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Lucius Gregory Meredith (Oct 04 2020 at 22:27):Just a quick check: is the category of graphs and graph homomorphisms monoidal closed?
Notification Bot (Oct 04 2020 at 22:36):This topic was moved here from #general > bicompletions by Nathanael Arkor
Nathanael Arkor (Oct 04 2020 at 22:37):You need to specify what you mean by "graph", since there are multiple possible definitions (e.g. undirected/directed, single/multiple edges, loops or not, etc.).
Nathanael Arkor (Oct 04 2020 at 22:40):For undirected graphs without multiple edges or loops, the tensor product of graphs is the cartesian product, which has a right adjoint given by the construction of exponential graphs.
John Baez (Oct 04 2020 at 23:12):There are tons of categories of graphs, each with tons of monoidal structures. In fact there's a Wikipedia page that seems to list 10 monoidal structures on the category of simple graphs:
John Baez (Oct 04 2020 at 23:13):though I haven't checked that they're all monoidal structures, and the last one could easily not be.
John Baez (Oct 04 2020 at 23:14):If we take graph to mean quiver, which is what category theorists usually do, then the category of graphs is a presheaf category, hence a topos, hence cartesian closed.
Dan Doel (Oct 04 2020 at 23:29):The quiver link seems to be the graph product link.
John Baez (Oct 04 2020 at 23:33):Whoops. I meant Wikipedia article
Quiver (mathematics)
John Baez (Oct 04 2020 at 23:54):Fixed.
Mike Stay (Oct 04 2020 at 23:55):Ronnie Brown et al describe several different monoidal closed categories of graphs:
https://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1a1/pdf
Mike Stay (Oct 05 2020 at 00:17):While there are tons of monoidal structures on graphs, I don't think many of them give rise to closed categories.
John Baez (Oct 05 2020 at 04:13):Since people know all the symmetric monoidal closed structures on the category Cat (there are just two, up to isomorphism) maybe they know them all - or someone can figure out all of them - on the category Graph. (Let's take this to be the category theorist's favorite category of graphs: the category of quivers.)