-monoidal category are the same as for a monoidal category, except that the constraint morphisms are no longer required to be invertible. The constraints are given a specific orientation and satisfy Mac Lane's five…
Critical pair analysis provides a convenient and computable criterion of confluence, which is a fundamental property in rewriting theory, for a wide variety of rewriting systems. Bonchi et al. showed validity of critical pair analysis for…
Kan extensions provide a natural general framework for a variety of combinatorial problems. We have developed rewriting procedures for Kan extensions (over the category of sets) and this enables one program to address a wide range of…
Rewriting systems on words are very useful in the study of monoids. In good cases, they give finite presentations of the monoids, allowing their manipulation by a computer. Even better, when the presentation is confluent and terminating,…
In functional programming, monads are supposed to encapsulate computations, effectfully producing the final result, but keeping to themselves the means of acquiring it. For various reasons, we sometimes want to reveal the internals of a…
We describe a generic construction of non-wellfounded syntax involving variable binding and its monadic substitution operation. Our construction of the syntax and its substitution takes place in category theory, notably by using monoidal…
Indexed symmetric monoidal categories are an important refinement of bicategories -- this structure underlies several familiar bicategories, including the homotopy bicategory of parametrized spectra, and its equivariant and fiberwise…
Many techniques in computer vision, machine learning, and statistics rely on the fact that a signal of interest admits a sparse representation over some dictionary. Dictionaries are either available analytically, or can be learned from a…
In this report, we present a new programming model based on Pipelines and Operators, which are the building blocks of programs written in PiCo, a DSL for Data Analytics Pipelines. In the model we propose, we use the term Pipeline to denote…
Transformers have achieved great success in several domains, including Natural Language Processing and Computer Vision. However, its application to real-world graphs is less explored, mainly due to its high computation cost and its poor…
Given a graph $G=(V,E)$, a set of vertices $D\subseteq V $ is called a dominating set if every vertex in $V\backslash D$ is adjacent to a vertex in $D$, and a subset $B\subseteq V $ is called a nonblocking set if $V-B$ is a dominating set.…
A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We…
We suggest that the canonical parallel operation of processes is composition in a well-supported compact closed category of spans of reflexive graphs. We present the parallel operations of classical process algebras as derived operations…
Morphisms are homomorphisms under the concatenation operation of the set of words over a finite set. Changing the elements of the finite set does not essentially change the morphism. We propose a way to select a unique representing member…
Learning to compute, the ability to model the functional behavior of a circuit graph, is a fundamental challenge for graph representation learning. Yet, the dominant paradigm is architecturally mismatched for this task. This flawed…
Monadic programming presents a significant challenge for many programmers. In light of category theory, we offer a new perspective on the use of monads in functional programming. This perspective is clarified through numerous examples coded…
We study two kinds of categorical traces of (monoidal) dg categories, with particular interest in categories of Soergel bimodules. First, we explicitly compute the usual Hochschild homology, or derived vertical trace, of the category of…