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View article: Colored Petri Nets are Monoidal Double Functors
Colored Petri Nets are Monoidal Double Functors Open
We give a characterization of colored Petri nets as monoidal double functors. Framing colored Petri nets in terms of category theory allows for canonical definitions of various well-known constructions on colored Petri nets. In particular,…
View article: Relative fixed points of functors
Relative fixed points of functors Open
We show how the relatively initial or relatively terminal fixed points for a well-behaved functor $F$ form a pair of adjoint functors between $F$-coalgebras and $F$-algebras. We use the language of locally presentable categories to find su…
View article: Relative fixed points of functors
Relative fixed points of functors Open
We show how the relatively initial or relatively terminal fixed points for a well-behaved functor $F$ form a pair of adjoint functors between $F$-coalgebras and $F$-algebras. We use the language of locally presentable categories to find su…
View article: Proceedings Fifth International Conference on Applied Category Theory
Proceedings Fifth International Conference on Applied Category Theory Open
The Fifth International Conference on Applied Category Theory took place at the University of Strathclyde in Glasgow, Scotland on 18-22 July 2022. This conference follows the previous meetings at Leiden (2018), Oxford (2019), MIT (2020, fu…
View article: Beyond Initial Algebras and Final Coalgebras
Beyond Initial Algebras and Final Coalgebras Open
We provide a construction of the fixed points of functors which may not be inital algebras or final coalgebras. For an endofunctor F, this fixed point construction may be expressed as a pair of adjoint functors between F-coalgebras and F-a…
View article: How to Compose Shortest Paths
How to Compose Shortest Paths Open
The composition problem for shortest paths asks the following: given shortest paths on weighted graphs M and N which share a common boundary, find the shortest paths on their union. This problem is a crucial step in any algorithm which use…
View article: Composing Behaviors of Networks
Composing Behaviors of Networks Open
This thesis aims to develop a compositional theory for the operational semantics of networks. The networks considered are described by either internal or enriched graphs. In the internal case we focus on $\mathsf{Q}$-nets, a generalization…
View article: Categories of Nets
Categories of Nets Open
We present a unified framework for Petri nets and various variants, such as pre-nets and Kock's whole-grain Petri nets. Our framework is based on a less well-studied notion that we call $Σ$-nets, which allow finer control over whether toke…
View article: Categories of Nets
Categories of Nets Open
We present a unified framework for Petri nets and various variants, such as pre-nets and Kock's whole-grain Petri nets. Our framework is based on a less well-studied notion that we call $\Sigma$-nets, which allow finer control over whether…
View article: The Open Algebraic Path Problem
The Open Algebraic Path Problem Open
The algebraic path problem provides a general setting for shortest path algorithms in optimization and computer science. We explain the universal property of solutions to the algebraic path problem by constructing a left adjoint functor wh…
View article: Open Petri nets
Open Petri nets Open
The reachability semantics for Petri nets can be studied using open Petri nets. For us, an “open” Petri net is one with certain places designated as inputs and outputs via a cospan of sets. We can compose open Petri nets by gluing the outp…
View article: Why is Homology so Powerful?
Why is Homology so Powerful? Open
My short answer to this question is that homology is powerful because it computes invariants of higher categories. In this article we show how this true by taking a leisurely tour of the connection between category theory and homological a…
View article: Generalized Petri Nets
Generalized Petri Nets Open
We give a definition of $\mathsf{Q}$-$\mathsf{Net}$, a generalization of Petri nets based on a Lawvere theory $\mathsf{Q}$ for which many existing variants of Petri nets are a special case. This definition is functorial with respect to cha…
View article: Translating and Evolving: Towards a Model of Language Change in DisCoCat
Translating and Evolving: Towards a Model of Language Change in DisCoCat Open
The categorical compositional distributional (DisCoCat) model of meaning developed by Coecke et al. (2010) has been successful in modeling various aspects of meaning. However, it fails to model the fact that language can change. We give an…