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Structured cospans

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John C. Baez and Kenny Courser

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: A -> X, a structured cospan is a diagram in X of the form L(a) -> x <- L(b). If A and X have finite colimits and L is a left adjoint, we obtain a symmetric monoidal category whose objects are those of A and whose morphisms are isomorphism classes of structured cospans. This is a hypergraph category. However, it arises from a more fundamental structure: a symmetric monoidal double category where the horizontal 1-cells are structured cospans. We show how structured cospans solve certain problems in the closely related formalism of `decorated cospans', and explain how they work in some examples: electrical circuits, Petri nets, and
chemical reaction networks.

Keywords:
bicategory, cospan, double category, monoidal category, network

2020 MSC:
18B10, 18M35, 18N10

*Theory and Applications of Categories,*
Vol. 35, 2020,
No. 48, pp 1771-1822.

Published 2020-10-29.

http://www.tac.mta.ca/tac/volumes/35/48/35-48.pdf

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