The n-Category Café

Right, it was exactly that $n$Lab page’s sticking to actions on sets that prompted me to think about actions on groupoids.

I’d love to actually get a good feel for pseudo (co)limits. Ordinary (co)limits make perfect sense, but it’s a struggle up there at the 2-categorical level to transmute the thicket of diagrams into intuition :) I guess it just takes practice…

This general discussion on limits in spaces and limits in categories reminds me of a few things which I’d like to jot down (and maybe expand on in the nLab at some point).

(1) In his metric spaces paper, Lawvere defines the general notion of “point of a Cauchy completion” in terms of an adjoint pair of bimodules. Such a point can be considered either as a topological limit (of a Cauchy sequence or something similar), or it can be considered as an (enriched) categorical limit of a special kind: what is called an “absolute” limit (i.e., a limit that is preserved by any functor). Let me say quickly in passing that the notion of absolute limit is self-dual: absolute limits coincide with absolute colimits, so this indecision whether to compare topological limits to categorical limits or to colimits should melt away in this context.

(How do you make an arrow with a vertical slash through the middle? I’d like to use that symbol category theorists like to use when discussing relations, spans, profunctors, that sort of thing. Instead, I’ll use an arrow with a bullet over it.)

Very quickly: if $X$ is a metric space in Lawvere’s sense, and $1$ is the one-pointed space, then a point $p$ in the Cauchy completion $\bar{X}$ is an adjoint pair of $V$-enriched bimodules (letting $V$ for now be the symmetric monoidal closed preorder $([0, \infty], \geq)$ with addition as tensor)

$$(p: 1 \stackrel{\bullet}{\to} X) \dashv (q: X \stackrel{\bullet}{\to} 1)$$

which when you work it out is a pair of $V$-enriched functors $p: X^{op} \to V$, $q: X \to V$ with unit and counit inequalities that look like this:

$$0 \geq \int^{x: X} p(x) + q(x) \qquad p(x) + q(y) \geq X(x, y)$$

The way to think of this is that $p(x)$ measures the distance $\bar{X}(x, p)$ in the Cauchy completion, and $q(x)$ measures the distance $\bar{X}(p, x)$, and so the second inequality expresses a triangle inequality condition in the Cauchy completion. As for the first inequality: the coend here is a sup in the preorder $([0, \infty], \geq)$, hence an infimum wrt the usual ordering, and so it says that the two distances $\bar{X}(x, p)$, $\bar{X}(p, x)$ both become arbitrarily small as $x$ ranges over $X$. In other words, that $X$ is dense in $\bar{X}$ (in the topological sense).

On the categorical side, we have embeddings

$$X \hookrightarrow \bar{X} \hookrightarrow V^{X^{op}}$$

and so by Yoneda, points of the Cauchy completion can be thought of as special types of enriched colimits of representables. They are “absolute” colimits, i.e., they are preserved by any $V$-functor $f: X \to Y$. This isn’t too hard to see: roughly speaking, any $V$-functor $f: X \to Y$ may be regarded as a bimodule $X \to V^{Y^{op}}$ [by composing with the Yoneda embedding], and as such is a left adjoint in the bicategory of enriched bimodules; its right adjoint takes $y$ in $Y$ to $Y(f-, y): X^{op} \to V$. Therefore, since left adjoints compose, composition with $f$ gives a map

$$\bar{X} = Ladj_{Bim}(1, X) \stackrel{f \circ -}{\to} Ladj_{Bim}(1, Y) = \bar{Y}$$

We can think of this as saying that any functor $f$ preserves these special types of colimits of representables. Finally, these absolute colimits are essentially the same as absolute limits because we could equally well think of points $p$ of $\bar{X}$ as given by their right adjoint bimodules $q$, which sit inside $(V^X)^{op}$, the free $V$-enriched completion of $X$ (as opposed to $V^{X^{op}}$, the free cocompletion), and play a game dual to what we said above.

If this seems very abstract, it may help to think of concrete examples for various $V$. When $V = Set$, the Cauchy completion is the Karoubi envelope or idempotent-splitting completion of $C$, which we can think of either as closing up a category $C$ with respect to equalizers of diagrams

$$\pi, 1: c \stackrel{\to}{\to} c$$

where $\pi$ is idempotent, or closing up with respect to taking coequalizers of the same diagrams – it comes out the same either way, and it is well known that such split (co)equalizers are preserved by any functor. Or, if we take $V = Ab$, then the absolute limit-closure is closing up under these types of equalizers and finite products; the absolute colimit-closure is closing up under coequalizers of these diagrams and finite coproducts; either way the result is the same, and all such limits/colimits are preserved by any $Ab$-enriched functor.

(2) Hm, I was going to make some other points about general topology and general category theory, but this comment has already gotten longer than I’d originally planned. I was going to describe some interesting work by Walter Tholen and his colleagues (such as Maria Clementino and Dirk Hofmann – I may be omitting other key players) on general unifying categorical contexts in which to put metric spaces, topological spaces, multicategories and other things on a common footing. Instead, I’ll just supply one reference and maybe discuss it another time.

I slightly prefer to say ‘colimit’. This is just a matter of convention […]

Yes. I had only chosen conventions such that the the limit of analysis maps to the limit of category theory. Which is kind of fun.

maybe pondering this tiny issue will help us understand some big questions a bit better someday…

Yes, that’s why I am interested in this, too. It is the feeling that behind these terminological similarities there is a deeper meaning hidden which can help understand the true structure behind some phenomena.

This gets even better with ends and coends. As you know. Not only do these have the suggestive integral notation, but also satisfy a bunch of laws which remind us of integration. Such as the “Fubini theorem” for coends.

For instance for $F : \mathbb{N} \to C$ some functor (with $\mathbb{N}$ regarded as a discrete category) we have that

$$\int^{n \in \mathbb{N}} F(n)$$

is the coproduct of all $F(n)$. (I am just saying this for completeness, in case anyone else is reading this who does not know coends.) If $C$ is a category with cardinality, i.e. a bimonoidal category equipped with a bimonoidal functor

$$|\cdot| : (C, \times, \sqcup) \to (\mathbb{R},\times, +)$$

then in this notation “cardinality commutes with the integral sign” in that

$$\left| \int^{n \in \mathbb{N}} F(n) \right| = \int^{n \in \mathbb{N}} |F(n)| \,.$$

So with $$|\cdot | : (Groupoids, \times,+) \to (\mathbb{R},\times, +)$$

being groupoid cardinality we can write your favorite example with $F : \mathbb{N} \to Groupoids$ given by $F : n \mapsto \mathbf{B} S_n = I // S_n$ as

$$\left| \int^{n \in \mathbb{N}} I//S_n \right| = \int^{n \in \mathbb{N}} |I//S_n| = \int^{n \in \mathbb{N}} 1/n! = e$$

and its nice that it is the same integral sign everywhere, even though it means different things inside and outside the cardinality operation.

Alternatively, the groupoid cardinality can be built into the integration process automatically.

If we generalize the discrete integration domain category $\mathbb{N}$ to any groupoid $X$ and if we assume that the functor $F : X \to C$ is “free”, then

$$\left| \int^{x \in X} F(x) \right| = \int^{x \in \bar X} |F(x)| d\mu_X(x) \,,$$

where $\bar X$ denotes the set of isomorphism classes of $X$ and $d\mu_X$ is the function

$$d \mu_X : x \mapsto \frac{1}{|Aut(x)|}$$

which assigns the groupoid measure to $x$.

So in this setup then we would get the above example as the integral over the functor $1 : FinSet \to FinSet$ constant on the singleton set

$$\left| \int^{n \in FinSet} 1 \right| \mapsto \int^{n \in \bar FinSet = \mathbb{N}} 1 d\mu_{FinSet}(n) = \int^{n \in \mathbb{N}} 1/n! = e$$

only that the first step is not an equality, since the constant functor is not free.

Of course Tom Leinster then generalized this statement from groupoids to categories. I had spent a bit of time trying to see which categories are such that this kind of coend over them automatically reproduces measures $d\mu_X$ that should appear in certain discrete path integrals, since it feels that

maybe pondering this tiny issue will help us understand some big questions a bit better someday…

You might not like this, but I think the following satisfies all your criteria.

Write $|C|$ for the cardinality (Euler characteristic) of a finite category $C$. (Restrict to groupoids if you like, but everything I’m about to say works more generally for categories.) Then define the distance between finite categories $C$ and $D$ to be the absolute value of $|C| - |D|$.

There might be more to this than meets the eye. There’s a kind of subtraction operation on categories, as follows. Given a pair of categories $C, D$ and a pair of functors $$F, G: C \stackrel{\to}{\to} D,$$ there’s a ‘category of elements’ $E$, formed by taking the disjoint union of $C$ and $D$ then adjoining a new arrow $c \to F(c)$, and similarly a new arrow $c \to G(c)$, for every $c \in C$. Composition is defined in the way that’s obvious once you think about it. Write $E = D -_{F, G} C$: it’s

‘$D$ minus $C$, relative to $F$ and $G$’.

Little theorem: $$|D -_{F, G} C| = |D| - |C|.$$ In particular, the left-hand side is independent of the choice of $F$ and $G$. The distance between $C$ and $D$ is its absolute value.

Since Tom has a notion of cardinality for categories, and a closely related notion of distance for categories, and notion of cardinality for topological spaces, it might be interesting to compare some known notions of distance for topological spaces.

I was just looking at that paper last night. In that spirit, you might think of a kind of ‘distance’ between categories that’s completely unlike what David was after.

First let’s consider metric spaces. In Lawvere’s approach, they’re categories enriched in $V = [0, \infty]$. There’s a thing called ‘Gromov-Hausdorff distance’ measuring the distance between two abstract metric spaces. This makes $V-\mathbf{Cat}$ itself into a metric space, that is, a $V$-category.

Question: for which $V$ can $V-\mathbf{Cat}$ be sensibly construed as a $V$-category?

(Point of precision: $V-\mathbf{Cat}$ has as its objects the $V$-categories with a set of objects.)

I don’t know the answer. However, the answer is also ‘yes’ when $V = \mathbf{Set}$. In that case, a $V$-category is a locally small category, and $V-\mathbf{Cat} = \mathbf{Cat}$, which is itself a locally small category.

In the case $V = [0, \infty]$, the $V$-category structure on $V-\mathbf{Cat}$ was given, for $V$-categories $A$ and $B$, by $$V-\mathbf{Cat}(A, B) = d_{GH}(A, B)$$ where $d_{GH}$ is Gromov–Hausdorff distance. In the case $V = \mathbf{Set}$, it’s given by $$V-\mathbf{Cat}(A, B) = \{ functors A \to B \}.$$ So the analogue of Gromov–Hausdorff distance is: the set of functors. ‘Distance’ is no longer a real number — it’s a set. It’s not really reasonable to think of it as any kind of distance, e.g. the ‘distance’ from $A$ to $A$ is the set of endofunctors of $A$, which might be highly nontrivial.

(Of course, you might prefer to take the category of functors instead of the mere set. This still doesn’t deserve to be called ‘distance’.)

I thought this is what I said. I must have phrased it in an ambiguous or incorrect way.

Maybe you need to update the version on the web. The version of the file currently available, dated Dec 28, still contains the wrong definition 8, with no changes: there it still says:

all objects of $\Psi$ isomorphic to $x$

Sorry for jumping in without knowing the context for this discussion at all. But presumably, you’ve also made plenty of use of the 2-inverse image? This is the groupoid whose objects consist of pairs $(\phi, f)$, where $f$ is an isomorphism

$f: \nu(\phi) \simeq x,$

and some obvious morphisms. More generally, one forms the 2-product

$\Psi\times_X A,$

given another map $F:A \rightarrow X$ of groupoids, whose objects are triples, $(\phi, a, f)$, where $f$ is an isomorphism from $\nu(\phi)$ to $F(a)$. This has the nice property that maps

$h:B \rightarrow \Psi$

and

$g:B\rightarrow A$

together with an isomorphism

$\nu\circ h \simeq F\circ g$

determine a map

$B \rightarrow \Psi\times_X A.$

I’m sure these constructions are familiar to most people here, but I thought I’d take the opportunity to review them, because they are among the small things that add up to give category theory its real fizz.

Minhyong wrote:

But presumably, you’ve also made plenty of use of the 2-inverse image? This is the groupoid whose objects consist of pairs $(\phi, f)$, where $f$ is an isomorphism $f: \nu(\phi) \simeq x$.

Yes! In fact I like that construction so much that it took me an embarrassingly long time to realize that it’s not what we need here. (We’re taking a groupoid over a groupoid $X$ and turning it into a vector in the vector space whose basis consists of isomorphism classes in $X$.)

So you call this the ‘2-inverse image’? Influenced by topology, one might call it the ‘homotopy fiber’. I don’t know what most people call it. I just know what it is.

Here’s a nice way to summarize the difference between this construction and the ‘essential inverse image’ that I’m using in this paper. The essential inverse image consists of objects $\phi$ with the property that $\nu(\phi)$ is isomorphic to $x$. The 2-inverse image consists of objects equipped with the structure of an isomorphism from $\nu(\phi)$ to $x$.

This is all part of the elaborate yoga of Postnikov towers and “properties, structure and stuff”, explained Lectures on cohomology and $n$-categories.