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Asked a question at graph of a functor.
I finally replied to Mike in the query box at graph of a functor
Continued the old discussion at graph of a functor.
Better would be to move that discussion here to the nForum, which is much more suited for hosting discussions.
(We had those query-box discussions a lot in the early days of the nLab but eventually switched to moving them to the nForum. )
Ah thanks for the tip. Here’s the discussion (I answered my own question in the sense that taking the 2 sided fibration associated the representable or corepresentable profunctor of a functor yield different notions of graph, but provisionally I don’t see why either should be preferred over the other. (Please correct me if I am wrong.))
Mike Shulman: It’s not obvious to me that this is the best thing to call the graph of a functor; there are lots of other graphy things one can construct from a functor that all reduce to the usual notion of the graph of a function. To start with, there is of course also the induced opfibration oven C×Dop, would you call that the “opgraph”? But actually, the two-sided fibration D←P→C (an opfibration over C and a fibration over D) looks to me more like a graph. And then there is of course the other profunctor induced by f, which gives a fibration over C×Dop, an opfibration over Cop×D, and a two-sided fibration from C to D.
Urs Schreiber: I would be inclined to loosely say “graph” for all of these and to introduce terminology like “opgraph” when it really matters which specific realization we mean. Because all these seem to be so similar to me that I am not sure if it is worth distinguishing them a lot. For instance, wouldn’t an analogous discussion be possible concerning what we call Fop:Cop→Dop given a functor F:C→D? I don’t actually know what a standard term is, does one say “opfunctor” for this? But I’d say it doesn’t matter much either way, calling Fop just a functor which effectively is the functor F doesn’t do much harm.
Colin Zwanziger: Aren’t we better off defining graph of a function as a span to avoid an arbitrary choice of ⟨1,f⟩ or ⟨f,1⟩ and then treating the two-sided fibration as the graph of a functor?Edit: Actually, we would still have to choose whether we were taking the graph of the representable or corepresentable profunctor induced by the functor, since these yield different spans. But we have that two functors F and G are adjoint iff (Lawvere’s definition) the (graph of F)_A and (graph of G)_B agree. One level down we would have two functions f and g are adjoint (=inverse) iff (graph of f)_A and (graph of g)_B agree, but the two notions of graph turn out to be the same at this level.
Sounds like we all agree that there are multiple things that could be called the “graph of a functor”. So someone should edit the page.
Removed query box and generalized the discussion accordingly to talk about all four graphs of a profunctor and all eight (!) graphs of a functor.
Mike Shulman: It’s not obvious to me that this is the best thing to call the graph of a functor; there are lots of other graphy things one can construct from a functor that all reduce to the usual notion of the graph of a function. To start with, there is of course also the induced opfibration oven $C\times D^{op}$, would you call that the “opgraph”? But actually, the two-sided fibration $D \leftarrow P \to C$ (an opfibration over $C$ and a fibration over $D$) looks to me more like a graph. And then there is of course the other profunctor induced by $f$, which gives a fibration over $C\times D^{op}$, an opfibration over $C^{op}\times D$, and a two-sided fibration from $C$ to $D$.
Urs Schreiber: I would be inclined to loosely say “graph” for all of these and to introduce terminology like “opgraph” when it really matters which specific realization we mean. Because all these seem to be so similar to me that I am not sure if it is worth distinguishing them a lot. For instance, wouldn’t an analogous discussion be possible concerning what we call $F^{op} : C^{op} \to D^{op}$ given a functor $F : C \to D$? I don’t actually know what a standard term is, does one say “opfunctor” for this? But I’d say it doesn’t matter much either way, calling $F^{op}$ just a functor which effectively is the functor $F$ doesn’t do much harm.
Colin Zwanziger: Probably the graph of a functor should be a two-sided fibration, though there are still 2 equally good choices corresponding to the representable or corepresentable profunctors induced by the functor.
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