The n-Category Café

Let $V$ be a monoidal category, which I’ll assume is symmetric, closed, complete or cocomplete as necessary. Take a pair of $V$-categories $A$ and $B$, and a $V$-functor

$$F: A \to B.$$

Then we get a new functor

$$N_F: B \to [A^{op}, V]$$

defined by

$$(N_F(b))(a) = B(F a, b).$$

This is sometimes called the nerve functor induce by $F$, because of the first example in the following list.

Examples   In the first few examples, I’ll take the enriching category $V$ to be $Set$.

• Let $F$ be the inclusion $\Delta \hookrightarrow Cat$, which takes a finite nonempty totally ordered set $[n] = \{0, \ldots, n\}$ and realizes it as a category in the usual way. Then $N_F: Cat \to [\Delta^{op}, Set]$ sends a small category $D$ to its nerve, which is the simplicial set whose $n$-simplices are paths in $D$ of length $n$.

• Let $F$ be the functor $\Delta \to Top$ that sends $[n]$ to the topological $n$-simplex $\Delta^n$. Then $N_F: Top \to [\Delta^{op}, Set]$ sends a topological space $X$ to its “singular simplicial set”, whose $n$-simplices are the continuous maps $\Delta^n \to X$.

• Let $F: A \hookrightarrow B$ be an inclusion of partially ordered sets, viewed as categories in the usual way. Then $N_F : B \to [A^{op}, Set]$ is given by

  $$(N_F(b))(a) = \begin{cases} 1 &\text{if}   b \leq a \\ 0 &\text{otherwise}. \end{cases}$$

  For example, let $B$ be the set $\mathbb{Z}$ of integers, ordered by divisibility. Let $A \subseteq \mathbb{Z}$ be the set of (positive) primes. Then $N_F$ is the functor $\mathbb{Z} \to Set^{\{primes\}}$ that sends $n \in \mathbb{Z}$ to

  $$([p | n])_{\text{primes}   p}$$

  where $[p|n]$ is Iverson notation: it’s $1$ if $p$ divides $n$ and $0$ otherwise.

• If $F$ has a right adjoint $G$, then $N_F(b) = A(-, G b): A^{op} \to V$. In particular, $N_F(b)$ is representable for each $b \in B$.

• Take the inclusion $Field \hookrightarrow Ring$, where $Ring$ is the category of commutative rings. Now take opposites throughout to get $F: Field^{op} \hookrightarrow Ring^{op}$. The resulting nerve functor

  $$N_F: Ring^{op} \to [Field, Set]$$

  sends a ring $R$ to $Ring(R, -)|_{Field}$ (by which I mean the restriction of $Ring(R, -): Ring \to Set$ to the category of fields). More explicitly,

  $$N_F(R) = \sum_{p \in Spec(R)} Field(Frac(R/p), -).$$

  Here $Spec(R)$ is the set of prime ideals $p$ of $R$, and $Frac(R/p)$ is the field of fractions of the integral domain $R/p$. In particular, $N_F(b)$ is a coproduct of representables for each $b \in B$. (In the terminology of Diers, one therefore says that $Field \hookrightarrow Ring$ is a “right multi-adjoint”. A closely related statement is that it’s a parametric right adjoint. But that’s all just jargon that won’t matter for this post.)

Now let’s consider $V = (\mathbb{R}^+, \geq)$ with its additive monoidal structure, so that $V$-categories are generalized metric spaces and $V$-functors are the functions $f$ between metric spaces that are variously called short, distance-decreasing, contractive, or 1-Lipschitz: $d(f(a), f(a')) \leq d(a, a')$ for all $a, a'$. I’ll just call them “maps” of metric spaces.

• Take a map $f: A \to B$ of metric spaces. The induced map $N_f: B \to [A^{op}, \mathbb{R}^+]$ is given by

  $$(N_f(b))(a) = d(f(a), b).$$

  Unlike in the first post, I’m not going to assume here that our metric spaces are symmetric. so the “op” on the $A$ matters, and $d(a, f(b)) \neq d(f(b), a)$ in general.

  We’ll be particularly interested in the case where the $V$-functor $F$ is full and faithful. Most of the examples above have this property. In the metric case, $V = \mathbb{R}^+$, being full and faithful means being distance-preserving, or in other words, an inclusion of a metric subspace. In that case, it’s normal to drop the $f$. So we’d usually then write

  $$(N_f(b))(a) = d(a, b).$$

Next I’ll define the potential function of a $V$-functor. For this we need some of the apparatus of magnitude. Roughly speaking, this means we have a notion of the “size” of each object of our base monoidal category $V$.

More exactly, we take a field $k$ and a function $|\cdot|: ob(V) \to k$, to be thought of as assigning to each object $S$ of $V$ its size ${|S|} \in k$. And to make everything work, we assume our size function $|\cdot|$ has reasonable properties:

$$S \cong T \implies |S| = |T|, \quad |S \otimes T| = |S| \cdot |T|, \quad |I| = 1,$$

where $\otimes$ is the monoidal product on $V$ and $I$ is its unit.

The basic definitions are these. A weighting on a finite $V$-category $A$ is a function $w_A : ob(A) \to k$ such that

$$\sum_{b \in A} |A(a, b)| w_A(b) = 1$$

for all $a \in A$, and a coweighting $w^A$ on $A$ is a weighting on $A^{op}$. Although it’s not true that every finite $V$-category has a unique weighting, it’s a harmless enough assumption that I’ll make it here. The magnitude of a finite $V$-category $A$ is

$$|A| = \sum_{a \in A} w_A(a) = \sum_{a \in A} w^A(a) \in k.$$

It’s a tiny lemma that the total weight $\sum_{a \in A} w_A(a)$ is equal to the total coweight $\sum_{a \in A} w^A(a)$, so you can equivalently define the magnitude to be one or the other.

There’s a whole lot to say about why this is a worthwhile definition, and I’ve said it in detail many times here before. But here I’ll just say two short things:

• Taking $V = FinSet$, $k = \mathbb{Q}$, and $|\cdot|$ to be cardinality, the magnitude of a finite category is equal to the Euler characteristic of its classifying space, under suitable hypotheses guaranteeing that the latter is defined.

• Take $V = \mathbb{R}^+$, $k = \mathbb{R}$, and $|\cdot|$ to be $s \mapsto e^{-s}$ (because that’s just about the only function converting the tensor product of $V$ into addition, as our size functions are contractually obliged to do). Then we get a notion of the magnitude of a metric space, a real-valued invariant that turns out to be extremely interesting.

That’s the magnitude of enriched categories. But we can also talk about the magnitude of enriched presheaves. Take a finite $V$-category $A$ and a $V$-presheaf $X: A^{op} \to V$. Its magnitude is

$$|X| = \sum_{a \in A} w^A(a) |X(a)| \in k.$$

When I wrote about the magnitude of functors before, I concentrated on covariant functors $A \to V$, which meant that the weights in the weighted sum that defines $|X|$ were, well, the weights $w_A(a)$. But since we’ve now changed $A$ to $A^{op}$, the weights have become coweights.

Let me briefly recall why the magnitude of presheaves is interesting, at least in the case $V = FinSet$:

• If our presheaf $X$ is a coproduct of representables then $|X|$ is $|colim(X)|$, the cardinality of the colimit of $X$.

• The magnitude of presheaves generalizes the magnitude of categories. The magnitude of a category $A$ is equal to the magnitude of the presheaf $A^{op} \to Set$ with constant value $1$.

• In the other direction, the magnitude of categories generalizes the magnitude of presheaves: $|X| = |\mathbb{E}(X)|$ for all presheaves $X$. Here $\mathbb{E}(X)$ means the category of elements of $X$, also called the (domain of the) Grothendieck construction.

  It’s enlightening to think of this result as follows. If we extend the codomain of $X$ from $Set$ to $Cat$ then $\mathbb{E}(X)$ is the colax colimit of $X: A^{op} \to Cat$. So $|X| = |colaxcolim(X)|$. You can compare this to the more limited result that $|X| = |colim(X)|$, which only holds when $X$ has the special property that it’s a coproduct of representables.

  There’s also another closely related result, due to Ponto and Shulman: under mild further hypotheses, $|X| = \chi(hocolim(X))$. Mike and I had a little discussion about the relationship here.

  Now here’s something funny. Which do you think is a more refined invariant of a presheaf $X: A^{op} \to FinSet$: the magnitude $|X|$, which is equal to $|colaxcolim(X)|$ and $\chi(hocolim(X))$, or the cardinality $|colim(X)|$ of the strict colimit?

  From one perspective, it’s the magnitude. After all, we usually think of lax or pseudo 2-dimensional things as more subtle and revealing than strict 1-dimensional things. But from another, it’s the cardinality of the strict colimit. For instance, if $A$ is the one-object category corresponding to a group $G$ then $X$ is a right $G$-set, the cardinality of the colimit is the number of orbits (an interesting quantity), but the magnitude is always just the cardinality of the set $X$ divided by the order of $G$ (relatively boring and crude).

  In this post I’ll keep comparing these two quantities. Whichever seems “more refined”, the comparison is interesting in itself.

(I stuck to the case $V = FinSet$ here, because for an arbitrary $V$, there isn’t a notion of “colimit” as such: we’d need to think about weighted colimits. It’s also harder to say what the category of elements or homotopy colimit of an enriched presheaf should be.)

Now here’s something new. The potential function of a $V$-functor $F: A \to B$ is the function

$$h_F: \ob(B) \to k$$

defined by

$$h_F(b) = |N_F(b)|$$

for all $b \in B$. Expanding out the definitions, this means that

$$h_F(b) = |A(F-, b)| = \sum_{a \in A} w^A(a) |B(F a, b)|,$$

where $w^A$ is the notation I introduced for the coweighting on $A$. If $F$ is full and faithful then $h_F \equiv 1$ on the set of objects in the image of $F$.

Examples   The definition I just gave implicitly assumes that $\ob(A)$ is finite. But I’ll relax that assumption in some of these examples, at the cost of some things getting less than rigorous.

Again, I’ll begin with some examples where $V = FinSet$.

• What’s the potential function of the inclusion $\Delta \hookrightarrow Cat$? To make sense of this, we need a coweighting on $\Delta$, and I’m pretty sure there isn’t one. So let’s abandon $\Delta$ and instead use its subcategory $\Delta_{inj}$, consisting of all the finite nonempty totally ordered sets $[n] = \{0, \ldots, n\}$ but only the injective order-preserving maps between them. This amounts to considering only face maps, not degeneracies.

  The coweighting on $\Delta_{inj}$ is $[n] \mapsto (-1)^n$. So, the potential function $h$ of $\Delta_{inj} \hookrightarrow Cat$ is given, on a category $D$, by

  $$h(D) = \sum_{n \geq 0} (-1)^n |Cat([n], D)|.$$

  Here $Cat([n], D)$ is the set of paths of length $n$ in $D$, which we often write as $D_n$. So

  $$h(D) = \sum_{n \geq 0} (-1)^n |D_n|.$$

  Even if the category $D$ is finite, this is a divergent series. But morally, $h(D)$ is the alternating sum of the number of $n$-cells of $D$.

  In other words, the potential function of $\Delta_{inj} \to Cat$ is morally Euler characteristic.

  As I hinted above, it’s interesting to compare the potential function with what you get if you take the cardinality of the colimit of $B(F-, b)$ instead of its magnitude. I’ll call this the “strict potential function”. In this case, it’s

  $$D \mapsto |colim_{[n] \in \Delta_{inj}} D_n| = |\pi_0(D)|$$

  — the cardinality of the set of connected-components of $D$. So while the potential function gives Euler characteristic, the strict potential function gives the number of connected components.

• Similarly, the potential function of the usual functor $\Delta_{inj} \hookrightarrow Top$, sending $n$ to $\Delta^n$, is given by

  $$D \mapsto \sum_{n \geq 0} (-1)^n |Top(\Delta^n, D)|$$

  for topological spaces $D$. Again, this formally looks like Euler characteristic: the alternating sum of the number of cells in each dimension. And much as for categories, the strict potential function gives the number of path-components.

• For the inclusion $\{primes\} \hookrightarrow (\mathbb{Z}, |)$ that we saw earlier, the potential function $h: \mathbb{Z} \to \mathbb{Q}$ is what’s often denoted by $\omega$, sending an integer $n$ to the number of distinct prime factors of $n$. The strict potential function is the same.

• An easy example that works for any $V$: if $F: A \to B$ has a right adjoint then since $A(F-, b)$ is representable, its magnitude is equal to the cardinality of its colimit, which is $1$. So $h_F: \ob(B) \to k$ has constant value $1$, as does the strict potential function.

  More generally, if $A(F-, b)$ is a coproduct of $n$ representables then $h_F(b) = n$, and the same is true for the strict potential function.

• For the inclusion $F: Field^{op} \hookrightarrow Ring^{op}$, we saw that

  $$N_F(R) = \sum_{p \in Spec(R)} Field(Frac(R/p), -)$$

  for every ring $R$. This is a coproduct of representables, so by the previous example, $|N_F(R)| = |Spec(R)|$. That is, the potential function is given by

  $$h_F: R \mapsto |Spec(R)|$$

  (as is the strict potential function). Usually $Spec(R)$ is infinite, so this calculation is dodgy, but if we restrict ourselves to finite fields and rings then everything is above board.

• For an opfibration $F: A \to B$, one can show that the potential function $h_F$ is given by

  $$h_F(b) = |F^{-1}(b)|.$$

  I should explain the notation on the right-hand side. The category $F^{-1}(b)$ is the fibre over $b \in B$, consisting of the objects of $A$ that $F$ sends to $b$ and the maps in $A$ that $F$ sends to $1_b$. We can take its magnitude (at least if it’s finite and lightning doesn’t strike), which is what I’ve denoted by $|F^{-1}(b)|$.

  I won’t include the proof of this result, but I want to emphasize that it doesn’t involve finding a coweighting on $A$. You might think it would, because the definition of $h_F(b)$ involves the coweighting on $A$. But it turns out that it’s enough to just assume it exists.

• Now take $V = \mathbb{R}^+$, so that $V$-categories are generalized metric spaces. The potential function of a map $f: A \to B$ of metric spaces is the function $h_f : B \to \mathbb{R}$ given by

  $$h_f(b) = \sum_{a \in A} w^A(a) e^{-d(f(a), b)}.$$

  In particular, if $f$ is the inclusion of a subspace $A \subseteq B$, then it’s natural to write $h_A$ instead of $h_f$, and we have

  $$h_A(b) = \sum_{a \in A} w^A(a) e^{-d(a, b)}.$$

  This is equal to $1$ on $A$ but can take other values on $B \setminus A$. In suitable infinite contexts, sums become integrals and $w^A$ becomes something like a measure or distribution, in which case the formula becomes

  $$h_A(b) = \int_A e^{-d(a, b)} \, d w^A(a).$$

  This is exactly the formula for the potential function in Part 1, with one difference: there, I used weightings on $A$, and here, I’m using coweightings. It’s coweightings that one should use. In the previous post, I assumed that all metrics were symmetric, which means that weightings and coweightings are the same. So there’s no inconsistency.

  (Of course, one could take duals throughout and use weightings on $A$ instead. But we’ll see that whichever choice you make, you end up having to consider weightings on one of $A$ and $B$ and coweightings on the other.)

  What about the strict potential function, sending a point $b \in B$ to the “cardinality of the colimit of $B(f-, b): A \to \mathbb{R}^+$”? Well, I put that phrase in inverted commas because we’re now in an enriched context, so it needs a bit of interpretation. “Cardinality” is okay: it becomes the size function $|\cdot| = e^{-(\cdot)}: \mathbb{R}^+ \to \mathbb{R}$. “Colimit” wouldn’t usually make sense in an enriched world, but we’re saved by the fact that the monoidal unit of $V = \mathbb{R}^+$ (namely, $0$) is terminal. The colimit of a $V$-functor into $V$ is just its infimum. So the strict potential function of a map $f: A \to B$ of metric spaces is just

  $$b \mapsto e^{-\inf_{a \in A} d(f(a), b)} = e^{-d(f A, B)}.$$

  I showed you an example of the difference between the potential function and the strict potential function in Part 1, although not with those words. If we take $f$ to be the inclusion $\{-1, 1\} \hookrightarrow \mathbb{R}$ then the potential function is

  

  whereas the strict potential function is

  

  These two functions are identical on $(-\infty, -1] \cup [1, \infty)$, but different on the interval $(-1, 1)$. If you’ve ever wondered what the difference is between strict and lax colimits, here’s an example!

By definition, the potential function of an enriched functor $F: A \to B$ is given by

$$h_F(b) = \sum_{a \in A} w^A(a) |B(F(a), b)|,$$

but a slightly different viewpoint is sometimes helpful. We can take the function $w^A: ob(A) \to k$ and push it forward forward along $F$ to obtain a new function

$$\begin{array}{cccc} F_* w^A: &ob(B) &\to &k, \\ &b &\mapsto &\sum_{a \in F^{-1}(b)} w^A(a). \end{array}$$

Really it’s best to think of $w^A$ as not a function but a measure taking values in $k$ (although these are the same thing on a finite set). Then this is just the usual pushforward measure construction. In any case, the formula for the potential function now becomes

$$h_F(b) = \sum_{b' \in B} (F_* w^A)(b') |B(b', b)|,$$

which has the advantage that everything is taking place in $B$ rather than $A$. In the case where $f: A \to B$ is an embedding of a subspace of a metric space, $f_* w^A$ is just the extension of the measure $w^A$ on $A$ to all of $B$, which we’d usually just write as $w^A$ (with a wink to the audience). In integral notation, the last formula becomes

$$h_F(b) = \int_B e^{-d(x, b)} \, d w^A(x),$$

which is what we had in the last post.

I’ve now explained what potential functions are. But what are they good for?

Last time, I explained that they’re very good indeed for helping us to calculate the magnitude of metric spaces. The key was that

$$|A| = \int_B h_A \, d w_B$$

for a subspace $A$ of a metric space $B$. And as I recounted, that key unlocks the door to the world of PDE methods.

So you might hope something similar is true for enriched categories in general: that there’s a formula for the magnitude in terms of the potential function. And there is! For any $V$-functor $F: A \to B$, it’s a theorem that

$$|A| = \sum_{b \in B} h_F(b) w_B(b).$$

This is an easy calculation from the definitions.

(If you’re really paying attention, you’ll notice that we used the coweighting on $A$ to define the potential function, and now we’re using the weighting on $B$. That’s just how it turns out. One is a weighting, and the other is a coweighting.)

Examples

• For a $V$-functor $F: A \to B$ that has a right adjoint, we saw that the potential function $h_F$ has constant value $1$, so this formula tells us that

  $$|A| = \sum_{b \in B} w_B(b).$$

  But the right-hand side is by definition the magnitude of $B$, so what this formula is saying is that

  $$|A| = |B|.$$

  In other words, if there’s an adjunction between two categories, their magnitudes are equal!

  This has been known forever (Proposition 2.4), and is also intuitive from a homotopical viewpoint. But it’s nice that it just pops out.

• Less trivially, we saw above that for an opfibration $F: A \to B$, the potential function is $h_F: b \mapsto |F^{-1}(b)|$. So

  $$|A| = \sum_{b \in B} w_B(b) |F^{-1}(b)|.$$

  In other words, the magnitude of the “total” category $A$ is the weighted sum over the base $B$ of the magnitudes of the fibres. (Special case: the magnitude of a product is the product of the magnitudes.)

  Now this has been known forever too (Proposition 2.8), but I want to emphasize that the proof is fundamentally different from the one I just linked. That proof constructs the weighting on $A$ from the weightings on the base $B$ and the fibres. Now that’s an easy and informative proof, but what we’ve just done is different, because it didn’t involve figuring out the weighting or coweighting on $A$. So although the result isn’t new or difficult, it’s perhaps grounds for optimism that the method of potential functions will let us prove new things about enriched categories other than metric spaces.

What new things? I don’t know! This is where I’ve got to now. Maybe there are applications in the metric world in which $f: A \to B$ is nothing like an inclusion. Maybe there are applications to graphs, replacing the PDE methods used for subspaces of $\mathbb{R}^n$ by discrete analogues. Maybe the potential function method can be used to shed light on the tricky result that the magnitude of graphs is invariant under certain Whitney twists (Theorem 5.2), and more generally under the sycamore twists introduced by Emily Roff (Theorem 6.5). Let’s find out!