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Stream: learning: questions

Topic: Linear sums in categories

Fernando Chu (Jul 29 2023 at 15:08):

In order theory, the following definition is useful
image.png

This can be generalized to a functor $Cat \times Cat \to Cat$. Is this useful somewhere? Does this have a right adjoint?

Todd Trimble (Jul 29 2023 at 17:30):

See Ordinal Sums and Equational Doctrines by Lawvere. It plays a role similar to simplicial joins.

John Baez (Jul 29 2023 at 21:09):

So the idea is that given two categories $X$ and $Y$ we form the coproduct $X + Y$ and then create a larger category that has the same objects and all the morphisms in $X + Y$, together with a unique morphism from any $x \in X$ to any $y \in Y$, and the more or less obvious rule for composing these? I feel there should be a slicker way to say what I just said, involving $X + Y$ and its obvious functor to category $2 = \{0 \to 1\}$. If I had more energy I'd look at Lawvere's paper, which would help.

James Deikun (Jul 29 2023 at 21:53):

My guess is it's the category where each $x \in X$ is the apex of a cone over $Y$ and each $y \in Y$ is the apex of a cocone under $X$.

John Baez (Jul 29 2023 at 22:01):

Actually here's what I was groping toward: the [[comma category]] of the functors $f: X \to 2$, $g: Y \to 2$ where $2$ is the 'walking arrow' category $0 \to 1$ and objects of $X$ get mapped to $0$ while objects of $Y$ get mapped to $1$.

James Deikun (Jul 29 2023 at 22:23):

I don't think that's the same thing though. I think that ends up being the same as $X \times Y$.

John Baez (Jul 29 2023 at 22:28):

Ugh. A comma object is too "limity", I'm looking for something colimity.

David Egolf (Jul 29 2023 at 22:56):

I suspect this is a relevant section from the paper by Lawvere mentioned above:
ordinal sum

See also: https://ncatlab.org/nlab/show/ordinal+sum#ordinal_sum_of_categories

Interestingly, that nlab article also notes that the ordinal sum may be viewed as "the collage of A and B along the terminal profunctor". Since this situation involves adding in morphisms between two categories, I was wondering if profunctors could be involved!

John Baez (Jul 29 2023 at 23:34):

Thanks! I like that nLab description. If you think of a profunctor from A to B as giving a set of arrows ("heteromorphisms") from each object of A to each object of B, the terminal profunctor will give just one arrow from each object of A to each object of B - a nice example of the dictum "the terminal thing has just one of everything it needs to have".

John Baez (Jul 29 2023 at 23:35):

So then yes, taking the [[collage]] of the terminal profunctor from A to B will have the desired effect.

Todd Trimble (Jul 30 2023 at 02:24):

David Egolf said:

I suspect this is a relevant section from the paper by Lawvere mentioned above:
ordinal sum

See also: https://ncatlab.org/nlab/show/ordinal+sum#ordinal_sum_of_categories

Interestingly, that nlab article also notes that the ordinal sum may be viewed as "the collage of A and B along the terminal profunctor". Since this situation involves adding in morphisms between two categories, I was wondering if profunctors could be involved!

The diagram shown in the nLab article, of shape

$A \stackrel{\pi_1}{\leftarrow} A \times B \stackrel{i_0}{\to} A \times 2 \times B \stackrel{i_1}{\leftarrow} A \times B \stackrel{\pi_2}{\to} B,$

whose colimit defines the ordinal sum, is the exact same shape as the diagram one would use to form the simplicial sum, say in the category of topological spaces or the category of simplicial sets, where one replaces the Cat-interval $2$ with the topological interval $I$. That's what I meant in my earlier comment when I said the ordinal sum is similar to the simplicial join.

Todd Trimble (Jul 30 2023 at 02:31):

For example, the simplicial join of a space $B$ with a point $A = 1$ is the cone on $B$.

Todd Trimble (Jul 30 2023 at 02:40):

In response to a question above: I don't believe the ordinal sum has a right adjoint, either in joint arguments nor in separate arguments. However, if I'm not mistaken, it does provide a monoidal product, whose unit is the initial category/space/whatever.

Alexander Kurz (Jul 30 2023 at 14:15):

Just to add that the collage gives us a way to represent relations in enriched categories. In ordinary (more specifically regular) categories one typically uses spans to represent relations, but in enriched categories spans work only in exceptional cases, while cospan (collages, cographs) work more generally. An example where both spans and cospans can be used to represent relations (profunctors) are poset and also cat-enriched categories.

Fernando Chu (Jul 30 2023 at 14:53):

Cool! Thanks for the answers and comments everyone.

Matteo Capucci (he/him) (Jul 31 2023 at 13:02):

This is the [[collage]] of the terminal profunctor between two categories (i.e. the profunctor $\top : \cal C \to D$ such that $\top(c,d)=1$ for every $c: \cal C$ and $d: \cal D$).

This should satisfy @John Baez's thirst for colimits :)