The n-Category Café

Thanks! Fixed.

What do you call a typo caused by clicking rather than typing? A clicko, maybe.

I have a friend who likes to use the word “braino” for an analogous purpose.

It seems kind of odd to just dismiss these as “centipede mathematics” when there are so many examples.

Exactly what the nLab says: “This a weakening of the concept of cartesian monoidal category, which might seem like pointless centipede mathematics were it not for the existence of interesting examples and applications.”

“the projection $X \otimes I \to X$ should be the identity”?

Yes, that’s true in any monoidal category with projections. The definition is set up so that whenever you have an equation involving the monoidal structure and the projections that’s true in every cartesian monoidal category, it’s also true in every monoidal category with projections. Since the projection $X \times 1 \to X$ in a cartesian monoidal category is always the identity (or strictly, the unit iso in the monoidal structure), the same is true in monoidal categories with projections.

I just dug out that problem sheet of Peter Johnstone’s that I mentioned (the one I learned this theorem from). This problem implicitly gives a definition of monoidal category with projections, and it appears to be very parsimonious. If I’m translating between notations correctly, it’s simply this:

A monoidal category with projections is a monoidal category $V$ together with natural transformations $$(\pi^1_{X, Y}\colon X \otimes Y \to X)_{X, Y}, \qquad (\pi^2_{X, Y}\colon X \otimes Y \to Y)_{X, Y}$$ such that $\pi^1_{I, I}$ is the unit isomorphism $I \otimes I \to I$.

Apparently, all the other properties follow from this one! Well, I guess what happens is this. I sketched the proof that every mon cat with projections is semicartesian. That proof only needs the naturality plus this property of $\pi^1_{I, I}$. But then it’s clear that in a semicartesian monoidal category, the projections satisfy a whole host of other properties, like the one about $X \otimes Y \otimes Z \to X$ that I mentioned in my post.

I don’t understand your $\mathcal{C} \amalg \{I\}$ example. You have a monoidal category $\mathcal{C}$; let’s call its unit object $J$. Then you adjoin a new “unit” $I$. What’s the tensor product on $\mathcal{C} \amalg \{I\}$? In particular, what’s $I \otimes J$? Or concretely, what monoidal category do you get from your construction if you take $\mathcal{C}$ to be the terminal category?

I am glad you posted this, because it turns out that almost every “nonlinear” category I care about is semicartesian

Thanks!

I’ve just (re)learned from Johnstone’s problem sheet that there’s a family of examples that I care about, and that isn’t on the nLab page. For any $p \in [1, \infty]$, there’s a $p$-norm-ish tensor product $\times_p$ on the category of metric spaces. It’s given on point-sets by taking the cartesian product, and on distances by taking a $p$th root of a sum of $p$th powers. (I guess you know what I mean!) That’s only the cartesian monoidal structure when $p = \infty$, which gives the max of distances. But it’s semicartesian for all $p$, since the unit object is always the terminal space $\{\bullet\}$.

This equivalence between monoidal categories with projections and those whose unit is terminal is somewhat puzzling to me, because the dual statement is not true.

Wait, what? How can this be? $(C, \otimes, I)$ is monoidal if and only if $(C^{op}, \otimes, I)$ is monoidal; “initial” is dual to “terminal”, and “injections” $X \to X \otimes Y$ are dual to “projections” (assuming I’ve understood the meaning of these terms).

What am I missing?

I mentioned your claim that

the category of stochastic matrices is enriched over convex spaces with tensor product

over at the thread that has motivated this one.

The question to ask then is whether there are interesting homology theories to be found in these probabalistic/stochastic cases based on Mike’s recipe.

What’s a minimal dense subcategory of the category of convex spaces? Perhaps those non-cancellative semi-lattice examples make this a little trickier.

We don’t actually need a dense subcategory; with the current perspective all we need is a coefficient system defined on some small subcategory that we can extend to the whole category by left Kan extension. One obvious interesting small subcategory of convex spaces that pops to my mind is the category of simplices, which means (if I’m not making a mistake) that any convex-space-enriched category should have homology with coefficients in any cosimplicial abelian group.

However, as I said in the other thread (which is, I suppose, where we actually ought to discuss this), it’s hard for me to see what that homology will actually look like without a more explicit description of the tensor product of convex spaces. Can anyone help? For instance, what is the tensor product of an interval with itself? Or more generally of two simplices?

when John mentions the category of convex sets as being semicartesian…

You link to a paper by the three of you where neither ‘semicartesian’, ‘convex set’, ‘convex space’, nor even ‘product’ is mentioned.

Some other place?

That’s more or less right. Metric spaces are categories enriched in a semicartesian category $([0, \infty]^{op}, +)$ in which the monoidal unit is terminal (and there may be more axioms to account for, e.g., we may want to include symmetry). Similarly, buildings whose “distances” are measured in a Coxeter group $W$ may be defined to be certain categories enriched in a semicartesian category $(W_\infty^{op}, \cdot)$ in which the monoidal unit is terminal, and there are more axioms to account for. (This $W_\infty$, derived from the Coxeter group $W$, is something Jim Dolan had playfully dubbed the “Murphy monoid”; there’s a kind of shaggy dog story there having to do with “Murphy’s law”. In my notes I use the more sober name Bruhat monoid. It’s a certain ordered monoid, or to be more precise an ordered $\ast$-monoid.)

There’s a certain perspective on enrichment that enters here, which I learned from Jim Dolan. Let’s take the simple case of enrichment in an ordered monoid $M$. Then the data of an $M$-enriched category is a set $X$ together with a function $d: X \times X \to M$, satisfying the usual axioms

$$d(x, y) \cdot d(y, z) \geq d(x, z)$$

$$e \geq d(x, x)$$

where the order relation on $M$ is written $\geq$ and $e$ is the identity element of $M$.

Another way of saying it is this: consider the poset of up-sets of $M$, $2^M$, equipped with Day convolution coming from $M^{op}$. This is a quantale or monoidal sup-lattice. Consider the set of binary relations on $X$, $2^{X \times X}$. Under relational composition, this is also a monoidal sup-lattice. Then the enrichment axioms on $d$ can be rephrased as saying that

$$2^d: 2^M \to 2^{X \times X}$$

is a lax monoidal functor.

It’s super-easy to get turned around here, so let me check that’s right. Take the case of principal $prin(u)$ up-sets in $M$ (we are writing the order relation as $\geq$, so actually that’s $\{m \in M: u \geq m\}$). Then $2^d(prin(u)) = \{(x, y) \in X \times X: u \geq d(x, y)\}$. Lax monoidality says (in part) that

$$2^d(prin(u)) \circ 2^d(prin(v)) \subseteq 2^d(prin(u v))$$

(where $\circ$ denotes relational composition) which is just a way of saying that if $u \geq d(x, y)$ and $v \geq d(y, z)$, then $u v \geq d(x, z)$. This is the triangle inequality in disguise.

This may look like an exercise in pointless abstract nonsense, except that the precise axioms for buildings (which are not just enriched categories of this sort; there are more axioms to account for) can be neatly reformulated in this categorical language. The notes at my nLab page, referred to before, give some details (although the proofs that the categorical description exactly matches the usual Tits definition have not yet been recorded there).

(If there’s interest in this, we can continue the discussion.)

In general, if $V$ is a small monoidal category, then I think we can similarly say that enrichment in $V$ amounts to a structure of lax monoidal functor of the form

$$Set^d: Set^V \to Set^{X \times X}$$

where the monoidal structure on $Set^{X \times X}$ is given by composition of endospans. Basically the lax monoidal structural constraint comes down to a family of functions of the form

$$V(u, d(x, y)) \times V(v, d(y, z)) \to V(u \otimes v, d(x, z)).$$

Wow, very cool! In the comment thread David linked to, Todd said

as of now, I don’t know what all this might be good for, aside from possible aesthetic appeal to those of a categorical bent.

I guess at the moment, with homology of enriched categories in mind, we can fantasize about a homology and magnitude theory for buildings (which Simon speculated about in that same thread).

The “certain perspective on enrichment” is surprising to me and I don’t really understand it yet. Does it relate directly to some other abstract definition of enrichment without dropping all the way down to the explicit version? Is it at all related to enriched categories as lax functors of bicategories into $\mathbf{B}V$, or the “Categories enriched on two sides” of Kelly-Labella-Schmitt-Street (which I also never really understood)? Does the function $d$ have to be given as data, or can you express it in terms of a lax monoidal functor $Set^V \to Set^{X\times X}$ with properties? Of course a monoid in $Set^{X\times X}$ is just an unenriched category, and also the same as a lax monoidal functor out of $1$. And $Set^{X\times X}$ is also just an endo-hom-category of the bicategory $Span(Set)$. Hmm.

Also, though, do I need to understand this definition of enrichment to think about buildings as enriched categories? In particular, can the additional building axioms be reformulated directly in more ordinary enriched-category language (even if they aren’t as neat)?

And can we work through some concrete small example to see what the resulting “homology of a building” looks like? Maybe the case of projective planes that you sketched here?

Mike, thanks for your interest and your questions, as always!

I’ll have to think about them. But let me just say what those additional axioms are that give buildings. (Really, I had better try to reconstruct the proof that these do the job. For now I’m going on memory that they do.)

One bit of low-hanging fruit is accounting for symmetry (as in symmetry for metric spaces). Here we need the notion of $\ast$-quantale or $\ast$-monoidal category. Of course the $\ast$-operator on relations $2^{X \times X}$ is relational opposite (permute the two factors $X$). On the Bruhat monoid side, where $W_\infty(D)$ is presented in terms of a Coxeter diagram or matrix $D$, we have generators $s_i$ satisfying relations of the form $s_i^2 = s_i$ and $s_i s_j s_i \ldots = s_j s_i s_j \ldots$ where both sides are alternating words of length $m_{i j}$ (given by $D$). We say a word $w$ is reduced if it cannot be written in shorter form using these relations, and the $\ast$-operator on $W_\infty(D)$ may be described as taking a word $w$ to its opposite gotten by writing it backwards. This is well-defined.

In general, if $d: X \times X \to M$ gives an enrichment of $X$ in a $\ast$-monoid $M$, then we say $X$ is symmetric if $2^d: 2^M \to 2^{X \times X}$ commutes with the $\ast$-operators on these quantales. Of course if a monoid $M$ is commutative to begin with (as in the case $([0, \infty], +)$), then we can take the $\ast$ to be the identity, and this leads for example to the symmetry axiom for metric spaces.

(Does a symmetry condition enter somewhere in the description of presheaves/sheaves as categories enriched in a bicategory that is due to Bob Walters? I have a vague memory this is so.)

The other axioms for buildings (which I still don’t have a good understanding of) involve the biclosed structure of $2^M, 2^{X \times X}$. At least I can say them quickly, and might as well throw them out there for your consideration.

Definition: Given a Coxeter diagram $D$, a $D$-building consists of a function $d: X \times X \to W_\infty(D)$ such that the map $2^d: 2^{W_\infty(D)} \to 2^{X \times X}$ is a strict morphism of biclosed $\ast$-monoidal categories (not just lax).

This means that $2^d$ preserves the residuation operators $/$, $\backslash$ on the nose, as well as the tensor product. (One residuation is the $\ast$-conjugate of the other, so it suffices that one of the residuations be preserved.)

All this seems to generalize just fine to more general enriched contexts (although we would want preservation of the pseudo kind, not strict), but I don’t recall what it means exactly even for the case of metric spaces, although I recall spending some time thinking about it.

It won’t help too much here on the $n$-Café, but here’s part of the program for the Australian Category Theory seminar at Macquarie University on Wednesday 21 September 2016:

4:00 pm in C5A 307.
Speaker: James Dolan.
Title: Incidence geometries as enriched categories.