You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
Jonathan Beardsley (Dec 16 2023 at 00:53):I gather that there is a tricategory of spans of categories. I'm interested in showing that "partially monoidal categories" i.e. categories that have a partially defined monoidal structure, are pseudomonoids of some form in some ?-category of spans of categories. Does anyone know of a reference for this sort of thing? In particular, is it possible to just ignore the tricategory structure?
Jonathan Beardsley (Dec 16 2023 at 00:54):I've seen people say things like "partial monoidal categories are pseudomonoids in the bicategory of categories and partial functors" but I'd like to try to say this just in Span(Cat) without having to build up the bicategory of partial functors
Jonathan Beardsley (Dec 16 2023 at 00:56):For instance, I was trying to write down the necessary "associativity" for such a thing being a "pseudomonoid" but I realized I probably don't have a map of spans "on the nose." Instead, I seem to have a map of spans which is mediated by some equivalences.
Jonathan Beardsley (Dec 16 2023 at 01:05):Maybe I should say something like.... what does it mean to say a "diagram commutes," in particular, what is the correct notion of a "map of spans?"
Patrick Nicodemus (Dec 16 2023 at 01:52):A bicategory itself is one kind of partial monoidal category, isn't it? Is your partial monoidal bicategory a bicategory or is it more general than that
Jonathan Beardsley (Dec 16 2023 at 02:28):Oh I'm only starting with partially monoidal 1-category.
Evan Patterson (Dec 16 2023 at 05:50):Jonathan Beardsley said:
Does anyone know of a reference for this sort of thing? In particular, is it possible to just ignore the tricategory structure?
Yes. Since Cat is a 2-category, as opposed to a genuinely weak bicategory, it has an underlying category and so there is a perfectly fine double category Span(Cat) as well as a bicategory Span(Cat). There are ways to upgrade these to three-dimensional structures but nothing forces you to do that.
Evan Patterson (Dec 16 2023 at 05:53):Jonathan Beardsley said:
Maybe I should say something like.... what does it mean to say a "diagram commutes," in particular, what is the correct notion of a "map of spans?"
There are different notions of a "map of spans" depending on whether you work in a bicategory or a double category of spans. In a bicategory of spans, a map from a span to another span (with the same feet) is a map making the two triangles commute. In a double category of spans, a map from a span to another span (now with possibly different feet) consists of maps , , and making the two squares commute.
Jonathan Beardsley (Dec 16 2023 at 06:00):Evan Patterson said:
Jonathan Beardsley said:
Maybe I should say something like.... what does it mean to say a "diagram commutes," in particular, what is the correct notion of a "map of spans?"
There are different notions of a "map of spans" depending on whether you work in a bicategory or a double category of spans. In a bicategory of spans, a map from a span $x \leftarrow s \rightarrow y$ to another span $x \leftarrow t \rightarrow y$ (with the same feet) is a map $s \to t$ making the two triangles commute. In a double category of spans, a map from a span $x \leftarrow s \rightarrow y$ to another span $w \leftarrow t \rightarrow z$ (now with possibly different feet) consists of maps $x \to w$, $y \to z$, and $s \to t$ making the two squares commute.
Oh nice. Thank you. The latter seemed to be what I was getting but I thought somehow that wasn't a thing.
Patrick Nicodemus (Dec 17 2023 at 11:54):Jonathan Beardsley said:
Oh I'm only starting with partially monoidal 1-category.
Yes, there's nothing 2-categorical going on in my message, my point is to forget about the 2-categorical structure of bicategories and view them 1-categorically. I mean if you forget about the 0-cells of a bicategory, and regard the 1-cells of the bicategory as objects in a category , with "vertical" 2-cells between them as morphisms, then horizontal composition of 1-cells and 2-cells is a partially defined monoidal product on the "total"1-category of 1-cells. The 0-cells just control the domain of definition of the monoidal product.
It just so happens that there are no morphisms at all between objects that fail to have the same domain and codomain 2-cell, but this makes no difference to understanding it as a 1-category.
Patrick Nicodemus (Dec 17 2023 at 12:00):This "monoidal category" would have multiple non-isomorphic units, which is very weird, but it makes sense that if you cannot define because the pair is not in the domain of the monoidal product, you cannot argue that , so it's natural that the unit fails to be unique.
Jonathan Beardsley (Dec 18 2023 at 17:31):Patrick Nicodemus said:
Jonathan Beardsley said:
Oh I'm only starting with partially monoidal 1-category.
Yes, there's nothing 2-categorical going on in my message, my point is to forget about the 2-categorical structure of bicategories and view them 1-categorically. I mean if you forget about the 0-cells of a bicategory, and regard the 1-cells of the bicategory as objects in a category , with "vertical" 2-cells between them as morphisms, then horizontal composition of 1-cells and 2-cells is a partially defined monoidal product on the "total"1-category of 1-cells. The 0-cells just control the domain of definition of the monoidal product.
It just so happens that there are no morphisms at all between objects that fail to have the same domain and codomain 2-cell, but this makes no difference to understanding it as a 1-category.
I see. Sorry, I didn't read carefully. I hadn't thought about whether or not the particular partial monoidal category I was looking at was just a special kind of bicategory (I wasn't aware of this characterization). It's a pretty simple category though, so it wouldn't surprise me. I'll have to look at it.
John Onstead (Jan 29 2024 at 15:47):I've recently been thinking about topics related to this post and I wanted something cleared up. What is the precise relationship between the "partialization" of an algebraic structure and the "oidification" of the corresponding categorical structure? IE, what is the relationship between partial monoids and categories, partial groups and groupoids, partial monoidal categories and bicategories, etc.? Specifically, if you are given categories like ParMon(Set) (the category of partial monoids) and Cat, is there an equivalence of categories between them; if not, what kinds of functors exist between them?