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: How do you call a category where arrows are vectors?

Ignat Insarov (May 14 2025 at 11:46):

Take a directed graph with edges $E$ and a function $h: E → M$, where $M$ is a vector space, or a semi-module, or a commutative monoid with $0$ and $+$. Now, take the reflexive and transitive closure as usual to get a free category, but also extend $h$ like so:

• send every reflexive arrow $\operatorname {id}_X$ to the identity element $0 ∈ M$
• send every transitive arrow $g ∘ f$ to $h (g) + h (f)$

Now, to avoid possible confusion, consider any two arrows with the same source and target that $h$ sends to the same vector to be equal. So, essentially arrows are vectors, though technically equipped with source and target assignment (just as functions are in categories of sets and functions).

Now, as seems obvious:

• define identity arrows to be those arrows with equal source and target that $h$ sends to $0$
• define composition of arrows by addition of the underlying vectors, so that $h (g ∘ f) ≔ h (g) + h (f)$

The requirements of category are fulfilled at once, as $0$ is the identity of the associative $+$ in any vector, semi-module or commutative monoid.

How do I call categories so constructed? Is there any literature that describes them?

Ignat Insarov (May 14 2025 at 11:49):

David Spivak and Brendan Fong describe something similar in chapter 2 of the book Seven Sketches in Compositionality, but as far as I recall they only consider the case of a single number where I want to have a set of vectors, and the whole machinery of enrichment they spin up is not needed in my construction.

John Baez (May 14 2025 at 12:12):

This construction is in the paper @Adittya Chaudhuri and I are writing! It's probably not new. If I understand what you're talking about, it works for any monoid $M$: I don't see commutativity being used.

We can describe it like this. Any monoid $M$ gives a category called its [[delooping]] $BM$. This has one object, one morphism for each $m \in M$, and composition is the monoid operation.

There's a forgetful functor from categories to graphs

$U : \mathsf{Cat} \to \mathsf{Gph}$

So, we also get a graph $U BM$ with one vertex and one edge for each element of $M$.

Now the fun starts: a graph $G$ with edges labeled by elements of $M$ is the same as a map of graphs

$h: G \to U BM$

The forgetful functor $U$ has a left adjoint

$F : \mathsf{Gph} \to \mathsf{Cat}$

By the way adjoints work, the map of graphs

$h: G \to U BM$

gives a functor between categories

$\tilde{h} : F G \to BM$

And I believe this is what you want. $\tilde{h}$ maps each morphism in the free category on $G$ to an element of your monoid $M$, and since it's a functor it obeys

$\tilde{h} (g \circ f) = \tilde{h}(g) \tilde{h}(f)$

Matteo Capucci (he/him) (May 14 2025 at 14:23):

another perspective on this is that you're defining an enrichment in $M$; indeed such enrichments correspond to lax functors from a category (such as $FG$) to $BM$ (aka [[polyad]]). since $M$ is a mere monoid your lax functor is in fact strict.

Morgan Rogers (he/him) (May 15 2025 at 11:00):

Note that neither of these constructions perform the identification of arrows that you requested @Ignat Insarov ; is that important?

Ignat Insarov (May 15 2025 at 11:20):

Maybe we can split the functor $\tilde h$ through a category $\mathcal X$, like so: $FG \overset {\tilde h_1} → \mathcal X \overset {\tilde h_2} → BM$. Now, if we require that $\mathcal X$ have the same objects as $FG$ and that $\tilde h_2$ be full and faithful, $\mathcal X$ should be exactly as I wanted. Right?