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Stream: theory: category theory

Topic: Conjectures on generalized equipments

James Deikun (Dec 28 2022 at 15:59):

It seems like there ought to be some definition of "operad with shapes" and "framed thickening" that allows generalizations of some classic definitions and results using categories and equipments. An "operad with shapes" should basically be:

1. A category of shapes $S(O)$.
2. Operations $o$, each of which has a domain $\mathrm{dom}(o)$ consisting of multiple shapes pasted together, and a codomain $\mathrm{cod}(o)$ consisting of a single shape, and specifications for each boundary facet of $\mathrm{cod}(o)$ identifying it as $o'(a,b,c,d,...)$ where $a,b,c,d,...$ are appropriate facets of $\mathrm{dom}(o)$.
3. Compositions, equations of the form $o(o'(a,b,...),o''(c,d,...),...)=p(a,b,c,d,...)$ (nesting depth is arbitrary). Compositions are "dependent" on whichever other compositions are needed to make them well-typed.
4. Associativity of compositions. This can be expressed in such a way that the RHS is always a simple composition and the LHS is a "composition of compositions", which is important.
5. Maybe some scaffolding needed to identify what "units" and "restrictions" are.

James Deikun (Dec 28 2022 at 16:01):

Say we have a specific "operad with shapes" $O$. Then the "framed thickening" $T(O)$ should roughly consist of:

1. Shapes inherited from $O$.
2. New shapes that come from the operations of $O$. The new shape for $o$ has $\mathrm{cod}(o)$ as "front", $\mathrm{dom}(o)$ as "back", and as "sides" the new shapes for specifications of $o$ glued to their arguments in "back" and corresponding boundary cell in "front". Note that since every shape has at least an identity operation, you get at least one new "thick" version of each inherited shape.
3. New operations that come from the compositions of $O$. The domain comes from the LHS, the codomain from the RHS, and the specifications from the dependencies. The compositions of multiply-nested identities give at least an $k$-ary operation for each $k$ on the new "simply thick" cells.
4. New compositions that come from the associativity equations of $O$. These at least make the new $k$-ary operations above into an unbiased associative "operation".
5. New associativity equations that the above compositions should satisfy by construction.
6. Some rules for existence of "units" and "restrictions" TBD.

James Deikun (Dec 28 2022 at 16:28):

Again let us take a specific "operad with shapes $O$, where only a single shape $\mathrm{Obj}$ is standalone (0-dimensional). Then, conjectured:

1. $O$-algebras assemble into a large $T(O)$-algebra $O\text{-Alg}$.
2. For a $T(O)$-algebra $X$, there is a natural way to define $X$-enriched $O$-algebras following Leinster.
3. $X$-enriched $O$-algebras assemble into a large $T(O)$-algebra $O\text{-Alg}_X$ (with (1) as a special case).
4. $\mathrm{Obj}$-shapes in $O\text{-Alg}_X$ are identified with $X$-enriched $O$-algebras.
5. $\mathrm{Id}_{\mathrm{Obj}}$-shapes ("new arrows") in $O\text{-Alg}_X$ are identified with maps of $X$-enriched $O$-algebras.
6. Other inherited shapes in $O\text{-Alg}_X$ are identified with some appropriate variant of "cofibrations" or "barrels".
7. You can "embed" $X$ in $O\text{-Alg}_X$ following David Jaz Myers.
8. You can "lift" an $O$-algebra $A$ to a "vertically trivial" $T(O)$-algebra $L(A)$.
9. Maps from $L(A)$ to $O\text{-Alg}$ "display" maps into $A$.

Nathanael Arkor (Dec 28 2022 at 16:57):

Do you have any concrete examples other than categories that you're trying to generalise?

James Deikun (Dec 28 2022 at 17:22):

Sets are one concrete example; the idea of a $C$-enriched set can be defined and then the Myers embedding factors through the Yoneda embedding, and functors from $\mathrm{Disc}(X)$ to $\mathrm{Set}$ display functions. Objects and arrows in $\mathrm{Set}$ are identified with sets and functions respectively, as expected, and similarly with $C$-enriched sets. Other examples include multicategories and virtual equipments; I've been trying to figure out what a "3-equipment" is and was struck by how much of what I was doing didn't really require a lot of special properties out of virtual equipments themselves.

James Deikun (Dec 28 2022 at 17:27):

Strict or Batanin $\omega$-categories are probably also examples that should work, so it might be fun to work out what happens.

James Deikun (Dec 28 2022 at 17:54):

To help (myself or whoever else wants to do it) with $\omega$-cats, I'm going to work out what $L(A)$ looks like:

1. For each inherited cell shape $s$ in $O$, $L(A)$ inherits the $s$-cells from $O$.
2. For each equation $o(a,b,c,...)=x$ that holds in $A$, $L(A)$ gets an $o$-shaped "new" cell.
3. Composition of the "new" cells is by substitution.
4. Units and restrictions should be uniquely determined by the thinness of the data.

James Deikun (Dec 28 2022 at 18:13):

What $\mathrm{Str\omega{}Cat}$ should look like:

1. Objects = pro-$0$-arrows = strict $\omega$-cats.
2. pro-$(n+1)$-arrows = two pro-$n$-arrows glued along hetero-$(n+1)$-cells that they respectively act on. I think these compose and have collages, which makes defining squares easier.
3. arrows = $0$-squares = strict $\omega$-functors
4. $n$-squares = strict $\omega$-functors from the collage of the composite of their source to the collage of their target, respecting the collage structure.

James Deikun (Dec 28 2022 at 18:16):

I think the "display" conjecture works fine like this, at least. It tells me some useful things about what "units" means, actually!

Mike Shulman (Dec 28 2022 at 18:43):

I presume you've looked at the literature on [[generalized multicategories]]? I would expect that once you make this precise, it would fit into those frameworks.

Mike Shulman (Dec 28 2022 at 18:44):

Your "framed thickening" sounds like the Baez-Dolan slice construction, is that what you have in mind to generalize?

James Deikun (Dec 28 2022 at 18:50):

It'd be nice to be able to do all this in the generality of Cruttwell-Shulman but so far I've only been able to make sense of it with categories of shapes that algebras are presheaves on.

I thought of the Baez-Dolan slice construction in connection with "framed thickening" but it's a bit different -- it forms higher cells and the "front" and "back" are similar, but the difference is there are now "sides".

Mike Shulman (Dec 28 2022 at 18:52):

Well, I wouldn't expect this to work in the generality of an arbitrary monad.

Mike Shulman (Dec 28 2022 at 18:52):

Is there something precise in the literature that you're trying to generalize with your "framed thickening"?

James Deikun (Dec 28 2022 at 18:54):

The constructions I'm trying to generalize are sets -> categories, categories -> virtual equipments, and my in-progress virtual equipments -> "3-equipments".

James Deikun (Dec 28 2022 at 18:56):

And yeah, an arbitrary monad would be almost certainly too much to ask, but monad with conditions in a way that would work in more settings would be nice.

Mike Shulman (Dec 28 2022 at 19:01):

How is what you want different from Leinster's $T'$ construction?

James Deikun (Dec 28 2022 at 19:11):

TBH I'm not sure of the exact difference; there must be one to get virtual equipments instead of virtual double categories but it has to do with units and restrictions which are the least worked-out parts of the story.

Mike Shulman (Dec 28 2022 at 19:14):

Hmm, I've never thought of the units and restrictions as incorporated in the notion of generalized multicategory, but an extra condition on top of that. So maybe what you want is a general notion of "unit and restriction" that applies to any $T'$?

James Deikun (Dec 28 2022 at 19:18):

Perhaps so. The example of $\mathrm{Str\omega{}Cat}$ shows that the "units" part can be "interesting"; there you need units for every dimension of pro-globe to make the display construction work.

James Deikun (Dec 28 2022 at 19:48):

I think Leinster's $M$ might be my $L$ ...

James Deikun (Dec 28 2022 at 21:48):

Okay, I spotted a difference between $T'$ and $T(O)$ ... for the former a $\mathrm{fc}'$-multicategory-enriched $\mathrm{fc}$-category will have a set of objects and sets of proarrows (w/pro-pro-arrow extents) while for the latter it would have a set of objects and pro-proarrows of proarrows. This happens because the $\mathrm{fc}$ monad lives on $\mathrm{Quiver}$ instead of $\mathrm{Set}$.

James Deikun (Dec 29 2022 at 00:03):

Maybe the/a function of "units" and their preservation here is to tweak the terminal $T(O)$-algebra to make it act like a "point". In this connection, might be interesting to see if there's a way to extend the "monoids and modules" construction to any Leinster generalized multicategory or else any "operad with shapes". Starting with the characterization "objects of $\mathrm{Mod}(A)$ are $O$-algebra maps $1 \to A$", is there a systematic way to define all the higher cells? (In the cases of sets and ordinary categories, $1$ already is a point, so that's a good sign ...)

James Deikun (Dec 29 2022 at 00:09):

Perhaps for inherited shapes $s$ the $s$-cells are maps from the "terminal $T(O)$-algebra with distinguished $s$-cell"? This might even work for new shapes but it seems more iffy to me...

James Deikun (Dec 29 2022 at 00:49):

If that works ... I think the structure corresponding to "with units" would be something like "for every shape $s$, every $s$-cell in the algebra embeds into a copy of the terminal object with distinguished $s$-cell as the distinguished $s$-cell in a universal way" ... the "in a universal way" is too vague though, I don't know what to really put there.

James Deikun (Dec 29 2022 at 00:59):

It does seem, though, that for virtual double categories, adding units and their corresponding opcartesian cells is enough to turn the walking object and proarrow into their terminal versions, but constructing some of the 2-cells for the walking arrow might require restrictions?

Mike Shulman (Dec 29 2022 at 01:30):

I don't have time to follow everything, but if you want a way to incorporate units into the generalized-multicategory structure you could look to [[augmented virtual double categories]] as a model.

James Deikun (Dec 29 2022 at 02:20):

I didn't even follow everything myself, as my "terminal algebra with distinguished $s$-cell" is actually no such thing but more like a "terminal $s$-barrel", and restrictions don't help make walking arrow-barrels terminal. (When I think of the fact that the ones besides $1$ are just terminal barrels though it seems less important to make all of them work though ... maybe only the object case even needs to work.)

I don't know whether [[augmented virtual double categories]] would help or hurt though as where the empty-targeted cells come from and when to make them is yet another thing to explain!