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: learning: questions

Topic: When/why are horn fillers universal?

James Deikun (Jan 14 2023 at 00:03):

In opetopic sets the universality conditions for horn fillers are put in by hand. However, a lot of other higher-dimensional structures defined by horn filling conditions seem to have nice universal properties on the fillers too. How does this work?

Amar Hadzihasanovic (Jan 14 2023 at 08:43):

It comes down to whether, in whatever model you are using, universality is "algebraic" or not, in the sense that universal cells in a fibrant object are or are not classified by morphisms from a "representing" object.

When this is the case, you can avoid specifying "these cells need to be universal in the filler", just replace the "representing object of a generic cell" with the "representing object of a universal cell" whenever needed.

Amar Hadzihasanovic (Jan 14 2023 at 08:46):

For example, in quasicategories as a model of $(\infty, 1)$-categories, every simplex of dimension 2 or higher is "universal"/a weakly invertible cell, so universal n-simplices for n > 1 in fibrant objects are just classified by the "generic" n-simplex, and you don't need to specify anything.

Amar Hadzihasanovic (Jan 14 2023 at 08:51):

However, if you try to use simplicial sets to model $(\infty, n)$-categories for $n > 1$, then universal cells stop being algebraic; instead, universality is specified itself by lifting properties, different from those of generic cells.
So what people do is introduce a "marking" of cells that are meant to be universal, and have different lifting conditions depending on the marking.
This is how the complicial model of higher categories works, and similarly the comical model based on marked cubical sets.

Amar Hadzihasanovic (Jan 14 2023 at 08:54):

The "marking" is what you call "put in by hand" universality conditions... I have not seen opetopic models presented in terms of "marked opetopic sets" but I think that it would be one way of specifying the same data.

Amar Hadzihasanovic (Jan 14 2023 at 08:55):

AFAIK my own model of higher categories based on diagrammatic sets is the only model of this kind where universal cells are "algebraic" in all dimensions.

Amar Hadzihasanovic (Jan 14 2023 at 09:01):

(Of course the marking itself is a way of "algebraicising" universality, since the generic shape with a marked top-dimensional cell becomes a representing object for universal cells of that shape and dimension. When I say "non-algebraic" I mean "non-algebraic wrt the pure, unmarked shapes".)

James Deikun (Jan 14 2023 at 14:03):

Ah, so basically:

• Horn-filling conditions basically can't distinguish a cell from a cell of the same "shape" in the same context, so
• if you need some cells to have a "universality" that some cells in an otherwise identical context don't, then
• you need to have some "shadow cell" attached to it to identify it, which
• is often abbreviated away by "marking" some cells?

Amar Hadzihasanovic (Jan 14 2023 at 14:22):

Yeah, that's it: if you already have a “model” of a universal cell in your category, then you can just attach this model; if you don't have such a “model”, then you have to introduce markings (which takes you out of your original shape category and into a category of “marked shapes”).

James Deikun (Jan 14 2023 at 14:25):

Which is annoying when you're looking for a way to specify some data geometrically since the marking seems somewhat algebraic. Like, you can't ensure everything that should be marked is, because horn filling conditions are like Horn clauses (heh), "whenever this conjunction of conditions obtains so does this other one", you can't say "if (whenever X obtains Y obtains) then Z obtains".

James Deikun (Jan 14 2023 at 14:28):

(So you can't say "whenever a cell always acts like it has a marking, it actually is marked" as a horn filling condition.)

James Deikun (Jan 14 2023 at 14:33):

Is there any connection between this need for auxiliary shapes and, say, non-fullness of inclusions? It seems clear that needing auxiliary shapes doesn't imply non-fullness of the inclusion but does the other direction hold?

Amar Hadzihasanovic (Jan 14 2023 at 18:57):

James Deikun said:

(So you can't say "whenever a cell always acts like it has a marking, it actually is marked" as a horn filling condition.)

Yeah, no, that's an extra property of fibrant objects. In the “complicial set” model these ones are called “saturated”.

Amar Hadzihasanovic (Jan 14 2023 at 19:03):

James Deikun said:

Is there any connection between this need for auxiliary shapes and, say, non-fullness of inclusions? It seems clear that needing auxiliary shapes doesn't imply non-fullness of the inclusion but does the other direction hold?

You mean whether “every morphism between fibrant objects is a 'functor'”?
Well, definitely if universal cells are “algebraic”, then they are automatically preserved by every morphism of fibrant objects: if a universal cell $x$ of shape $S$ in $X$ fibrant is represented by a morphism $\tilde{x}: S^U \to X$, then a morphism $f: X \to Y$ with $Y$ fibrant takes $x$ to the cell represented by $\tilde{x};f: S^U \to Y$ which is universal.

Amar Hadzihasanovic (Jan 14 2023 at 19:05):

And then if composites and units are exhibited by universal cells, this should imply that any morphism sends composites to composites and units to units.

Amar Hadzihasanovic (Jan 14 2023 at 19:06):

So yeah, by contraposition if you don't have fullness, then you don't have “algebraic universal cells” and you need markings/auxiliary shapes.

Amar Hadzihasanovic (Jan 14 2023 at 19:08):

I would be very surprised if there were cases in which you don't have “algebraic universality”, yet all morphisms preserve universal cells, but it's not obvious that it is impossible.

Amar Hadzihasanovic (Jan 14 2023 at 19:09):

Of course “not all morphisms preserve universals” may be your intended behaviour, for example in representable multi(bi)categories where generic morphisms are lax (monoidal) functors, and the ones that preserve universals are the strong functors.

James Deikun (Jan 14 2023 at 23:12):

Amar Hadzihasanovic said:

I would be very surprised if there were cases in which you don't have “algebraic universality”, yet all morphisms preserve universal cells, but it's not obvious that it is impossible.

In the case of [[virtual equipments]] all the virtual double category functors between them preserve the Cartesian squares for the restrictions (but not the opCartesian squares for the units). So at least the inclusion of virtual equipments in virtual double categories with units/unit preserving functors is full, and it seems like they are distinguished as the virtual double categories (with units) over 1 by restriction-lifting functors. Yet I can't see how to describe restriction-lifting functors, or restrictions full stop, by a horn-filling condition without an extra shape for the Cartesian squares.

James Deikun (Jan 15 2023 at 01:56):

Also is a big part of the whole [[monads with arities]] apparatus about finding a way to pick a "basis of shapes" where a particular monad is expressible with horn filling conditions?

James Deikun (Jan 15 2023 at 02:03):

(Though I think that involves the unique/strict version iirc.)

James Deikun (Jan 15 2023 at 17:50):

I think it would be relatively okay to have to deal with "saturated fibrant cofibrant" objects if "saturated" can be defined in an adaptable way with some kind of good formal properties. I think a reasonably good test of such a thing is writing down completeness under a few kinds of limits and 2-limits in Cat/Str2Cat, starting with the simple but essential "terminal object" ... without saturation though it feels like getting the costs of algebraic structure without the advantages.