{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T13:39:22Z","timestamp":1762522762576},"reference-count":48,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2022,9,1]],"date-time":"2022-09-01T00:00:00Z","timestamp":1661990400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Compositionality"],"abstract":"One goal of applied category theory is to understand open systems. We compare two ways of describing open systems as cospans equipped with extra data. First, given a functor L<\/mml:mi>:<\/mml:mo>A<\/mml:mi><\/mml:mrow>\u2192<\/mml:mo>X<\/mml:mi><\/mml:mrow><\/mml:math>, a `structured cospan' is a diagram in X<\/mml:mi><\/mml:mrow><\/mml:math> of the form L<\/mml:mi>(<\/mml:mo>a<\/mml:mi>)<\/mml:mo>\u2192<\/mml:mo>x<\/mml:mi>\u2190<\/mml:mo>L<\/mml:mi>(<\/mml:mo>b<\/mml:mi>)<\/mml:mo><\/mml:math>. We give a new proof that if A<\/mml:mi><\/mml:mrow><\/mml:math> and X<\/mml:mi><\/mml:mrow><\/mml:math> have finite colimits and L<\/mml:mi><\/mml:math> preserves them, there is a symmetric monoidal double category whose objects are those of A<\/mml:mi><\/mml:mrow><\/mml:math> and whose horizontal 1-cells are structured cospans. Second, given a pseudofunctor F<\/mml:mi>:<\/mml:mo>A<\/mml:mi><\/mml:mrow>\u2192<\/mml:mo>C<\/mml:mi>a<\/mml:mi>t<\/mml:mi><\/mml:mrow><\/mml:math>, a `decorated cospan' is a diagram in A<\/mml:mi><\/mml:mrow><\/mml:math> of the form a<\/mml:mi>\u2192<\/mml:mo>m<\/mml:mi>\u2190<\/mml:mo>b<\/mml:mi><\/mml:math> together with an object of F<\/mml:mi>(<\/mml:mo>m<\/mml:mi>)<\/mml:mo><\/mml:math>. Generalizing the work of Fong, we show that if A<\/mml:mi><\/mml:mrow><\/mml:math> has finite colimits and F<\/mml:mi>:<\/mml:mo>(<\/mml:mo>A<\/mml:mi><\/mml:mrow>,<\/mml:mo>+<\/mml:mo>)<\/mml:mo>\u2192<\/mml:mo>(<\/mml:mo>C<\/mml:mi>a<\/mml:mi>t<\/mml:mi><\/mml:mrow>,<\/mml:mo>\u00d7<\/mml:mo>)<\/mml:mo><\/mml:math> is symmetric lax monoidal, there is a symmetric monoidal double category whose objects are those of A<\/mml:mi><\/mml:mrow><\/mml:math> and whose horizontal 1-cells are decorated cospans. We prove that under certain conditions, these two constructions become isomorphic when we take X<\/mml:mi><\/mml:mrow>=<\/mml:mo>\u222b<\/mml:mo>F<\/mml:mi><\/mml:math> to be the Grothendieck category of F<\/mml:mi><\/mml:math>. We illustrate these ideas with applications to electrical circuits, Petri nets, dynamical systems and epidemiological modeling.<\/jats:p>","DOI":"10.32408\/compositionality-4-3","type":"journal-article","created":{"date-parts":[[2022,9,1]],"date-time":"2022-09-01T20:52:43Z","timestamp":1662065563000},"page":"3","source":"Crossref","is-referenced-by-count":6,"title":["Structured versus Decorated Cospans"],"prefix":"10.46298","volume":"4","author":[{"given":"John C.","family":"Baez","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of California, Riverside CA, USA 92521"},{"name":"Centre for Quantum Technologies, National University of Singapore, Singapore 117543"}]},{"given":"Kenny","family":"Courser","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of California, Riverside CA, USA 92521"}]},{"given":"Christina","family":"Vasilakopoulou","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Patras, Greece 265 04"}]}],"member":"25203","published-online":{"date-parts":[[2022,9,1]]},"reference":[{"key":"0","unstructured":"AlgebraicPetri team, GitHub repository. https:\/\/github.com\/AlgebraicJulia\/AlgebraicPetri.jl."},{"key":"1","doi-asserted-by":"publisher","unstructured":"A. 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