# May 18, 1995 {#week53}

Near the end of April I was invited by Ronnie Brown to Bangor, Wales for
a very exciting get-together. Readers of "This Week's Finds" will
know I'm interested in $n$-categories and higher-dimensional algebra
these days. Brown is the originator of the term "higher-dimensional
algebra" and has been sort of a prophet of the subject for many years.
Tim Porter at Bangor also works on this subject; I'll try to say a bit
more about his stuff next week. And visiting Bangor at the time were
John Power and Ross Street, two category theorists who do a lot of work
on $n$-categories. So I had a chance to learn some more higher-dimensional
algebra and category theory and see what these folks thought of my crazy
ideas.

1) Ronald Brown, "Out of line", _Royal Institution Proceedings_ **64**, 207--243.

Brown is very interested in explaining mathematics to the public, and
this paper is based on a talk he gave to a general audience. It is a
very accessible introduction to what mathematics is really all about,
and what higher-dimensional algebra is about in particular. "Out of
line" is a pun, of course, because not only does higher-dimensional
algebra seek to burst free of certain habits of "linear thinking" that
tend to go along with symbol string manipulation, it also has been a bit
outside the mainstream of mathematics until recently.

Now, when I speak of "linear thinking" I am not indulging in some
vague new-agey complaint against rationality. I mean something very
precise: the tendency to focus ones energy on operations that are easily
modelled by the juxtaposition of symbols in a line. The primordial
example is addition: we have a bunch of sticks in a row:
$$\vert\vert\vert\vert\vert$$
and we say there are "5" sticks and write
$$1+1+1+1+1=5.$$
Fine. But when we have a $2$-dimensional array of sticks:
$$\begin{gathered}\vert\vert\vert\vert\\\vert\vert\vert\vert\end{gathered}$$

                            ||||
                            ||||

we are in a hurry to bring the situation to linear form by making up a
new operation, "multiplication", and saying we have $2 \times 4$ sticks. This
isn't so bad for plenty of purposes; it's not as if I'm against times
tables! But certain things, particular in topology, can get obscured by
neglecting operations that correspond most naturally to
higher-dimensional forms of juxtaposition, and Brown's paper explains
some of these, and how to deal with these problems. The point is not to
avoid linear notation; it's to avoid falling into certain mental traps
it can lead you into if you're not careful!

2) A. J. Power, "Why tricategories?", preprint available as `ECS-LFCS-94-289` from Laboratory for Foundations of Computer Science, University of Edinburgh. Also available at `http://www.ima.umn.edu/talks/workshops/SP6.7-18.04/power/power.pdf`

When I mentioned this paper to a friend, she puzzledly asked "'Why try
categories?'?" And indeed, one must have tried categories and enjoyed
them before moving on to bicategories, tricategories and that great
beckoning terra incognita of mathematics, $n$-category theory.

In a sense I already know "why tricategories". I think they're great,
and in a paper with James Dolan --- summarized in
["Week 49"](#week49) --- I did my best to get everyone else
interested in general $n$-categories. For me, the great thing about
$n$-category theory is that it strives to formalize the notion of
"process" in all its recursive splendor. An $n$-category is a
mathematical structure containing not only objects, which one might
think of as "things", and morphisms, which one might think of as
"processes leading from one thing to another", but also $2$-morphisms,
which are "processes leading from one process to another", and then
3-morphisms, etc., on up to $n$-morphisms.

In physics and topology applications, the $k$-morphisms can often be
thought of as $k$-dimensional geometrical objects, since (as the Greeks
knew) the process of motion of a point traces out a $1$-dimensional
figure, and similarly the motion of a $1$-dimensional figure traces out a
2-dimensional surface... and $n$-dimensional spacetime is in some rough
sense "traced out" by the motion of $(n-1)$-dimensional spacelike slices
through time. If you think this is vague, you're right --- and
that's why one needs $n$-category theory, to make it precise! When one
understands $n$-categories (which so far we really do only up to $n = 3$)
one sees that the possibilities inherent in $n$-dimensional topology match
up very nicely with one might have stumbled on not knowing topology at
all, but just playing around with this iterated notion of processes
between processes between processes... This "natural correspondence"
between purely algebraic concepts and topological ones is what makes
topological quantum field theory tick, and I can't help but feel that
it will have quite a bit to say about physics eventually.

Power, however, gives a quite different set of reasons for being
interested in tricategories. These are drawn from computer science and
logic, and they make me realize yet again how poor and outdated my
education in logic was, and how much interesting stuff there is going on
in the subject!

At a completely naive level, one might already expect that relation
between "processes" and "things" is subtle and interesting in
computer science. For after all, we can think of a program either as a
process that turns one thing into another, or as data, a sort of thing,
which other programs can act on. Power does not really emphasize this
issue explicitly, but I can't help remarking on it, especially because
it reminds me of the curious fact that in mathematical physics, "time
is the last dimension".

That is, in topological quantum field theory, the $n$-morphisms in an
$n$-category, which are the processes having no further processes going
between them, represent the passage of time. And indeed, for many
practical purposes it appears that $n = 4$ is where things leave off,
since spacetime appears $4$-dimensional. On the other hand, nobody knows
any mathematical reason why one has to stop at any given $n$. So
ultimately we should try to understand "$\omega$-categories", having
$n$-morphisms for all $n$ greater than or equal to zero (0-morphisms being
simply objects, and $1$-morphisms being morphisms). This corresponds
philosophically to the notion that every "process" can also be
regarded as a "thing" which other processes can transform. Moreover,
we should also try to understand "$\mathbb{Z}$-categories", having $n$-morphisms
for all integers $n$, even negative ones! In this world, where there is no
bottom as well as no top, every "thing" can also be regarded as a
"process".

But I digress. Power is actually more interested in a different (though
perhaps related) hierarchy. Sometimes people like to say computers just
do stuff with bunches of numbers, but that's pretty misleading. Of
course computers *can* do things with numbers, but that's one of the
simpler mathematical things they can do. A number is an element of a set
(the set of real numbers, or some set of more computer-manageable
numbers.) And computers have no problem dealing with elements of sets.
But computers can also deal with sets themselves --- and more fancy
mathematical objects.

Many mathematical objects are sets, or bunches of sets, equipped with
operations satisfying equational laws. For example, a group is a set
equipped with a product and inverse operation satisfying various laws.
Sometimes these operations are only defined if certain conditions hold,
of course. For example, a category is a set of "objects" and a set of
"morphisms", together with various operations like composition of
morphisms, but one can only compose two morphisms $f\colon x\to y$ and $g\colon w\to z$ if $y = w$. Other examples might include graphs, partially ordered
sets... and all sorts of things computer scientists know and love.

We could call all of these "sets with essentially algebraic
structure." Mathematically sophisticated computer scientists want to be
able define data types corresponding to arbitrary sorts of sets with
essentially algebraic structure, and to play around with them easily. So
they need to ponder such things in considerable generality.

Note that in all cases, there is not just a bunch of objects to play
with --- like "groups" or "partially ordered sets" --- but a
*category* in which the morphisms are structure-preserving maps between
the objects in question. For example, there is a category $\mathsf{Grp}$ whose
objects are groups and whose morphisms are group homomorphisms.

The categories one gets this way are of a certain sort. Power calls them
"categories of models of finite limit theories". To define this
requires a bit of know-how, but it's basically simple. For example,
suppose I were trying to explain the definition of a category to a
computer scientist; I might say, every category has a set $\mathrm{ob}$ of objects
and a set $\mathrm{mor}$ of morphisms; every morphism has an object called its
source (or domain), so there is a function
$$\operatorname{source}\colon\mathrm{mor}\to\mathrm{ob}$$
and similarly every morphism has an object called its target (or
codomain) so there is a function
$$\operatorname{target}\colon\mathrm{mor}\to\mathrm{ob}.$$
Now, we can compose a morphism $f$ and a morphism $g$ to get $fg$ if $\operatorname{target}(f) = \operatorname{source}(g)$, so we have a composition function
$$\operatorname{composition}\colon C\to\mathrm{mor}$$
defined only on the subset $C$ of $\mathrm{mor}\times\mathrm{mor}$ that is the biggest subset
making the following diagram commute:
$$
  \begin{tikzcd}
    C \ar[r,"p_1"]
      \ar[d,swap,"p_2"]
    &\mathrm{mor} \ar[d,"\operatorname{target}"]
  \\\mathrm{mor} \ar[r,"\operatorname{source}"]
    &\mathrm{ob}
  \end{tikzcd}
$$
where $p_1\colon(f,g)\mapsto f$ and $p_2\colon(f,g)\mapsto g$.

Now category theorists have a slick way of dealing with these functions
defined only a subset satisfying equational conditions; instead of
talking about the "biggest subset" $C$ they would say that $C$ is the
"limit" of the diagram
$$
  \begin{tikzcd}
    &\mathrm{mor} \ar[d,"\operatorname{target}"]
  \\\mathrm{mor} \ar[r,"\operatorname{source}"]
    &\mathrm{ob}
  \end{tikzcd}
$$
If you don't get this, don't worry; in a sense it's just another way
of talking about the same thing, with the advantage of being infinitely
more general, since one can talk about the limit of any diagram, though
here we will only need limits of *finite* diagrams.

Then, after having lined up these ingredients (and I have left some
out!), I could go ahead and say what equational laws they need to
satisfy, like associativity of composition; and if I wanted I could
write all these laws out using commutative diagrams, too! Then I would
have laid out the "theory of categories" --- a complete description
of the operations in a category and the laws they obey.

The theory of categories is a typical example of a "finite limit
theory", because what I really did above, in describing the "theory of
categories", is describe a CATEGORY, say $\mathsf{Th}$, having objects called $\mathrm{ob}$
and $\mathrm{mor}$, and morphisms called $\operatorname{source}$, $\operatorname{target}$, $\operatorname{composition}$, and so on,
such that various diagrams commute! Moreover, we should think of $\mathsf{Th}$ as a
category with all finite limits, that is, one in which all finite
diagrams have limits. That allows us to deal with things like the object
$C$ above, which are defined as limits of finite diagrams.

So we have this thing $\mathsf{Th}$, the "theory of categories". And then, any
*particular* category is a "model" of this theory $\mathsf{Th}$. A "model"
assigns to each object in $\mathsf{Th}$ a particular set --- for example, "mor"
above gets assigned the set of morphisms in some particular category $\mathcal{C}$
--- and assigns to each morphism in $\mathsf{Th}$ a particular function --- for
example, "composition" above gets assigned the function representing
composition in $\mathcal{C}$. Moreover, this assignment satisfies a bunch of utterly
obvious consistency conditions which one summarizes by saying that a
"model of the theory $\mathsf{Th}$ is a functor from $\mathsf{Th}$ to $\mathsf{Set}$ that preserves
finite limits". In logic, you know, a model of a theory is something
that assigns to each thingie in the theory an actual thingie, in such a
way that all the stuff the theory SAYS is true about these thingies, IS
true!

Now if you are with me thus far you either know this stuff better than I
do, or else I congratulate you, because the example I picked was
deliberately self-referential and confusing --- I was using category
theory to describe the theory of categories, and also, the theory $\mathsf{Th}$
itself was a category! But the world of thought does have a funny way of
wrapping back on itself like that... so it's good to get used to it.

In fact there is a big literature on "sets with essentially algebraic
structure" and "categories of models of finite limit theories"...
this is a branch of logic they never taught me about in school, but it
definitely exists, and Power gives some references to it:

3) P. Gabriel and F. Ulmer, _Lokal praesentierbare Kategorien_, in Springer Lecture Notes in Math **221** (1971).

    G. Kelly, Structures defined by finite limits in the enriched context I, _Cahiers de Top. et. Geom. Diff._ **23** (1982), 3--41.

    Michael Makkai and Robert Pare, "Accessible categories: the foundations of categorical model theory", in _Contemp. Math._ **104** (1989).

But let's dig in a bit further, since really the fun is just starting.
Now, I told you what a model of one of these finite limit theories Th
was, but not what a morphism between models is! Well, if a model is a
sort of functor, a morphism between them should be a sort of natural
transformation between functors; that's how it usually goes. So there
is really a category $\mathsf{Mod}(\mathsf{Th})$ of models of one of these theories $\mathsf{Th}$. If
$\mathsf{Th}$ were the theory of categories as above, $\mathsf{Mod}(\mathsf{Th})$ would be the category
of (small) categories, which is called $\mathsf{Cat}$. To take a less fiendish
example, if $\mathsf{Th}$ were the theory of groups, $\mathsf{Mod}(\mathsf{Th})$ would the category
$\mathsf{Grp}$.

But now suppose one wanted to build a computer language that could not
only deal with all sorts of data types corresponding to different "sets
with essentially algebraic structure", but also various "categories
with essentially algebraic structure". For if one likes category theory
well enough to do a lot of computer science using it, it makes sense to
let the computer itself join the fun, by creating a language in which
it's easy to talk about categories. After all, our eventual goal with
computers is to have them completely replace computer scientists, right?

Well, in a way "categories with essentially algebraic structure"
aren't terribly different from sets with essentially algebraic
structure. Roughly, the idea is that instead of having a data type that
consists of a bunch of sets with functions between them satisfying some
equational laws, we have a data type consisting of a bunch of
categories, functors between them, and natural transformations between
THEM satisfying equational laws. What this means is that if we try to
copy the above stuff, instead of a "theory" we will have a
"2-theory" $\mathsf{Th}$, which is some sort of $2$-category, and then a model of
this would be a 2-functor from $\mathsf{Th}$ to $\mathsf{Cat}$. We want to wind up getting a
$2$-category $\mathsf{Mod}(\mathsf{Th})$ of models of $\mathsf{Th}$.

But actually carrying this out is a bit tricky, and much of Power's
paper goes into the details of various proposed schemes. Of course there
is no reason in principle to stop here, other than our limited
understanding of $n$-categories, sheer bewilderment, or boredom. Reasoning
about $n$-categories always tends to drag in $(n+1)$-categories, because the
collection of all $n$-categories with some particular structure (such as
the "essentially algebraic structures" I've focussed on here, but
also other sorts) typically forms an $(n+1)$-category. This is how Power
motivates tricategories. Right now we are stuck at $n = 3$, but there are
good reasons to expect that pretty soon we'll go beyond that. In fact,
Power and Street showed me a sketch of their ideas on
tetracategories....