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Stream: learning: questions

Topic: What to look for in a category of algebraic structures?

Carlos Zapata-Carratala (Jul 06 2022 at 17:59):

CONTEXT: In my ongoing research on ternary and higher-arity structures we came across a ternary algebra called a semiheap: a set with a ternary operation $(S,(\cdot \cdot \cdot))$ such that
$((abc)de) = (a(dcb)e) = (ab(cde))$
These structures can be regarded as generalizing many known algebraic strcutures such as groups, torsors, afine spaces, involuted monoids, dagger categories (in the -oid case), etc. We have found a natural notion of morphism that deviates from the standard homomorphism of universal algebras (in particular, it generalizes splittings in a non-trivial way involving two maps in different directions between semiheaps,) which has allowed us to solve many uncomfortable questions about ternary notions of invertibility, units and equivalence.

QUESTION: Having found this previously unindentified category of semiheaps we are wondering what are natural questions one may ask of a category of algebraic structures. What do you think is important to claim that a category of algebraic strucutures is more "correct" than another? We are mostly interested in finding categorical arguments to claim that our definition of "the category of semiheaps" is somehow better or "more natural" than the usual automatic definition via homomorphisms of universal algebras (it should be mentioned that isomorphisms of universal algebras give isomorphisms of semiheaps but the converse is not true in general).

John Baez (Jul 06 2022 at 18:05):

The usual concept of morphism between algebraic structures - familiar from universal algebra or the study of [[algebraic theories]] - is known to always give a category that's complete, cocomplete, etc. etc.

In fact, if you list enough "etc." here you get theorems saying that categories with all these properties are equivalent to categories of algebraic structures with the usual morphisms! The classic result along these lines is Birkhoff's theorem, and there's an updated version for algebraic theories.

John Baez (Jul 06 2022 at 18:06):

If you're using some other notion of morphism then it would be good to fit this other notion into a general context. If it's good, it's probably an example of a good general idea. But here I'm talking as a category theorist: this is the philosophy of category theory.

Carlos Zapata-Carratala (Jul 06 2022 at 19:04):

John Baez said:

If you're using some other notion of morphism then it would be good to fit this other notion into a general context. If it's good, it's probably an example of a good general idea. But here I'm talking as a category theorist: this is the philosophy of category theory.

By this you mean finding the corresponding analogue of general "algebraic theories" that fit our notion of morphism for the particular case of semiheaps?

John Baez (Jul 06 2022 at 19:08):

Something like that. I'd have to see some details to know precisely which way to go (and I don't have much time for this right now, alas). Is your notion of morphism a pair of maps $f: X \to Y, g: Y \to X$ obeying some conditions?

Carlos Zapata-Carratala (Jul 06 2022 at 19:51):

John Baez said:

Something like that. I'd have to see some details to know precisely which way to go (and I don't have much time for this right now, alas). Is your notion of morphism a pair of maps $f: X \to Y, g: Y \to X$ obeying some conditions?

Yes, we define a morphism of semiheaps
$\varphi_\psi : (A, (\cdot \cdot \cdot)_A) \to (B, (\cdot \cdot \cdot)_B)$
as a pair of maps $\varphi : A \to B$ and $\psi : B \to A$ such that
$\varphi ((a \, \psi (b) \, a')_A) = (\varphi(a) \, b \, \varphi(a'))_B$.

John Baez (Jul 06 2022 at 20:05):

Okay, thanks - that formula has a certain charm, and maybe it will ring a bell for someone here.

Mike Shulman (Jul 06 2022 at 21:41):

Reminds me of some kind of internal adjunction between closed categories...

Morgan Rogers (he/him) (Jul 07 2022 at 08:41):

I would be curious to know how they interact with homomorphisms. You could consider investigating a double category with homomorphisms as vertical morphisms and... intermorphisms? (did you come up with a good name yet?)... as horizontal morphisms.
As for constructing the category with these things on their own, I would recommend abstracting the natural structure on these. Does the collection of intermorphisms from $A$ to $B$ have some structure? Possibly a ternary operation inherited from that on $A$ or $B$? Are there analogues of natural transformations between these things, or distinguished instances which arise when $A$ or $B$ has extra structure? Does the forgetful functor to $\mathrm{Set}$ have nice properties? (is it faithful, for instance?) You already mentioned limits and colimits as a possible direction for investigation, but what universal properties make sense for these which aren't directly limits or colimits?

Morgan Rogers (he/him) (Jul 07 2022 at 08:46):

To give you some direction here, one potential goal would be to work out how much these morphisms "see" of the internal structure of the semiheaps. Intermorphisms from the trivial semiheap correspond to "capturing elements", that is $a \in A$ such that $(aa'a)_A = a$ for all $a' \in A$, for instance; is there an object that does a bit better at representing the forgetful functor to $\mathrm{Set}$?

Oscar Cunningham (Jul 07 2022 at 10:15):

Sometimes you can reimagine what you think of as the 'elements' of the objects of a category, in order to make it more algebraic. For example take $\mathbf{Rel}$, the category of sets with relations as morphisms. This category isn't obviously algebraic. But a relation between two sets is the same thing as a sup-preserving map between their powersets. So $\mathbf{Rel}$ is equivalent to $\mathbf{FreeSupLat}$, the category of free sup-lattices (i.e. powersets), which is algebraic in the sense that it's a full subcategory of $\mathbf{SupLat}$, which is monadic over $\mathbf{Set}$.

Oscar Cunningham (Jul 07 2022 at 10:19):

Morgan Rogers (he/him) said:

is there an object that does a bit better at representing the forgetful functor to $\mathrm{Set}$?

I don't think any object will do a good job of representing a forgetful functor. An important example of semiheaps is groups with the operation $(xyz)=xy^{-1}z$. But then there are no morphisms from any semiheap into a group of larger cardinality, because as we vary b in $\varphi ((a \, \psi (b) \, a')_A) = (\varphi(a) \, b \, \varphi(a'))_B$ the right hand side will take on every value in the group, whereas the left hand side will take on at most $|A|$ values.

Morgan Rogers (he/him) (Jul 07 2022 at 11:00):

That's a nice example @Oscar Cunningham ! But the forgetful functor is contravariantly represented by the trivial semiheap, since there is a unique function $\phi: A \to 1$ and all of the sections to this function are valid choices for $\psi$..!

Morgan Rogers (he/him) (Jul 07 2022 at 11:04):

@Carlos Zapata-Carratala I hope this observation^ is helpful :wink:
Worth noting that this forgetful functor only represents $\psi$ and so is not faithful (since $\psi$ doesn't determine $\phi$).

Carlos Zapata-Carratala (Jul 07 2022 at 19:52):

Morgan Rogers (he/him) said:

I would be curious to know how they interact with homomorphisms. You could consider investigating a double category with homomorphisms as vertical morphisms and... intermorphisms? (did you come up with a good name yet?)... as horizontal morphisms.
As for constructing the category with these things on their own, I would recommend abstracting the natural structure on these. Does the collection of intermorphisms from $A$ to $B$ have some structure? Possibly a ternary operation inherited from that on $A$ or $B$? Are there analogues of natural transformations between these things, or distinguished instances which arise when $A$ or $B$ has extra structure? Does the forgetful functor to $\mathrm{Set}$ have nice properties? (is it faithful, for instance?) You already mentioned limits and colimits as a possible direction for investigation, but what universal properties make sense for these which aren't directly limits or colimits?

We only recently found a strong motivation to define these alternative semiheap morphisms so we didn't think of a name yet. "Intermorphisms" sounds like a good name a priori.

Carlos Zapata-Carratala (Jul 07 2022 at 19:53):

Many thanks for all your comments. These give us very useful prompts to guide our exploration of these structures.