The n-Category Café

I’ve thought about it a bit. One approach to defining a “quasi–double-category” is to look for horn-filling conditions on a bisimiplicial set. One could also pick a notion of $(\infty,2)$-category and generalize Wood’s definition of proarrow equipment. However, I think my inclination would be to fill in the analogy

operad : dendroidal set :: virtual double category : ??

and then write down horn-filling conditions on the ??s to obtain a notion of quasi-vdc. The universal properties of units, composites, and cartesian fillers should then also be expressible in this language, analogously to the notion of Cartesian fibration between quasicategories. A quasi-vdc with units and cartesian fillers one might then call an “$(\infty,2)$-category with a (virtual) proarrow equipment.”

As for what ?? is, probably the general stuff about nerves and familial representability would be helpful. However, I think one could also give a direct definition in terms of the equipment of $T$-spans, where $T$ is the “free category” monad on directed graphs.

look for horn-filling conditions on a bisimiplicial set.

Or maybe on a dendroidal simplicial set?

(Non-planar dendroidal, that is.)

Modelling how the virtual double category pairs the aspects of a multicategory with that of a category “running parallely”.

There is a known model structure on dendroidal simplicial sets that localizes the simplicial direction such as to be Quillen equivalent to the standard Moerdijk-Cisinki model structure. This is useful for many computations in the ordinary model structure.

I am not sure if it is known what this models without the localization in the simplicial direction.

In other words, my remark here probably amounts to asking/suggesting if a dendroidal refinement of $\Theta$-spaces might be a way to model double $(\infty,r)$-categories, i.e. in terms of localizations of dendroidal presheaves on $\Theta_r$.

Even if not, that would be something interesting to look at. The model structure should be the evident one.

(NB:as usual, the links provided are just for the record and for other readers. )

First of all, are the “finite lists whose elements can be permuted” here the same as multisets (free commutative monoids)?

If you read the paper (hint, hint), you’ll find that the answer is no. The relevant monad is the “free symmetric monoidal category” monad. So “can be permuted” was a rough way of talking, not about unordered lists, but about lists equipped with an isomorphism to each of their permutations.

Conversely, are the “objects labeled by an integer” of graded categories related to free abelian groups?

Not exactly, but they are related to free Z-sets.

(That’s a use of “conversely” that I haven’t seen before… are free abelian groups the converse of free commutative monoids? (-: )

For many years I’ve regretted the name ‘fc-multicategory’ and hoped for a better alternative.

I’m glad that we aren’t stepping on your toes by trying to rename them! (-:

I have one reservation, which is that it makes a multicategory sound a bit like a defective or wannabe monoidal category.

This is true. One could argue that a multicategory is, in fact, a not-quite-as-good alternative to a monoidal category. Certainly, there are things that one can do with monoidal categories that one can’t do with multicategories. I also think that “virtual” is not as bad as “pseudo” in this respect, because in ordinary language, “virtually X” means “almost X” or “might as well be X” (TheFreeDictionary says “virtual” means “existing or resulting in essence or effect though not in actual fact, form, or name”), whereas “pseudo” has the connotation of a fake imitation.

That said, if you’ve got a better suggestion, I’m all ears. I do like the idea of attaching an adjective to “monoidal category” (resp. “double category”) to obtain a name for “multicategory” (resp. “fc-multicategory”). But I’m not especially attached to “virtual,” if anyone has a better suggestion.

As you know (but lots of people don’t), fc-multicategories are T-multicategories in the case where T is the free category monad fc on the category of directed graphs.

That’s true, if “T-multicategory” is meant in your sense (which Geoff and I call a T-monoid in a virtual equipment of spans). But fc-multicategories are also T-multicategories where T is the free double category monad on the virtual equipment of Cat-graphs and Cat-graph-profunctors, where now “T-multicategory” is meant in the slightly different sense of Hermida (which Geoff and I call a normalized T-monoid). Likewise, ordinary multicategories are T-monoids where T is the free-monoid monad on (spans of) sets, but are also normalized T-monoids where T is the free-monoidal-category monad on categories and profunctors. And quite generally, T-monoids can be identified with normalized Mod(T)-monoids for any monad T on a virtual equipment.

However, some other notions of multicategory can only be defined as normalized T-monoids for some T which is not of the form Mod(S). For instance, symmetric multicategories are normalized T-monoids, where T is the free-symmetric-monoidal-category monad on categories and profunctors, and this T is not of the form Mod(S). For that reason, Geoff and I decided that “normalized T-monoid” is a better definition of “generalized multicategory,” since it is actually more general. So when I said that we call T-multicategories “virtual T-algebras,” I meant that we call normalized T-monoids “virtual T-algebras.” Thus, since fc-multicategories are normalized T-monoids for the free-double-category monad T, they are “virtual double categories.” A “virtual category” would be a normalized T-monoid for the free-category monad, which is actually a pretty weird sort of structure. (I hope you agree that fc-multicategories should be called “virtual double categories” and not “virtual categories”? Since they are manifestly double-category-like.)

All of this is explained in the paper (hint, hint). It’s possible that it’s not explained very well, though; we had a lot of trouble trying to get this across clearly. So if you have any suggestions about how to improve the exposition, we’d love to hear them.

Tom wrote:

For many years I’ve regretted the name ‘fc-multicategory’…

Good! I haven’t paid much attention to them, precisely because their name leaves me unable to tell what they’re like, and therefore unable to guess when they might be the thing I need.

(I also get them mixed up with $T$-multicategories, which are somewhat easier to fathom, as long as one is the elite club of people who know that ‘$T$’ means ‘monad’, and that multicategories involve the ‘free monoid’ monad.)

Tom wrote:

I believe this is why the term rig was coined, even though the synonymous term semiring already existed.

I don’t know why it was coined, but there are three additional reasons I like the term ‘rig’. First, the term ‘semiring’ suggests an analogy

$$semiring:ring::semigroup:group$$

but this is sadly misleading, since a semiring lacks only negatives, while a semigroup lacks not only negatives but also the identity element.

(A semigroup is slightly less than half a group: it should be called a ‘$(\frac{1}{2}-\epsilon)$-group’, while a monoid could be called a ‘$(\frac{1}{2} + \epsilon)$-group’.)

Second, the term ‘rig’ always gets a laugh when you explain how it means ‘ring without negatives’. And this laugh makes the definition impossible to forget.

Third — but this one is a lot less important — I think it’s fun to point out that $Card$, the class of all cardinals, is a big rig.

If only we could get away with that ;)

In other words, whether the Makkai-Lurie notion of ultracategory is related to the Clementino-Tholen notion of ultracategory?

I wrote that they were different on the $n$Lab having seen statements in the latter like

In an ultracategory $A$ we can think of the hom-set $A(\xi, y)$ as the set of all ‘proofs’ for the validity of the statement ‘$\xi$ converges to $y$’,

for an ultrafilter $\xi$ on the objects of $A$.

Is there some relation between these versions of ultracategory?

Since a Makkai-Lurie ultracategory has an operation indexed by ultrafilters and a Clementino-Tholen ultracategory has morphisms whose domains are indexed by ultrafilters, one might hope that they are related analogously to monoidal categories and ordinary multicategories, respectively, with the Makkai-Lurie ultrapowers being characterized by a universal property inside a Clementino-Tholen ultracategory. I expect that isn’t quite right, but maybe one or the other of the definitions can be modified to make it true.

I was hoping that it would be possible to construct a line in Table 1 of your A unified framework for generalized multicategories which has Makkai-Lurie ultracategory in the ‘Pseudo $T$-algebra’ column.

In Lurie’s paper it is shown that a set, $X$, with an ultrastructure is a compact Hausdorff topology on $X$. So maybe I need to think about the relationship of this with the other characterization of a compact Hausdorff space as an algebra of the ultrafilter monad.

How do ultrafilters on indexing sets of points in a space relate to ultrafilters on the set of points of a space?

Yes, exactly.

An ultrafilter on an indexing set of points can be pushed forward to an ultrafilter on the set of points itself….

So one might consider the ultrafilter monad on $Span(Set)$. $T$-monoids are Clementino-Tholen ultracategories and pseudo $T$-algebras are something like Makkai-Lurie ultracategories.

Why consider things indirectly via indexing sets? Is it a size issue, e.g., to handle large categories?

Is there not an issue with the principle of equivalence for categories and ultrafilters on their sets of objects, e.g., in the case of a single object and infinitely many isomorphic copies?

I suspect the point is more about the lack of idempotence. E.g. when considering meets and joins in a poset, it suffices to consider the input to be a subset of its elements, since meets and joins are idempotent. But when considering (co)limits — even just (co)products — in a category, we can’t assume that we have a subset of objects of the category, since we might want to take the product of a bunch of copies of a single object.

This is essential for Makkai-Lurie, since their canonical example is an ultraproduct, and those are not idempotent: an ultrapower of a single object is generally not the same as that object. Which suggests to me that Clementino-Tholen were on the wrong track for categorifying the ultrafilter monad on $Rel$. As always, it’s important to have examples to justify a definition.

Okay, now I’ve finally had a look at Lurie’s definition, and I think I can say something more intelligent relating it to generalized multicategories.

Let $Set$ be the (large) category of small sets and $CAT$ the (“very large”) category of large categories. Let $\mathcal{U}$ be the large category whose objects are small sets $S$ equipped with an ultrafilter $\mu$, and whose morphisms $(S,\mu) \to (T,\nu)$ are functions $f :S\to T$ such that $\nu = f_\ast(\mu)$. For a large category $X$, let $U X$ be the large category $$U X = \sum_{(S,\mu)\in \mathcal{U}} X^S.$$ Thus the objects of $U X$ are triples $(S,\mu,M)$ where $(S,\mu)\in\mathcal{U}$ and $m:S\to X$. Its morphisms $(S,\mu,M) \to (T,\nu,N)$ are pairs $(f,\xi)$ where $f:(S,\mu)\to (T,\nu)$ in $\mathcal{U}$ and $\xi$ is a transformation from $N\circ f:S\to X$ to $M:S\to X$, i.e. for each $s\in S$ we have $\xi_s:N_{f(s)} \to M_s$.

I claim that $U$ is a pseudomonad on $CAT$ and extends to the double category $PROF$. The unit $X\to U X$ sends $x\in X$ to $(1,\eta,x)$ where $x:1\to X$ and $\eta$ is the unique ultrafilter on $1$. For the multiplication, an object of $U U X$ consists of $(S,\mu)\in \mathcal{U}$, a family of sets $(T_s)_{s\in S}$ equipped with ultrafilters $\nu_s$, and a function $M:(\sum_{s\in S} T_s) \to X$; the multiplication $U U X \to U X$ sends this to the set $\sum_{s\in S} T_s$ equipped with $M$ and the ultrafilter $\int \nu d\mu$ defined for $A\subseteq \sum_{s\in S} T_s$ $$A\in \int \nu d\mu \iff \{ s\in S \mid A \cap T_s \in \nu_s \} \in \mu.$$ I haven’t checked the pseudomonad laws or the extension to $PROF$, but I’d be surprised if anything goes wrong.

Thus, we get a notion of “$U$-multicategory”, a.k.a. normalized $U$-monoid in $PROF$, which includes pseudo $U$-algebras as the representable case. In fact, as noted in Cruttwell-Shulman Theorem 9.2, normalized $U$-monoids also include normal colax $U$-algebras as a “weakly representable” case, and it’s the latter that I claim are very similar to Lurie’s ultracategories.

Suppose $\mathcal{M}$ is a normal colax $U$-algebra. Its algebra action $U\mathcal{M}\to \mathcal{M}$ sends each triple $(S,\mu,M)$ to an object $\int_S M d\mu$, as in datum (1) for a Lurie ultracategory. The functorial action of this map naturally combines the functoriality of Lurie’s (1) with the “ultraproduct diagonal” maps $\int_T M_t d(f_\ast\mu) \to \int_S M_{f(s)} d\mu$, for any $f:(S,\mu)\to (T,\nu)$ and $M:T\to \mathcal{M}$, from his Notation 1.3.3. Lurie includes the extra axiom that these maps are isomorphisms when $f$ is injective; we could regard this as an extra property of an ultracategory, or perhaps incorporate it into the monad $U$ with a coinverter.

The adjective “normal” means that the colax unitor relating the composite $\mathcal{M}\to U \mathcal{M} \to \mathcal{M}$ to the identity is an isomorphism. This map looks like $\int_1 M d\eta \cong M$, where $(1,\eta)$ is again the unique ultrafilter on $1$. This isomorphism is a special case of Lurie’s datum (2), while all of datum (2) can be recovered from this special case together with an invertible ultraproduct diagonal applied to the injection $s:1\to S$.

The colax associator of a colax $U$-algebra is a transformation $$\int_{\sum_{s\in S} T_s} M_{s,t} d(\int \nu d\mu) \to \int_S (\int_{T_s} M_{s,t} d\nu_s) d\mu.$$ If we specialize to the case when all the $T_s$ are the same set $T$ (though the ultrafilters $\nu_s$ may be different) and $M : (\sum_{s\in S} T) \cong S\times T \to \mathcal{M}$ factors through the projection $\pi : S\times T\to T$, and compose with the ultraproduct diagonal map associated to this projection, we get Lurie’s datum (3): $$\int_T M_t d(\pi_\ast \int \nu d\mu) \to \int_S (\int_T M_t d\nu_s) d\mu$$ Conversely, if we take $T = \sum_{s\in S} T_s$ and consider the ultrafilters $(i_s)_\ast \nu_s$, where $i_s : T_s \to \sum_{s\in S} T_s$ is the injection, then from Lurie’s (3) we recover the colax associator.

Lurie’s axiom (B) constructs the invertible ultraproduct diagonal maps from datum (3). His axiom (A) looks like one of the axioms relating the colax unitor and associator, while his axiom (C) looks like the associativity axiom for the colax associator — with the associator reformulated into datum (3) in both cases. I don’t see any version of the axiom relating the unitor and associator in Lurie’s definition; it would be something like that the map induced from the associator $$\int_{\sum_{s\in S} 1} M_s d(\int \eta d\mu) \to \int_S (\int_1 M_s d\eta) d\mu \cong \int_S M_s d\mu$$ is the isomorphism induced by the bijection $S \cong \sum_{s\in S} 1$. But perhaps this follows automatically from Lurie’s choice of which data to take as primitive and which as derived.

It would be some extra work to prove more carefully that Lurie’s ultracategories are equivalent to normal colax $U$-algebras (satisfying the invertibility condition) — so far we’ve just observed that the data in each can be constructed from the other and that the axioms look analogous. But this makes it seem reasonably likely to me that it’s true.

One more note. Lurie’s primary example of an ultracategory is a category with ultraproducts constructed in the usual categorical way as a directed colimit of products: $$\int_S M_s d \mu = \colim_{A\in \mu} \prod_{s\in A} M_s.$$ I suspect that if the products exist but the colimits don’t, this still defines a $U$-multicategory, with $$\hom((S,\mu,M),N) = \lim_{A\in \mu} \hom(\prod_{s\in A} M_s, N).$$

Great. I was worrying about ultraproducts.

Must take a look later at the left/right ultrastructure concepts (after Theorem 0.0.4 and before Warning 1.0.4), which I was imagining related to colax/lax algebra morphisms.

Then, regarding your final post, the issue in Remark 1.0.3.

Assuming all the above goes through, that raises plenty of questions for me:

• Does providing a generalized multicategory setting mean that some of Lurie’s results on ultracategories follow from general theory?

• Is the wider class of normalized $U$-monoids interesting in itself?

• Do we gain anything in terms of understanding the Makkai dual embedding of the 2-category of pretoposes into ultracategories?

• Is any light shone on the proximity of ultracategories to pyknoticity/condensedness, as in the claim that ultracategories are pseudopyknotic categories (here)?

• Is there something condensity-monad-esque lurking about, as I was hoping here?