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Stream: theory: category theory

Topic: Weak initial algebras?

Bruno Gavranović (Jan 15 2024 at 17:55):

It is known that starting with a category $\mathcal{C}$ and an endofunctor $F : \mathcal{C} \to \mathcal{C}$ we can form the category of $F$-algebras where objects are $F$-algebras and morphisms are $F$-algebra homomorphisms. And initial algebras, if they exist, are categorical semantics for inductive types.

I'm wondering what the higher dimensional generalisation of this to 2-categories is, and if someone can help point to some literature.

Namely, if instead we start with a 2-category $\mathcal{B}$ and a 2-endofunctor $F : \mathcal{B} \to \mathcal{B}$, it seems natural to form the 2-category where objects are still $F$-algebras, but morphisms are now lax $F$-algebra homomorphisms, i.e. where the appropriate diagram is defined only up to a 2-cell in $\mathcal{B}$. Likewise, 2-cells in $F$-Alg are 2-cells in $\mathcal{B}$ between the maps between the carriers.

Now, I imagine this calls for a generalisation of the concept of an initial algebra. I imagine we are to look at lax initial algebras (or quasi, I believe they might conicide in this case), i.e. an object in $F$-Alg such that there exists a 1-morphism to any other algebra, but where now instead the 2-cell is unique.

This is as far as I've gotten, and I'm wondering: how does one compute initial algebras here? I imagine one should genrealise the computation of the colimit of the initial chain to two dimensions, and I'm wondering has this been worked out -- and are there any interesting examples.

Sam Speight (Jan 15 2024 at 22:14):

Have a look at https://arxiv.org/abs/1402.0761.

John Onstead (Jan 16 2024 at 04:03):

This discussion brings up something I've been curious about for a while, namely, the universal property of the category of F-algebras. What universal property does it satisfy- for instance, if it is some form of 2-limit/2-colimit (as it almost certainly is given "all concepts are weighted limits") what is both the diagram functor and the weighting functor of this limit/colimit? And when we extend to the 2-categorical case, now we find ourselves with 4 different categories of F-algebras- the lax, oplax, pseudo, and strict versions. Which universal properties/3-limits do each of these categories satisfy (diagram and weighting) and how do they differ and compare with one another?

Bruno Gavranović (Jan 16 2024 at 13:37):

This is related to one other question I had: if strict initial algebras are categorical semantics of inductive types, what kind of inductive types do lax/oplax initial algebras are categorical semantics of?

Are there any enlightening examples?
I'm having trouble imagining what the extra degree of freedom (lax/oplax) could even model.

Sridhar Ramesh (Jan 16 2024 at 16:24):

John Onstead said:

This discussion brings up something I've been curious about for a while, namely, the universal property of the category of F-algebras. What universal property does it satisfy- for instance, if it is some form of 2-limit/2-colimit (as it almost certainly is given "all concepts are weighted limits") what is both the diagram functor and the weighting functor of this limit/colimit?

The category of F-algebras is the inserter from F to the identity.

Mike Shulman (Jan 16 2024 at 17:29):

And the various 2-categories of F-algebras are likewise the inserter from F to the identity in the "3-categories" whose 2-cells are lax, oplax, pseudo, or strict natural transformations. (I put "3-category" in quotes becuase lax and oplax transformations are not the 2-cells of a 3-category in the usual sense, but as soon as you make sense of what 3-dimensional structure they are the 2-cells of, the statement should become true.)

John Onstead (Jan 16 2024 at 22:43):

Thanks, inserters seem very interesting and I haven't learned much about them yet, so I will make sure to do so!
However, I am not sure what 3-dimensional structure a lax natural transformation would be a part of. If not a 3-category, maybe a tricategory? If not, it seems like this limit would be hard to take since you need some category to define the limit within! That's why I was thinking that maybe there's some concept of a "3-limit", maybe even one that categorifies the inserter from 2-categories to 3-categories (and so taking 3-cells more into account), one can take within the "typical" tricategory 2Cat of 2-categories or bicategories to get these categories.

Kevin Arlin (Jan 16 2024 at 22:52):

There are three so-called Gray monoidal products on the category of 2-categories (and strict 2-functors): they're adjoint to the closed structures where the morphisms in an exponential are respectively pseudo-, lax-, or oplax natural transformations. Categories enriched in Gray are a perfectly normal enriched category theory concept that also models 3-dimensional category theory more faithfully than strict 3-categories, and in particular you can talk about weighted limits in there in the perfectly usual way. This may be the way you'd want to go here.

Mike Shulman (Jan 16 2024 at 23:16):

Yes, just as there's a $\rm Gray_{pseudo}$-enriched category of strict 2-categories, strict 2-functors, pseudonatural transformations, and modifications, there's a $\rm Gray_{lax}$-enriched category of strict 2-categories, strict 2-functors, lax natural transformations, and modifications, and similarly in the oplax case. If you want to include bicategories and pseudofunctors, there's probably a suitably weak version of that too, but I don't know that it's been written down.

David Kern (Jan 19 2024 at 14:14):

This $\rm Gray_{lax}$-enriched bicategory should be the self-enrichment of $\rm Bicat$ on its Gray biclosed monoidal structure, right? In that case, this follows from the constructions of a Gray tensor product of $(\infty,2)$-categories (of which there are a few, in various models — that I haven't kept track of), specialised to bicategories — and if you really don't want to go to $\infty$-bicategories, the arguments for a "model-independent" characterisation of the Gray tensor product in Campion–Maehara's construction ought to specialise well to a characterisation for bicategories.

Kevin Arlin (Jan 19 2024 at 17:22):

Most models of $(\infty,2)$-categories don’t specialize directly to bicategories but to an analogue with a non-algebraically-defined composition. So the specialization isn’t immediate, although maybe the necessary links are in the literature.