Contents

### Context

#### Higher category theory

higher category theory

## 1-categorical presentations

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

A computad is a formal device (due to Ross Street, 1976) for describing “generators” of higher categories, and in particular for $n$-categories. They generalize quivers (directed graphs), which describe generators of ordinary (1-)categories. In a sense, $n$-computads are the “most general” structure from which one can generate a free $n$-category; they allow the free adjoining of $k$-cells whose source and target are arbitrary composites of previously adjoined $(k-1)$-cells.

Originally the $n$-categories under consideration were strict, but more recently the concept of $n$-computad has been expanded to take into account weak $n$-categories and other higher-categorical structures. The notion is tied to algebraic senses of higher categories, but computads can also be used as the input for geometric senses as well, and may aid in a comparison between the two approaches.

Computads are also called polygraphs, following Burroni; this term is especially used in parts of the literature related to rewriting theory.

## Definition

Each type of higher-categorical structure comes with its own notion of computad. Thus there are computads for strict $n$-categories, computads for weak $n$-categories, computads for double categories, and so on.

First we will define computads relative to an arbitrary globular operad $P$. This reproduces the original strict situation when $P$ is the terminal globular operad, whose algebras are strict ∞-categories, but it also applies to operads $P$ whose algebras are Batanin weak ∞-categories. The generalization to the weak case is originally due to Batanin; the following simpler reformulation of it is due to Richard Garner.

Let $P$ be a globular operad and let $P Alg$ denote the category of $P$-algebras and strict morphisms. Thus if $P$ is terminal, then $P Alg = Str \omega Cat$. We denote by $U_P$ the forgetful functor $P Alg \to GSet$ and by $F_P$ its left adjoint, where $GSet$ denotes the category of globular sets. Let $2_n$ denote the $n$-globe and $\partial_n$ its boundary (which is the pushout of two $(n-1)$-globes along their boundaries). These are globular sets and thus they generate free $P$-algebras $F_P 2_n$ and $F_P \partial_n$.

We now define, recursively in $n$, the category $n Cptd_P$ of $n$-computads relative to $P$, together with an adjunction

$n Cptd_P \underoverset{U_n}{F_n}{\rightleftarrows} P Alg .$
• When $n=0$, the category $(-1) Cptd_P$ is Set, $U_0\colon P Alg \to Set$ picks out the set of objects, and $F_0$ is its left adjoint, which constructs the free $P$-algebra on a set (considered as a globular set concentrated in degree 0).

• If $n\gt 0$, an n-computad consists of an $(n-1)$-computad $C$, a set $X$, and a function

$x\colon X\to P Alg(F_P \partial_n, F_{n-1} C) = GSet(\partial_n, U_P F_{n-1} C).$

We call $X$ the set of n-cells. Note that $x$ simply equips each $n$-cell with a pair of parallel $(n-1)$-cells in $F_{n-1} C$. In fancy language, the category $n Cptd_P$ is the comma category $Set / B_n$, where $B_n = P Alg(F_P \partial_n, F_{n-1} -)$.

• The functor $U_n\colon P Alg \to n Cptd_P$ sends a $P$-algebra $A$ to the $n$-computad defined by the $(n-1)$-computad $U_{n-1} A$ together with $X$ and $x$ defined by the following pullback square in Set:

$\array{X & \overset{}{\to} & P Alg(F_P 2_n, A)\\ ^x \downarrow && \downarrow\\ P Alg(F_P \partial_n, F_{n-1} U_{n-1} A) & \underset{}{\to} & P Alg(F_P \partial_n, A)}$

Here the bottom map is induced by the counit of the adjunction $F_{n-1}\dashv U_{n-1}$, while the right-hand map is induced by the inclusion $\partial_n \hookrightarrow 2_n$.

• The left adjoint $F_n$ of $U_n$ is defined by taking an $n$-computad $D = (C,X,x)$ to the following pushout in $P Alg$:

$\array{X\cdot F_P \partial_n & \overset{\overline{x}}{\to} & F_{n-1} C\\ \downarrow && \downarrow\\ X \cdot F_P 2_n & \underset{}{\to} & F_n D}$

Here $\cdot$ denotes a copower by a set, the top map $\overline{x}$ is the adjunct of $x$, and the left-hand map is again induced by the inclusion $\partial_n \hookrightarrow 2_n$. Adjointness is easy to verify using the universal properties of pullbacks and pushouts.

Finally, the category $\omega Cptd_P$ of $\omega$-computads is the limit of the diagram

$\dots \to 2 Cptd_P \to 1 Cptd_P \to 0 Cptd_P \to (-1) Cptd_P$

consisting of the functors $n Cptd_P \to (n-1)Cptd_P$ taking $(C,X,x)$ to $C$. The functors $U_n$ form a cone over this diagram and thus induce a functor $U_\omega\colon P Alg \to \omega Cptd_P$, which is easily seen to have a left adjoint which takes an object $(C_n)$ of $\omega Cptd_P$ to the colimit

$F_{-1} C_{-1} \to F_0 C_0 \to F_1 C_1 \to F_2 C_2 \to \dots.$

#### Examples

For simplicity, let $P$ be the terminal globular operad, such that $P$-algebras are strict ∞-categories.

• As in the definition, a 0-computad is just a set, and the free 0-category on a 0-computad is just itself, considered as a discrete $\omega$-category.

• A 1-computad consists of 0-computad $C$ (i.e. a set) together with another set $X$ and a function $X\to GSet(\partial_1, U_P F_0 C)$. Now $\partial_1$ is just a pair of objects, so this means that each element of $X$ is equipped with a pair of elements of $C$, which we call its source and target. Thus a 1-computad is just a quiver.

• The free 1-category on a 1-computad is the usual free category on a quiver. That is, its objects are the vertices of the graph and its morphisms are finite composable strings of edges in the quiver.

• A 2-computad is given by a quiver together with a set $C_2$ of 2-cells, each equipped with a source and a target which are composable strings of edges in the graph. For example, if the given quiver is a square

$\array{\bullet & \overset{f}{\to} & \bullet \\ ^h\downarrow && \downarrow^k\\ \bullet& \underset{g}{\to} & \bullet}$

then the free category it generates is the free noncommutative square, having two diagonals $k f$ and $g h$. We can then make a 2-computad by adding one 2-cell $\alpha$ with source $k f$ and target $g h$. The free 2-category on this 2-computad can then be drawn pictorially as

$\array{\bullet & \overset{f}{\to} & \bullet\\ ^h\downarrow & \swarrow_\alpha & \downarrow^k\\ \bullet & \underset{g}{\to} & \bullet}$
• Any $n$-globular set can be considered as an $n$-computad where for each $n$, the functions $s, t: C_n \rightrightarrows F_{n-1}(\mathcal{C})_{n-1}$ land inside $C_{n-1}$. The free $n$-category on this $n$-computad will then agree with the free $n$-category on the $n$-globular set we started with.

• Every oriental can be presented by a computad, as can every opetope, cube, as can most other geometric shapes for higher structures.

### On inverse diagrams

The above definition can easily be generalized by replacing the globe category by any direct category, i.e. the opposite of an inverse category. In more detail, let $\mathcal{I}$ be any inverse category, and consider the diagram category $\Set^{\mathcal{I}}$. Then any object $n\in \mathcal{I}$ has a representable functor $Y_n\in \Set^{\mathcal{I}}$ defined by $Y_n(x) = \mathcal{I}(n,x)$, which has a “boundary” subfunctor $\partial_n \subset Y_n$ consisting of all nonidentity morphisms $n\to x$ (as $x$ varies). These play the role of $2_n$ and $\partial_n$ above.

Let $P$ be a monad on $\Set^{\mathcal{I}}$ with induced adjunction $F_P : \Set^{\mathcal{I}} \rightleftarrows P Alg : U_P$. Since (by definition of “inverse category”) the objects of $\mathcal{I}$ have a well-founded relation given by the existence of nonidentity arrows, we will define by well-founded recursion on $n\in \mathcal{I}$ the category $n Cptd_P$ and an adjunction $n Cptd_P \underoverset{U_n}{F_n}{\rightleftarrows} P Alg$.

Assume, therefore, that this adjunction has been defined for all $k$ for which there is any nonidentity morphism $n\to k$. Let $(\lt n) Cptd_P$ be the limit of the $k Cptd_P$ for all such $k$, with an induced adjunction $F_{\lt n} \dashv U_{\lt n}$ to $P Alg$ (see below). Now define an $n$-computad to consist of a $(\lt n)$-computad $C$, a set $X$, and a function

$x\colon X\to P Alg(F_P \partial_n, F_{\lt n} C) = Set^{\mathcal{I}}(\partial_n, U_P F_{\lt n} C).$

The functor $U_n\colon P Alg \to n Cptd_P$ is defined as before: it sends a $P$-algebra $A$ to the $n$-computad defined by the $(\lt n)$-computad $U_{\lt n} A$ together with $X$ and $x$ defined by the following pullback square in Set:

$\array{X & \overset{}{\to} & P Alg(F_P Y_n, A)\\ ^x \downarrow && \downarrow\\ P Alg(F_P \partial_n, F_{\lt n} U_{\lt n} A) & \underset{}{\to} & P Alg(F_P \partial_n, A)}$

Similarly, its left adjoint $F_n$ takes an $n$-computad $D = (C,X,x)$ to the following pushout in $P Alg$:

$\array{X\cdot F_P \partial_n & \overset{\overline{x}}{\to} & F_{\lt n} C\\ \downarrow && \downarrow\\ X \cdot F_P Y_n & \underset{}{\to} & F_n D}$

This definition is not quite correct as written, because it assumes in the definition of $(\lt n)$-computads not just that the categories $k Cptd_P$ are defined with adjunctions to $P Alg$, but that they are related by suitable functors as $k$ varies that commute with these adjunctions. Thus, formally speaking it has to be phrased as a definition of a functor, rather than a function, by well-founded recursion, as explained here. We leave the details to the reader.

For example, if $\mathcal{I} = \mathcal{P}\times \mathcal{P}$, where $\mathcal{P} = (1 \rightrightarrows 0)$, then there is a monad on $\mathcal{I}$ whose algebras are double categories, and in this way we can obtain a notion of “double computad”.

The category of computads relative to a globular operad $P$ is sometimes a presheaf category, and when it isn’t, it “almost is.” At first glance, it may look as though it should always be a presheaf category, say $Set^{Ctp^{op}}$ where $Ctp$ is the category of “computopes.” A “computope” is, intuitively, one possible “shape” for an $n$-cell in an $n$-category (i.e. a $P$-algebra). For example, every “globe” is a computope, as is every simplex/oriental, every cube, and so on. It may feel at first as though a computad should be specified by giving a set of cells of each “shape” (i.e. for each computope) related by “face maps,” generalizing globular sets, simplicial sets, cubical sets, and so on.

However, when $P$ is the terminal globular operad whose algebras are strict $\omega$-categories, the presence of identities in the notion of free $n$-category prevents this from quite working, for sort of the same reason that strict n-categories are insufficient for $n\gt 2$. For instance, there is a 2-computad with one 0-cell $x$, no 1-cells, and two 2-cells $\alpha$ and $\beta$. The source and target of $\alpha$ and $\beta$ must then both be the identity 1-cell $id_x$ of $x$. Now in the free 2-category generated by this 2-computad, we have $\beta\alpha = \alpha\beta$, by the Eckmann-Hilton argument. If we define a 3-computad on top of this 2-computad with a 3-cell $\mu$ whose source (say) is $\beta\alpha = \alpha\beta$, then there can be no “face” maps from the computope-shape of $\mu$ to the computope-shape of $\beta$ and $\alpha$, since there is no way to distinguish $\alpha$ from $\beta$ (i.e. neither one is the “first” or the “second”).

This argument only kicks in for $n\ge 3$, so the categories of 0-computads, 1-computads, and 2-computads are presheaf categories. Moreover, the problem can also be avoided if $P$ is “suitably weak.” To define what this means, one considers the “slices” of the operad $P$. By truncation, any such $P$ induces a monad $P_k$ on the category of $k$-globular sets. Now the full subcategory of $P_n Alg$ on the objects whose underlying globular set is $(n-1)$-terminal (i.e. terminal in degrees $\lt n$) is monadic over the category of sets; the resulting monad on Set is called the $k$-slice of $P$. In Theorem 5.2 of Computads and slices of operads, Batanin shows that the category of $n$-computads of a Batanin-operad $P$ is a presheaf category if the $k$-slices of $P$ are strongly regular theories? for all $0\leq k \leq n-1$. This applies in particular to the case where $P$ is an operad for Batanin weak $\omega$-categories.

On the other hand, even in the strict case we can obtain presheaf categories of suitably restricted computads. For instance, if we consider only “many-to-one” computads in which the target of each $n$-cell consists of exactly one $(n-1)$-cell (rather than a free composite of such), we obtain a presheaf category, which is in fact equivalent to the category of opetopic sets. We also obtain a presheaf category if we restrict both source and target to not be identity arrows; see Henry 2017.

The monadicity of $\omega Cptd$ over $\omega Cat$ was first claimed by Batanin in his ‘98 paper. The proof however made use of the fact that $\omega Cptd$ was a category of presheaves, which was later found to be false. Métayer provided a corrected proof in

Quite generally, computads can be used to describe cofibrant replacements. Specifically, the $\omega$-categories freely generated by computads are precisely the cofibrant objects in the canonical model structure on strict ∞-categories. This is discussed in

• F. Métayer, Cofibrant complexes are free (arXiv)

• F. Métayer, Resolutions by polygraphs (tac)

The cofibrant resolution $(-)_{cof} \to Id : \omega Cat \to \omega Cat$ given by Métayer in these articles is the one counit of the adjunction $\omega Cptd \to \omega Cat$. In particular, this implies that every strict ∞-category is equivalent as an $\omega$-category to one that is freely generated by a computad. (Notice that these articles say “polygraph” for “computad”, following Burroni).

It is to be expected that similar results are true for Batanin weak $\omega$-categories, defined as algebras for some other “contractible” globular operad $P$.

Moreover, for any globular operad $P$, one can show that the “cofibrant replacement” obtained as above from the adjunction $\omega Cptd_P \rightleftarrows P Alg$ is the same as the “canonical” cofibrant replacement comonad obtained from the algebraic weak factorization system on $P Alg$ generated by the boundary inclusions $F_P \partial_n \to F_P 2_n$. This is proven in

In particular, the Kleisli category of this comonad is a good candidate for the category of weak functors between Batanin weak $\omega$-categories.

2-computads were originally defined by Ross Street in the paper

• Ross Street, Limits indexed by category-valued 2-functors.

The goal there was to describe which 2-categories are “finitely presented” (the presentation being given by a 2-computad) in order to describe the correct notion of “finite 2-limit”.

I don’t know the first use of “$n$-computads,” but Michael Makkai has studied them as a way to define opetopic sets, and shown that they are “almost” but not quite a presheaf category: