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Stream: learning: questions

Topic: Is this structure known?

Ellis D. Cooper (Aug 19 2020 at 21:30):

Say objects of a structure are finite sets. A morphism from finite set A to finite set B is a finite partially ordered set with A for its minimal elements and B for its maximal elements. Composition of A to B and B to C is the partially ordered set obtained by identifying the maximal elements of A to B with the minimal elements of B to C. "Tensor" product of two disjoint finite sets is their union. "Tensor" product of A to B and C to D is from A union C to B union D by forming the disjoint union of the two given finite partially ordered sets. Is this structure a monoidal category in some obvious way? What kind? Is there a "natural" way to define a finite partially ordered set whose elements are the morphisms from A to B?

Notification Bot (Aug 19 2020 at 23:40):

This topic was moved here from #Is this structure known? > stream events by Nathanael Arkor

Ellis D. Cooper (Aug 20 2020 at 11:02):

@Nathanael Arkor
Thank you, Sir.

fosco (Aug 20 2020 at 12:47):

Call $\mathcal K$ this category; given $A,B$ is the partial order in $\hom_{\mathcal K}(A,B)$ part of the specification? If yes, is it determined in a certain canonical way?

Mike Stay (Aug 20 2020 at 18:22):

Sounds very much like the structured cospans work of @John Baez and Kenny Courser. In that paper they glue together graphs with boundary vertices; I think you just need to use directed acyclic graphs to get your posets.

Mike Stay (Aug 21 2020 at 03:24):

@Ellis D. Cooper ^^