Category Theory
Zulip Server
Archive

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.

Stream: learning: questions

Topic: "Swapping" Morphisms and Objects

Sean Maley (Oct 23 2024 at 23:51):

In the spirit of "no question is dumb"

Is there a way to map from a category C with ob(C) and Mor(C), to a category D where ob(D)=Mor(C) and Mor(D)=ob(C)?

Alex Kreitzberg (Oct 24 2024 at 00:00):

I don't think there is a generic way, imagine the category 𝟚, $A \rightarrow B$

You'd have three objects in the resulting category, $id_A, \rightarrow,$ and $id_B$, but only two arrows, $A$ and $B$, but you need at least three arrows, because each object needs an identity.

That said, your question makes me think of string diagrams, where you visually flip the roles of what's a point, a line, and a region, so maybe there's an interesting duality here if you're careful.

Jean-Baptiste Vienney (Oct 24 2024 at 00:04):

Do you want a functor?

Sean Maley (Oct 24 2024 at 00:07):

@Alex Kreitzberg Yes this is my understanding. Can't get the simplest example to port over, but I wondered if higher categories or other advanced concepts (things I have not played with yet) had anything to offer.

Sean Maley (Oct 24 2024 at 00:08):

@Jean-Baptiste Vienney Would be nice, but by definitions not sure it can work out unless something creative is done.

John Baez (Oct 24 2024 at 00:13):

Basically the answer to the stated question is "no". But there are sneaky things you can do. A "category internal to Vect" is a category with a vector space of objects and a vector space of morphisms, where everything in sight is linear.

A nontrivial little fact is that a category internal to Vect is equivalent to a pair of vector spaces and a linear map between them:

$d: M \to O$

where $O$ is the vector space of objects and $M$ is the vector space, not of all morphisms, but of morphisms $f: 0 \to x$ for some object $x$.

Then by taking the dual vector spaces we get

$d^\ast: O^\ast \to M^\ast$

which by that little fact is equivalent to another category internal to Vect! The space of morphisms in this new category is the dual of the space of objects in the original category.

Jean-Baptiste Vienney (Oct 24 2024 at 00:15):

John Baez said:

Basically the answer to the stated question is "no".

But why? You can just map every object to a single chosen object and every morphism to the corresponding identity.

Jean-Baptiste Vienney (Oct 24 2024 at 00:17):

(Taking into acounts that morphisms in the second category are objects of the first and objects of the second are morphisms of the first but it should not change anything to the existence of these functors.)

Jean-Baptiste Vienney (Oct 24 2024 at 00:19):

But first you have to build such a category $D$.

Jean-Baptiste Vienney (Oct 24 2024 at 00:19):

But is this category $D$ given in the data of the problem or are you allowed to build any one?

Sean Maley (Oct 24 2024 at 00:23):

@Jean-Baptiste Vienney Trying to "preserve" everything from the first category, if at all possible. Collapsing like this isn't ideal (although tbf that may not have been clear from how it the question was posed).

Sean Maley (Oct 24 2024 at 00:24):

@John Baez Thank you, this is the kind of insight I was hoping for :thumbs_up:

John Baez (Oct 24 2024 at 01:35):

Jean-Baptiste Vienney said:

John Baez said:

Basically the answer to the stated question is "no".

But why?

As far as I can tell, @sean was asking if there's a way to take an arbitrary category C with object set ob(C) and morphism set mor(C) and construct a category D with object set ob(D)=mor(C) and morphism set mor(D)=ob(C).

We cannot. Let C be a category with 1 object and 2 morphisms. Then D would need to have 2 objects and 1 morphism. But there is no category with 2 objects and 1 morphism.

David Corfield (Oct 24 2024 at 10:48):

One might consider Poincaré dual string diagrams an attempt to reverse dimensions:
image.png
But of course these diagrams aren't categories of any kind. Which raises a question: what kinds of structured entity are these dual string diagrams?

Peva Blanchard (Oct 24 2024 at 11:26):

It also makes me think of hypergraph.
If I remember correctly a hypergraph is given by:

• a set $V$ of vertices
• a set $E$ of (hyper) edges
• and an incidence relation $I \subseteq E \times V$

In that case, swapping vertices and edges amount to taking the transpose of the incidence relation.

I remember that there is a notion of hypergraph category but I don’t know much about them.

John Baez (Oct 24 2024 at 16:54):

I know too much about hypergraph categories. :weary: But I've never tried to imagine the 'dual' of a hypergraph category. It's an interesting idea, but don't see how it would make sense, since we know how to compose morphisms, but not objects.

John Baez (Oct 24 2024 at 17:05):

Petri nets are very similar to hypergraphs, except that each hyperedge is oriented, going from some finite set of vertices to some other finite set of vertices. So we have two relations, $I \subseteq E \times V$ saying which edges go in to each vertex and $O \subseteq E \times V$ saying which edges go out of each vertex.

People have observed that you can dualize any Petri net and get a new Petri net, but nobody has ever done anything with this idea, as far as I know! I have wondered about it.