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I added some examples of virtual double categories that do not have composites described in Crutwell-Shulman.
@Mike Shulman, on this page you said virtual double categories can be defined as
a generalized multicategory, in the sense of Hermida, Cruttwell-Shulman, and others, relative to the monad T on graphs-internal-to-Cat whose algebras are double categories
did you mean “categories internal to Grph” rather than “graphs internal to Cat”? That seems to be how y’all defined it in Cruttwell-Shulman. I have a suspicion they turn out to be the same thing, but when you define the monad it is easier to think of it as categories internal to Grph because the monad is a lifting of the free category monad on Grph.
They are the same thing. I find it easier to think of graphs internal to Cat, because then the monad is just the special case of the free internal-category monad in the topos of graphs.
Hm, I tried to use tikzcd but it is not rendering properly. Am I doing something very wrong?
You need to remove the enclosing \[
\]
.
(This is an issue to be handed to an ambitious sysadmin as soon as we have a new one: So far the tikzcd
-support here is completely alien to the ambient Instiki: It interacts neither with display-math environments, nor with equation numbering nor with bullet listst nor with any other form of indentation.)
Let’s copy all references also to the author’s page, with a link back to the entry, such as here
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