The n-Category Café

At the heart of the explanation will be a generalized nerve construction:

Let $\mathbf{T}$ be a nice monad on a nice category. Then there are a canonically-defined category $\Delta_\mathbf{T}$ and ‘nerve’ functor $$\mathbf{Alg}(\mathbf{T}) \to [\Delta_\mathbf{T}^{op}, \mathbf{Set}]$$ with all the excellent properties of the ordinary nerve functor.

The ordinary nerve construction comes from taking $\mathbf{T}$ to be the free category monad on the category of directed graphs.

Ordinary nerves

As usual, $\Delta$ denotes a skeletal category of nonempty finite totally ordered sets. Note the ‘nonempty’: it matters! Following tradition, I’ll write its objects as $[0], [1], \ldots$, where $[n]$ is an $(n + 1)$-element totally ordered set.

Since ordered sets can be regarded as special categories, and order-preserving maps are then just functors, you can view $\Delta$ as a full subcategory of $\mathbf{Cat}$. The inclusion $\Delta \to \mathbf{Cat}$ induces a functor $$\begin{aligned} N: &\mathbf{Cat} &\to &[\Delta^\op, \mathbf{Set}] \\ &C &\mapsto&Hom(-, C), \end{aligned}$$ called the nerve functor.

The first excellent property of the nerve functor is that it is full and faithful. (In the jargon, $\Delta$ is a dense subcategory of $\mathbf{Cat}$.) Because of this, $\mathbf{Cat}$ is equivalent to a full subcategory of $[\Delta^\op, \mathbf{Set}]$, namely, the full subcategory consisting of all simplicial sets isomorphic to the nerve of some category. This subcategory is called the essential image of the nerve functor.

The second excellent property of the nerve functor is that there is an intrinsic description of its essential image. In fact, there are many such descriptions, and you probably know a couple (e.g. via Segal conditions, or unique filling of inner horns). I’ll give one in detail later. So you could define a category as a simplicial set conforming to one of these descriptions, and a functor as a map between such simplicial sets.

What’s not to love?

The nerve construction is simple, clean, and links category theory to simplicial sets, which are important in topology and other areas. Why wasn’t it love at first sight?

Well, there’s a difference between useful and natural. For example, the notion of triangulated category has certainly been useful, but almost no one who’s thought about it believes that the definition is ‘right’. Brutally put, it’s a hack: useful, but not natural. (I’d say the same of model categories.) There’s no doubt that the nerve construction has been found useful, but I wasn’t convinced that it was natural.

OK, but what does ‘natural’ mean? It’s a matter of aesthetics, and obviously there’s no precise answer, but here’s my stab at it. When you meet a new definition, there are two questions you might ask:

where does it come from?

and

why is the definition exactly that, not something slightly different?

If there’s a satisfactory answer to both, you can call the definition ‘natural’.

Let’s try this out on the nerve functor.

Where does it come from?   Since the nerve functor is induced by the inclusion $\Delta \to \mathbf{Cat}$, we’re really asking where $\Delta$ and its inclusion into $\mathbf{Cat}$ come from.

One answer might be that $\Delta$ is the free monoidal category on a monoid. But that’s just wrong: $\Delta$, as defined above and in most of the literature, does not include the empty set, whereas this free monoidal category $\mathbf{D}$ is a skeleton of the category of finite, possibly empty, totally ordered sets. Had you been shooting for $\mathbf{D}$, you’d have hit it — but presheaves on $\mathbf{D}$ are not the same as simplicial sets.

Another answer is geometric: think of triangles, tetrahedra, etc. and the maps between them. But how do you justify the ordering of the vertices? That doesn’t seem very geometric. And even if you can justify it, it would be more satisfactory if you could say where $\Delta$ came from without appealing to the visual sense — by thought alone.

Why is the definition exactly that, not something slightly different?  The nerve functor would continue to be full and faithful if you replaced the category $\Delta$ of finite nonempty totally ordered sets by the category of finite nonempty partially ordered sets, or changed ‘finite’ to ‘countable’, or dropped the nonemptiness, or any combination of these. (I’m sure you could also find intrinsic descriptions of the essential image in each of these cases.) You might wonder whether $\Delta$ was somehow minimal, but no: if you replace $\Delta$ by its full subcategory consisting of just $[0]$, $[1]$ and $[2]$, the nerve functor is still full and faithful.

Asking these questions is a substitute for innocence. When you meet an unmotivated definition, you want to ask ‘but why?’ However, unless you’re completely impatient, you’ll suspend judgement until you’ve seen a few results using the definition… and before you know it, you’ve forgotten that innocent first question.

In my pre-revelation years, I didn’t understand why the nerve construction was a natural thing. I didn’t actively think that it was unnatural — I just remained to be convinced.

Generalized nerves

The following theorem is what convinced me. I’ll state it roughly now and fill in most of the details afterwards.

Theorem   Let $\mathbf{T}$ be a nice monad on a nice category $\mathcal{E}$. Then there is a canonical small full subcategory $\Delta_\mathbf{T}$ of $\mathbf{Alg}(\mathbf{T})$ such that the induced functor $$N_T: \mathbf{Alg}(\mathbf{T}) \to [\Delta_\mathbf{T}^{op}, \mathbf{Set}]$$ is full and faithful. Its essential image consists of the presheaves on $\Delta_T$ preserving certain limits.

Call a presheaf on $\Delta_\mathbf{T}$ a $\mathbf{T}$-simplicial set. Then a $\mathbf{T}$-algebra can be regarded as a $\mathbf{T}$-simplicial set satisfying a limit-preservation condition. The slogan is this:

$\mathbf{T}$-simplicial sets are to $\mathbf{T}$-algebras
as
ordinary simplicial sets are to categories.

To help us make precise the vague parts of the theorem, let’s go back to the ordinary nerve construction, which corresponds to taking the free category monad $\mathbf{T}$ on the category $\mathcal{E}$ of directed graphs. First note that $\mathcal{E}$ is a presheaf category, since a directed graph is a presheaf on the small category $$\mathbf{E} = (0 \stackrel{\to}{\to} 1).$$ Convention: I’ll write the value of a presheaf at an object using a subscript; for instance, if $X$ is a directed graph (a presheaf on $\mathbf{E}$) then $X_0$ and $X_1$ mean $X(0)$ and $X(1)$. Now, the free category functor $T$ can be described as follows: for a directed graph $X$, $$\begin{aligned} (T(X))_0 & = & Hom([0], X), \\ (T(X))_1 & = & \sum_{n \in \mathbb{N}} Hom([n], X). \end{aligned}$$ Here $[n]$ means the directed graph $\bullet \to \bullet \to \cdots \to \bullet$ with $n$ arrows; the free category on it is the ordered set also called $[n]$. The summation means coproduct.

Here are the precise hypotheses. The category $\mathcal{E}$ is ‘nice’ if it is a presheaf category $[\mathbf{E}^{op}, \mathbf{Set}]$. The monad $\mathbf{T} = (T, \eta, \mu)$ is ‘nice’ if it satisfies the following two conditions:

• For each $e \in \mathbf{E}$, the functor $(T(-))_e$ is a coproduct of representables. This means that there are a set $I(e)$ and a family $(W_{e, i})_{i \in I(e)}$ of presheaves such that $$(T(X))_e = \sum_{i \in I(e)} Hom(W_{e, i}, X)$$ for all presheaves $X$.
  
  An equivalent condition is that $T$ preserves connected limits.
  
  Another equivalent condition is that the induced functor $\mathcal{E} \to \mathcal{E}/T(1)$ has a right adjoint; one then says that $T$ is a ‘parametric right adjoint’, as in Mark’s title.
• The unit $\eta$ and multiplication $\mu$ are cartesian natural transformations (that is, the naturality squares are not only commutative but pullbacks).

These conditions on a monad are very commonly met in higher-dimensional category theory. (Sometimes $T$ is only required to preserve pullbacks, not all connected limits.) I’ll give examples later.

We can now see exactly what $\Delta_\mathbf{T}$ is. Write $F: [\mathbf{E}^{op}, \mathbf{Set}] \to \mathbf{Alg}(\mathbf{T})$ for the free $\mathbf{T}$-algebra functor. Then $\Delta_\mathbf{T}$ is the full subcategory of $\mathbf{Alg}(T)$ consisting of the algebras $F(W_{e, i})$, for $e \in \mathbf{E}$ and $i \in I(e)$.

In the motivating example, the free category monad on directed graphs, $\Delta_\mathbf{T}$ is the full subcategory of $\mathbf{Cat}$ whose objects are $F([0])$ (from taking $e = 0$) and $F([0]), F([1]), F([2]), \ldots$ (from taking $e = 1$). In our notation, the category $F([n])$ is written as $[n]$, so $\Delta_\mathbf{T} = \Delta$. It makes no difference that the category $F([0])$ was listed twice. So we recover the usual nerve construction.

I won’t describe the limit-preservation condition, since it’s a bit more complicated, but it can be done explicitly in terms of the representing family $(W_{\bullet\bullet})$. In the motivating example, it says that a simplicial set $X$ is the nerve of a category if and only if for each $k, n_1, \ldots, n_k \in \mathbb{N}$, the colimit $$[n_1] +_{[0]} [n_2] +_{[0]} \cdots +_{[0]} [n_k] = [n_1 + \cdots + n_k]$$ in $\Delta$ is turned by $X$ into a limit in $\mathbf{Set}$. (Here $+_{[0]}$ means pushout over $[0]$.) In other words, for all $k$ and $n_i$, the canonical map $$X_{n_1 + \cdots + n_k} \to X_{n_1} \times_{X_0} X_{n_2} \times_{X_0} \cdots \times_{X_0} X_{n_k}$$ is an isomorphism. This is one of the well-known Segal-type characterizations of nerves of categories.

That’s how I learned to love the nerve construction. I loved it even more when I figured out some other examples.

Examples

$M$-sets  Let’s begin with a simple one. Fix a monoid $M$. The category of sets is certainly ‘nice’, as is the free left $M$-set monad $M \times -$ on $\mathbf{Set}$. If you work it out, you’ll see that $\Delta_\mathbf{T} = M^{op}$, so that a $\mathbf{T}$-simplicial set (presheaf on $\Delta_\mathbf{T}$) is a functor $M \to \mathbf{Set}$. The limit-preservation condition is vacuous, so we recover the basic observation that the category of left $M$-sets is the functor category $[M, \mathbf{Set}]$.

Strict $n$-categories  Fix $n \in \mathbb{N} \cup \{\infty\}$. Let $\mathcal{E}$ be the category of $n$-dimensional globular sets (also called $n$-graphs), which is ‘nice’ (a presheaf category). Let $\mathbf{T}$ be the free strict $n$-category monad on $\mathcal{E}$, which is also nice. Then $\Delta_T$ is the category whose objects are globular pasting diagrams of dimension at most $n$, viewed as strict $n$-categories, and whose maps are strict $n$-functors.

(To view a globular pasting diagram as an $n$-category, picture, say, a 2-dimensional such diagram. The 0-, 1- and 2-cells in the diagram freely generate a 2-category. This gives rise to an $n$-category for any $n\geq 2$ by adding just identities in higher dimensions.)

This category $\Delta_T$ has been much studied, especially in the case $n = \infty$. André Joyal seems to have been the first to do so, in an unpublished note ‘Disks, duality and $\Theta$-categories’. He wrote $\Theta$ for $\Delta_T$, and proposed that a weak $\infty$-category might be defined as a presheaf on $\Theta$ satisfying certain conditions. (For details, see Definition J of this or page 269 of this.) Clemens Berger showed that the nerve functor $\mathbf{Str}\infty\mathbf{Cat} \to [\Theta^{op}, \mathbf{Set}]$ is full and faithful, which is a special case of the general result above. I have the impression that Michael Makkai and Marek Zawadowski also proved this independently, but I don’t have a reference.

Weak $n$-categories  Several of the proposed definitions of weak $n$-category are of the form ‘a weak $n$-category is an algebra for a certain monad on a certain category’. The monad and the category are (almost?) always ‘nice’ in our sense. This means that a weak $n$-category can also be regarded as a presheaf-with-properties: that is, a presheaf on $\Delta_\mathbf{T}$ (where $\mathbf{T}$ is the free weak $n$-category monad) preserving certain limits.

People often make the distinction between ‘algebraic’ definitions of weak $n$-category (in which an $n$-category is a presheaf with structure) and ‘non-algebraic’ definitions (in which an $n$-category is a presheaf with properties). This result shows how algebraic definitions can be regarded as non-algebraic.

Multicategories  A multigraph consists of a set $X_0$ and, for each $a_1, \ldots, a_n, a \in X_0$, a set $Hom(a_0, \ldots, a_n; a)$. Let $\mathbf{T}$ be the free (non-symmetric) multicategory monad on the category of multigraphs. Both category and monad are ‘nice’, so the theorem applies and we have a nerve construction for multicategories. The category $\Delta_\mathbf{T}$ has as its objects all finite planar rooted trees.

Weber’s new work

Here I’ll say a tiny bit about how Mark’s paper goes beyond what I did — though I’m conscious that I’m missing out large parts of his work.

First, Mark’s proof of the result above is probably more efficient than mine. He makes effective use of a factorization system, whereas I just went at it directly and got a great long proof that I didn’t much fancy writing up.

Second, Mark seems to have a result that resembles the one above but works in greater generality. Here’s an important example. Over the last few years, Ieke Moerdijk and Ittay Weiss have developed the theory of dendroidal sets. The slogan is:

dendroidal sets are to symmetric multicategories
as
simplicial sets are to categories.

We almost did this example just now, except that we did non-symmetric multicategories. To get dendroidal sets, we’d want to use the free symmetric multicategory monad on multigraphs — but this monad isn’t ‘nice’ in the sense above. However, Mark’s theory does manage to capture this example.

It would be great if someone else wrote something about the parts of Mark’s paper that go beyond what I’ve explained.