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Max New (Jul 18 2022 at 00:07):A bifibration induces for each adjoint functors between the fibers .
In my work I use bifibrations where this adjunction is a co-reflection, i.e., is isomorphic to the identity. Is there a name for this kind of bifibration/is anyone aware of places this concept has been used?
John Baez (Jul 18 2022 at 00:16):Does the bifibration , which picks out the set of nodes of a graph, have this property?
Max New (Jul 18 2022 at 00:30):No, because you can take any graph G and map it to a one node graph with no edges and then push/pull to get back the discrete graph of G.
This probably indicates that you need a pretty restrictive notion of morphism in . I should mention that in my examples, E and B are preorders and so and are a Galois connection.
John Baez (Jul 18 2022 at 03:49):Thanks. Okay, yes, it sounds like a strong condition.
Max New (Jul 18 2022 at 13:17):Oh, I wasn't quite right above but the property is still not true. As an example start with { l -> r }, pushforward the unique function to a one-element set to get a vertex with a self loop, and then pull back and you will get a complete graph on two vertices
John Baez (Jul 18 2022 at 13:26):Okay, that's better. You fooled me with the first example. :upside_down:
John Baez (Jul 18 2022 at 13:27):Is it true that pulling back and then pushing forward is also not isomorphic to the identity for ? If I were awake I could figure it out.
John Baez (Jul 18 2022 at 13:29):I guess it's not. Start with the map from the 2-element set to the 1-element set. Take the graph with a single self-loop on the 1-element set. Pull it back to the 2-element set, then push forward. I think you get a graph with 4 self-loops on the 1-element set.
Max New (Jul 18 2022 at 13:31):I think a simpler example that works for non-multi-graphs would be to start with a vertex with a self loop and then pull back over an empty set of vertices and then push forward and you get a vertex with no self loop
John Baez (Jul 18 2022 at 13:32):Yes.
John Baez (Jul 18 2022 at 13:33):So, it seems nothing like this property holds for the bifibrations of the sort I've been thinking about. (If it did, I would want to know!)
Tom Hirschowitz (Jul 18 2022 at 13:50):@Max New Now I'm curious about how your bifibration looks like!
Max New (Jul 19 2022 at 19:11):It comes from my work on gradual typing https://arxiv.org/abs/1802.00061 That paper is phrased in terms of double categories that are pro-arrow equipments, and there's a way to define pro-arrow equipments using fibrations so I was wondering if there was a word for this additional axiom that we have.
John Baez (Jul 19 2022 at 19:19):As you probably know - but just in case you don't - when Shulman gets pro-arrow equipments from bifibrations in Framed bicategories and monoidal fibrations, in Definition 13.31 he introduces various special kinds of bifibrations, like Beck-Chevalley, strong Beck-Chevalley, weak Beck-Chevalley, etc. I needed to learn about these in my work on decorated and structured cospan double categories. But I don't think he mentions the kind you're talking about.
Nathanael Arkor (Jul 19 2022 at 19:48):This seems quite an unusual property to impose on an equipment: asking for every induced adjunction to be a coreflection is asking for every morphism in the underlying 2-category to be fully faithful. Accordingly, the condition is in a sense an equipment-theoretic version of left cancellative (2-)category. I would suspect that such structures are not particularly widespread.