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: theory: category theory

Topic: Categories of shapes

James Deikun (Sep 24 2021 at 14:36):

Given a colored planar operad $O$, one can construct what I've been calling its "category of shapes" $Shp(O)$. The objects are the operators of the operad; the morphisms $o \rightarrow p$ are factorizations $o = f(id, \ldots, id, p(g_1, \ldots, g_n), id, \ldots, id)$.

In the case of the $Assoc$ operad, the resulting category is the augmented simplex category $\Delta_0$. In the case of a category taken as colored operad, it is the opposite of the twisted arrow category (aka category of factorisations). The associated pages on the nLab don't seem to point to this generalization.

Has anyone heard of this construction in approximately this generality? Is there a nice non-component-based description of it?

Also, $\Delta_0$ is the walking strict $Assoc$-category equipped with an $Assoc$ algebra. Does this generalize? It seems to me that it does, but I don't have a nice snappy proof or even a way of phrasing it for the nontrivially-colored case.

Mike Shulman (Sep 24 2021 at 16:20):

What is $f$ in your factorization?

John Baez (Sep 24 2021 at 16:41):

Any operation in the operad, I guess.

James Deikun (Sep 24 2021 at 16:46):

That's correct. $f$ and the $g_i$ are arbitrary operations, part of the data of a morphism.

Mike Shulman (Sep 24 2021 at 16:47):

Huh, this doesn't seem to me like a very natural thing to consider, but it reminds me of the Baez-Dolan slice construction.

John Baez (Sep 24 2021 at 16:48):

It reminded me of that too... for some reason. :upside_down:

John Baez (Sep 24 2021 at 16:49):

But we look at more general ways of sticking together operations in an operad, not just those of this form.

James Deikun (Sep 24 2021 at 16:50):

The reason I started thinking about it was to serve as a canonical category of arities for a category over O in the Trimble sense ... or just an O-algebra for that matter.

John Baez (Sep 24 2021 at 16:51):

But in my paper with Dolan, we also showed that for any operad $O$, the concept of $O$-algebra makes sense in any $O$-category (meaning roughly a category with an $O$-action). We actually show an $n$-categorical generalization of this idea.

John Baez (Sep 24 2021 at 16:52):

So the "free $O$-category on an internal $O$-algebra" seems like a very interesting thing to study!

John Baez (Sep 24 2021 at 16:53):

Mainly I'd be interested to know what it is in a bunch of examples.

John Baez (Sep 24 2021 at 16:54):

Of course it's famous and wonderful that the augmented simplex category is the free monoidal category on a monoid object.... but what are some other examples?

James Deikun (Sep 24 2021 at 17:08):

Well, it's maybe less famous, but if I calculate the category of shapes for $\mathbb{Z}_2$ as an operad, I get a category with:

• two objects, $I$ and $\Sigma$
• an involution $\sigma_X$ on each object
• two morphisms $\omicron\, \iota : I \rightarrow \Sigma$ and their inverses
• composing with a $\sigma$ on either end swaps $\omicron$ and $\iota$.

I think this also satisfies the right universal property for a "free category with strict involution and an object with a strict involution" ...

James Deikun (Sep 24 2021 at 17:18):

The endomorphism operad of 2 probably gives a much better example, but it's much harder to calculate on either side of the conjecture ... in between are things like "$k$-adic semigroups" (exactly one operation of each arity $kn+1$ and composition the only thing it can be) ... it seems like mostly when people are interested in operads it's not in $\mathcal{Set}$ so it's hard to come up with celebrated examples without going enriched.

John Baez (Sep 24 2021 at 18:28):

How about the operad for commutative monoids?

I think the free commutative monoidal category on a commutative monoid is likely to be a bit less interesting than the free symmetric monoidal category on a commutative monoid... but they should both be interesting.

James Deikun (Sep 24 2021 at 19:55):

It seems like the free commutative monoidal category on a commutative monoid should look like ...

• objects the natural numbers
• morphisms generated by $c : 2 \rightarrow 1$ and $u : 0 \rightarrow 1$ subject to the relation $c \circ (1 + u) = 1$.

Associativity seems to come for free here as a result of commutativity.

The relation above plus the interchange law seem to add up to a lot of algebraic relations. In fact, it looks like given a formula for a morphism using $c$, $u$, identities, $+$, and composition, the result is already overdetermined by the number of $$c$$s, the number of $$u$$s, and the source and target.

This isn't looking familiar to me yet. It's simple but not quiiiite trivial...

James Deikun (Sep 24 2021 at 19:58):

Actually the cs and us cancel out, and they determine the difference between source and target, so i think the only thing that matters is the source and target. Maybe it is trivial ...

John Baez (Sep 24 2021 at 20:03):

Maybe the category you're getting here is what you get when you hit the augmented simplex with an "abelianization" functor?

Let me explain:

There's a morphism from the operad for monoids to the operad for commutative monoids, so any commutative monoid object in a symmetric monoidal category $C$ has an underlying monoid object, giving us a functor

$U : \mathrm{CommMon}(C) \to \mathrm{Mon}(C)$

where $\mathrm{CommMon}(C)$ is the category of commutative monoid objects in $C$, and similarly for $\mathrm{Mon}(C)$.

When $C$ is nice enough - like when it has colimits and the tensor product distributes over colimits, maybe? - $U$ should have a left adjoint

$L : \mathrm{Mon}(C) \to \mathrm{CommMon}(C)$

That's what I mean by "abelianization".

So, I'm wildly guessing that when you abelianize the "free monoidal category on a monoid object", you get the "free commutative monoidal category on a commutative monoid object".

James Deikun (Sep 24 2021 at 20:15):

It seems to end up just being a codiscrete/chaotic category though, unless I'm completely missing something. Given $f: m \rightarrow n, g: p \rightarrow q$ the interchange law gives $f + g = f \circ m + q \circ g = (f + q) \circ (g + m)$ and it works for either order so you can just slide all the generating arrows past each other freely, and then cancel either all the $u$ s or all the $c$ s with the unit law ... leaving no degrees of freedom.

James Deikun (Sep 24 2021 at 20:18):

(The symmetric case is definitely going to be more interesting though.)

James Deikun (Sep 24 2021 at 21:00):

I guess the free strictly-associative symmetric monoidal category has objects the natural numbers again, and its morphisms can be described by a string diagram calculus:

• Hom(n,m) has n strings in, m out
• Strings are all the same, no colors
• Strings can cross and it doesn't matter which is on top; two crossings are the same as none
• Strings can be capped on their input side, and these caps can be retracted toward the output side
• There is a joiner piece $\triangleright$ ; it is "symmetric" in the sense that a triangle with a crossing on the input side is the same as one without
• If one of the inputs of the triangle is capped, the whole thing can be replaced with a simple string.

James Deikun (Sep 24 2021 at 21:03):

Oh, also two triangles feeding into each other can be switched along the piece of string that connects them.

James Deikun (Sep 24 2021 at 21:05):

I think the upshot of this is that the category is going to be FinSet ...

James Deikun (Sep 24 2021 at 21:15):

In more detail ... the category FinSet, the monoidal product the coproduct, and the generic commutative monoid the single-element set.

James Deikun (Sep 24 2021 at 21:16):

(And the braiding, the unique nontrivial permutation on the two-element set...)

John Baez (Sep 24 2021 at 21:39):

That sounds good!

James Deikun (Sep 25 2021 at 12:19):

Hm, it seems the "category of shapes" construction actually can be extended to symmetric operads in such a way as to encompass this example too. Just add to the data of a morphism a permutation to be applied to the inputs, and the category of shapes of Comm turns out to be FinSet. The previous examples, considered now as symmetric operads, lead to the same categories as before up to equivalence, although they are now "fluffed up" quite a bit with the extra symmetrized operations. Taking a skeleton of "fluffy $\Delta_0$ " leads back to plain old $\Delta_0$ though.

James Deikun (Sep 27 2021 at 14:21):

Going by the above, a "category of shapes" for a multisorted Lawvere theory should be, if it makes sense:

• as objects, operations of the Lawvere theory
• as morphisms $o \rightarrow p,$ equations $o(\overline{x}) = \phi(p,\overline{x})$ where $p$ occurs linearly in $\phi$
• morphisms treated as equivalent if the theory proves the corresponding $\phi$ equivalent without knowing what $p$ is.

If this is correct then maybe it is possible to come up with more notable examples of 'free T-category on a T' using this system ...

James Deikun (Sep 27 2021 at 14:32):

(I'm not 100% sure $p$ should be linear in $\phi$ though ...)

James Deikun (Sep 27 2021 at 15:34):

Really not sure this one makes sense though ... don't seem to get anything much like the simplex category for monoids either way.