homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
symmetric monoidal (∞,1)-category of spectra
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.
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. Here 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 ω-categories, but it also applies to operads $P$ whose algebras are Batanin weak ω-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
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
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:
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 $P Alg$:
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
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
For simplicity, let $P$ be the terminal globular operad, such that $P$-algebras are strict ω-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
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
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.
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.
The adjunction $n Cptd \rightleftarrows n Cat$ should be monadic, at least in good cases. However, it is not clear to the author of this page whether or where a correct proof of this fact appears in the literature for any value of $n\gt 1$.
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 ω-categories. This is discussed in
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 ω-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
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:
Makkai, Harnik, and Zawadowski, Multitopic sets are the same as many-to-one computads, link.
Makkai, The word problem for computads, link.
Computads were studied by Burroni under the name “polygraph” in the framework of rewriting:
The following more recent papers are referred to above:
One should beware that there are some erroneous claims in some of Batanin’s papers; in particular the claim that computads relative to a globular operad are always a presheaf category. This was explicitly shown to be false in
Michael Makkai and Marek Zawadowski, The category of 3-computads is not cartesian closed, MR.
Eugenia Cheng, A direct proof that the category of 3-computads is not cartesian closed
Finally, we had some blog discussion about this at