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Bruno Gavranović (Jan 25 2023 at 17:31):Given an endofunctor , it's known that we can form a category whose objects are its algebras (and morphisms are algebra homomorphisms), and a category whose objects are its coalgebras (whose morphisms are coalgebra homomorphisms).
I've recently learned that there is also a notion of a coalgebra-to-algebra morphism (making the appropriate diagram commute) and I'm wondering whether all this data can be organised as some sort of a double, or n-fold category.
It looks like we want two different types of 0-cells (algebras, and coalgebras), where morphisms between the same types of cells would be homorphisms of algebras and respectively coalgebras. Then also we could have a morphism between different types of 0-cells - which would be the coalgebra-to-algebra morphisms. (There's also the algebra-to-coalgebra morphism) which follows in a similar way.
Question: what kind of structure is this? It's not a double category, as I initially suspected. We'd have at least 2 types of 0-cells and three types of 1-cells.
Nathanael Arkor (Jan 25 2023 at 18:08):Algebras and coalgebras for an endofunctor form a category. Is there any structure you're interested in that is not captured simply by the categorical structure?
Nathanael Arkor (Jan 25 2023 at 18:10):The objects may be distinguished by being in one of two classes, but the morphisms don't appear to have any interesting properties: their shape is uniquely determined by the class of their domain and codomain. So I think it's misleading to say that there are four kinds of 1-cell, as the kinds should be parameterised by the kind of their (co)domain.
Mike Shulman (Jan 25 2023 at 18:24):I think this category sits inside a larger and perhaps more natural-looking category consisting of objects with a relation to . Then sometimes the relation is functional and sometimes cofunctional.
Bruno Gavranović (Jan 25 2023 at 19:49):Nathanael Arkor said:
Algebras and coalgebras for an endofunctor form a category. Is there any structure you're interested in that is not captured simply by the categorical structure?
They both form individual categories, yes. But the coalgebra-to-algebra morphism isn't a morphism in either one of them.
Bruno Gavranović (Jan 25 2023 at 19:49):Mike Shulman said:
I think this category sits inside a larger and perhaps more natural-looking category consisting of objects $X$ with a relation to $FX$. Then sometimes the relation is functional and sometimes cofunctional.
Hmm, that sounds plausible.
Bruno Gavranović (Jan 25 2023 at 19:50):Nathanael Arkor said:
The objects may be distinguished by being in one of two classes, but the morphisms don't appear to have any interesting properties: their shape is uniquely determined by the class of their domain and codomain. So I think it's misleading to say that there are four kinds of 1-cell, as the kinds should be parameterised by the kind of their (co)domain.
I see. You're saying that once I know the type of the domain and codomain, the type of the morphism is fixed, unlike with, say, double categories.
Nathanael Arkor (Jan 25 2023 at 20:53):Bruno Gavranovic said:
Nathanael Arkor said:
Algebras and coalgebras for an endofunctor form a category. Is there any structure you're interested in that is not captured simply by the categorical structure?
They both form individual categories, yes. But the coalgebra-to-algebra morphism isn't a morphism in either one of them.
I mean that there is a category whose objects are either algebras or coalgebras, and whose morphisms are the appropriate notion of morphism in each case. But I agree that seeing this category as sitting inside a larger one seems more natural.
Bruno Gavranović (Jan 25 2023 at 23:44):Mike Shulman said:
I think this category sits inside a larger and perhaps more natural-looking category consisting of objects $X$ with a relation to $FX$. Then sometimes the relation is functional and sometimes cofunctional.
One issue with this is that we lose directionality as information available at the type level.
James Deikun (Jan 26 2023 at 00:15):One larger category you might embed this in is the category of dialgebras for all endofunctors on a specific category; in this case the type information would indeed be in the morphisms as there would be a "twisted" type that exchanged the functors and an "untwisted" type that kept them the same.
James Deikun (Jan 26 2023 at 00:17):This might be seen as a -graded category.
Bruno Gavranović (Jan 26 2023 at 00:54):Can you elaborate on the "twisted" type? What does it mean to exchange the functors?
James Deikun (Jan 26 2023 at 01:02):
Untwisted on left, twisted on right.
Bruno Gavranović (Jan 26 2023 at 10:46):Right, but a category of dialgebras chooses only one of them. How do you get the other one?
James Deikun (Jan 26 2023 at 12:19):Ah, well, it wouldn't be the standard category of dialgebras but a "super category" of them. (And you could also combine this with the other construction and have dialgebras in a suitable category of relations ... but you would still want the 'twisted' morphisms so you could formally change which side is which.)
Matteo Capucci (he/him) (Jan 26 2023 at 16:00):Bruno Gavranovic said:
It looks like we want two different types of 0-cells (algebras, and coalgebras), where morphisms between the same types of cells would be homorphisms of algebras and respectively coalgebras. Then also we could have a morphism between different types of 0-cells - which would be the coalgebra-to-algebra morphisms. (There's also the algebra-to-coalgebra morphism) which follows in a similar way.
Question: what kind of structure is this? It's not a double category, as I initially suspected. We'd have at least 2 types of 0-cells and three types of 1-cells.
Hylomorphisms (algebra to colagebra morphisms) form a profunctor . Then the category you're describing is the [[collage]] of this profunctor. It is basically plus the addition of hylomorphisms that link the two otherwise disjoint categories
Matteo Capucci (he/him) (Jan 26 2023 at 16:05):If you want morphisms also in the other direction you take the collage of the cohylomorphisms (I really hope they're not really called like this) profunctor , going the other way, and then you take another pushout:
image.png
Bruno Gavranović (Jan 27 2023 at 00:21):What's the intuition between thinking about the profunctor between the respective categories?
Matteo Capucci (he/him) (Jan 27 2023 at 10:23):Profunctors are the way you specify [[heteromorphisms]], that is morphisms between objects of different categories. Hylomorphisms are indeed morphisms between objects naturally residing in different categories. The collage collates homomorphisms and heteromorphisms in one single category.
Bruno Gavranović (Jan 29 2023 at 09:41):Ah I see. This makes sense, given the comment from @Nathanael Arkor about the type of morphism being fixed after I know the type of domain and codomain.
Bruno Gavranović (Jan 29 2023 at 09:42):James Deikun said:
Ah, well, it wouldn't be the standard category of dialgebras but a "super category" of them. (And you could also combine this with the other construction and have dialgebras in a suitable category of relations ... but you would still want the 'twisted' morphisms so you could formally change which side is which.)
What's a super category? :big_smile: Is your suggestion the same as @Matteo Capucci (he/him)'s?
Matteo Capucci (he/him) (Jan 29 2023 at 09:44):I think by 'supercategory' James means a category that contains dialgebras (as opposite to a subcategory which is contained)
James Deikun (Jan 29 2023 at 18:10):It's kind of both that, and more specifically a -graded category where the "bosonic" subcategory is the standard category of (all) dialgebras and the "fermionic" morphisms are the "twisted" ones above.
John Baez (Jan 30 2023 at 06:22):Wow, so really "super" in the sense of physics.
Matteo Capucci (he/him) (Jan 30 2023 at 11:30):To myself I was like 'imagine if he's actually meaning that super'... well :laughing:
Matteo Capucci (he/him) (Jan 30 2023 at 11:32):James Deikun said:
It's kind of both that, and more specifically a -graded category where the "bosonic" subcategory is the standard category of (all) dialgebras and the "fermionic" morphisms are the "twisted" ones above.
Is a -graded category a category enriched in -modules?
This reminds me of this nLab article I was looking at a few days ago. I wonder if one can really make sense of this construction in terms of that.
James Deikun (Jan 30 2023 at 12:53):A -graded category is a category equipped with a functor to .
Bruno Gavranović (Jan 30 2023 at 12:56):Hm, how does this functor give us these two types of morphisms?
James Deikun (Jan 30 2023 at 12:57):The two types of morphisms are the "even" ones that are mapped to 0 (aka id) and the "odd" ones that are mapped to 1.
James Deikun (Jan 30 2023 at 12:58):By functoriality you get that an odd morphism composed (either side) with an even morphism is odd, and composing two of the same is even.
John Baez (Jan 30 2023 at 18:20):Here a graded category is one where the objects come in two kinds, called 'even' and 'odd'. Usually it applies to a monoidal category where tensoring an object of the ith kind with one of the jth kind gives one of the (i+j)th kind, where i, j . But I don't know if that's happening here.
And usually it applies to a symmetric monoidal Ab-enriched category where the symmetry works a certain way: switching two 'odd' objects introduces a sign. But that's not happening here.
dusko (Feb 06 2023 at 09:50):Bruno Gavranovic said:
Given an endofunctor , it's known that we can form a category whose objects are its algebras (and morphisms are algebra homomorphisms), and a category whose objects are its coalgebras (whose morphisms are coalgebra homomorphisms).
I've recently learned that there is also a notion of a coalgebra-to-algebra morphism (making the appropriate diagram commute) and I'm wondering whether all this data can be organised as some sort of a double, or n-fold category.
It looks like we want two different types of 0-cells (algebras, and coalgebras), where morphisms between the same types of cells would be homorphisms of algebras and respectively coalgebras. Then also we could have a morphism between different types of 0-cells - which would be the coalgebra-to-algebra morphisms. (There's also the algebra-to-coalgebra morphism) which follows in a similar way.
Question: what kind of structure is this? It's not a double category, as I initially suspected. We'd have at least 2 types of 0-cells and three types of 1-cells.
One source of coalgebras over algebras is operational semantics of computation: languages or tests are given as algebras, and the meaning is given by coalgebras over algebras. This is how Gordon Plotkin has been modeling things since I think the 1900 and 90s, and that was summarized in a LICS paper he had with Daniele Turi. Then Ben Worrell and Mike Mislove and I noticed that any semantical correlation is based on evaluating algebras against coalgebras. That is in a paper I think under coalgebra on dusko.org. (I think the published version was truncated. My son was just born and I didn't even register the question.) But then later it turned out that tangent and cotangent bundles arise by testing coalgebras over algebras, and Bertried Fauser even worked out general relativity in that format. That is in a paper called Testing Smooth Coalgebra by Vector Algebra or something like that, which is surely on arxiv. And then that was also used in concept mining for recommender systems and recently to battle echo chambers. And when the coalgebras are the news stream and the algebra the user preferences, then everything gets saturated, and then you have proper coalgebras for a comonad over algebras for a monad. That is in the nucleus paper https://arxiv.org/abs/2004.07353. Aha and here is the smooth coalgebra: https://arxiv.org/abs/1402.4414.
But all that is under the assumption that we start from a problem, and a solution takes a shape of coalgebras over algebras (which turns out to happen whenever the problem is semantical). If we want to know what shape can coalgebras over algebras take in general, then I don't see why there should be a maximally general answer. Most structures can be generalized and if anything, at least the higher dimensions are not in a short supply in our little universe : )