Tangent categories

First encounter

The first time I encountered tangent categories was during the COVID19 pandemic (hopefully the first and also the last global pandemic I see in my lifespan). Stuck at home for months after being fired from a job as a receptionist in a hostel (I had a lot of fun doing that job), I was starting losing hope for my plan of travelling and working in Canada for at least a year.

One day, I was searching the web looking for something that could be related to a vague idea I was working on for a while. After a few hours searching for the right thing, I found a paper, written by Robin Cockett, a professor at the University of Calgary and Geoff Cruttwell, professor at Mount Allison University.

The paper, titled Differential structure, Tangent structure, and SDG, was 87 pages long and contained lots of ideas that, at that time, I couldn’t fully understand. Despite its intimidating title and its length (I wasn’t in academia at that time and my background in physics didn’t fully prepare me for a category theory paper), the paper was impressingly clear and simple in exposing the core ideas.

The main claim of the paper was that, in order to do geometry, the only thing you need is a category with a tangent structure. A box with the correct context. But was a tangent structure a sufficient context for such a bold claim? Could really be geometry done only using a notion of the tangent bundle functor?

This work, I discovered, was actually a generalization of another paper, written by Rosický in 1984. Rosický’s 11 pages paper gave an abstract axiomatization of the tangent bundle functor. Now, most of the difficult of category theory is put in the definitions. This is because, when you play with abstraction, it is quite easy to slip into the abyss of things too general. Dually, working with examples, might end into a theory that over fits the data, namely, that is too specific and not general enough.

This justifies my skepticism when I read the first time the definition of a tangent category: no mention on real numbers, let alone topology, or fibre bundle structures! All important structures that appear naturally in differential geometry. Could a tangent category be the right generalization of differential geometry?

Doing geometry with categories

If I have to motivate the definition of a tangent category, I could start by telling you about differential geometry. The problem is, any course on differential geometry takes the first week and a half only to introduce the most basic object of study, that is, a smooth manifold and this by assuming usually you have a solid background in topology and real analysis.

I believe that we can entirely forget about differential geometry, which is where tangent categories took inspiration from, and play a game. Let’s build together a theory of geometry.

The first ingredient of our theory is a class of objects that will be our geometric spaces. An example of a geometric space could be the Cartesian plane, or this room, or again, the surface of our planet, or maybe the entire observable universe.

We’re playing with abstraction, so instead of focusing entirely on one space, we shall collect all possible spaces of a certain type. Remember, bird view. If you are a differential geometer you might choose those spaces as smooth manifolds; if you are an algebraic geometer, you might want to take schemes, instead.

But regardless of the concrete model of spaces that you have in mind, the point is to have a notion of generic geometric space. Such a space, might not contain points, but, for the sake of our mental stability, we may indulge in thinking of those spaces as made by points.

Next, we need a way to transform points of a space into the points of another space. These are the morphisms of our tangent category, while the objects are in fact the spaces.

Now, imagine to sit on a point of one of these spaces. Differential geometry is all about the local experience of a space, in contrast with the global experience, in the same way, Earth looks flat in our daily experience, however, if you found the new Amazon and exploit enough your employees, you might afford a ticket to see the planet from above. In that scenario, you’ll see it as a spherical object floating in the darkness of the Milky way.

This distinction between local and global is very central in geometry and appears naturally in tangent categories, as well.

I was saying, imagine to seat at a certain point of your favourite space. You might not just want to seat, but walk around a bit. For this, you need a notion of direction. For this, you need that each point $x$ is associated to a space of directions, that we denote by $\mathsf{T}_xM$, where $M$ denotes the space you are in.

Directions are different than points for a few reasons. For starters, any point in a space is as important as any other point. Points are very democratic in that sense. Directions are not like that. There is always a special direction, called the zero direction, that we denote by $0_x\in\mathsf{T}_xM$. The zero direction is the direction of “staying at the same spot at all time”. The second difference is that, while points cannot be sum together, directions have a well-defined notion of sum. If $u$ and $v$ are two directions, their sum $u+v$ should be understood as the direction “moving in the direction $u$ while moving in the direction $v$ at the same time.

In some spaces you might also be able to “come back” to the original point if you move in a direction $u$. This means that for every direction $u$, there is a direction $-u$, such that, $u+(-u)=0_x=(-u)+u$. In a general tangent category, we might not have all negatives.

It is also important to realize that the space of directions $\mathsf{T}_xM$ at a point $x$ might be different than the space of directions $\mathsf{T}_yM$ at another point $y$. This is the case on a sphere. The directions I can move now that I live in Australia is very different than the directions my mom can move to, where she lives in Italy.

In a tangent category, the spaces of directions are all put together in a larger structure, called the tangent bundle of a space. The tangent bundle of $M$, denoted $\mathsf{T}M$ is another geometric space, whose points are all the directions of $M$ at each point of $M$. So, a point of $\mathsf{T}M$ should be thought as a pair $(x,u_x)$, where $x$ is a point of $M$ and $u_x$ is a direction of $M$ at $x$.

It is very important to remember that, in general, we cannot regard $\mathsf{T}M$ as the Cartesian product $M\times V$ for some global space of directions $V$. This would amount to establish that every space of directions is exactly the same. There’s a deep reason why this is not globally the case, related to the so-called hairy ball theorem, that establishes that the tangent bundle of the sphere $\mathbb{S}^2$ cannot be written as the product of $\mathbb{S}^2$ and something else.

The tangent bundle is not just an assignment, for every space $M$ of another space $\mathsf{T}M$. In fact, $\mathsf{T}$ is a functor, meaning that, for every morphism $f\colon M\to N$, there is a morphism $\mathsf{T}f\colon\mathsf{T}M\to\mathsf{T}N$, such that, $\mathsf{T}\mathsf{id}_M$ is still the identity $\mathsf{id}_{\mathsf{T}M}$ and such that $\mathsf{T}(g\circ f)=\mathsf{T}g\circ\mathsf{T}f$

A tangent category then comprises a collection of maps

$$p_M\colon\mathsf{T}M\to M\quad z_M\colon M\to\mathsf{T}M\quad s_M\colon\mathsf{T}_2M\to\mathsf{T}M$$

respectively called, the projection, the zero, and the sum. $p_M$ is the map that sends every direction $(x,u_x)$ to its base point $x$, while $z_M$ takes each point $x$ to its zero direction $0_x$ and $s_M$ sums two directions $(x,u_x)$ and $(x,v_x)$ at the same base point together to give $(x,u_x+v_x)$. These maps capture the zero and the sum of directions.

Finally, there are two, more mysterious maps:

$$l_M\colon\mathsf{T}M\to\mathsf{T}\mathsf{T}M\quad c_M\colon\mathsf{T}\mathsf{T}M\to\mathsf{T}\mathsf{T}M$$

The firs one is called the vertical lift and the second one, the canonical flip. The vertical lift satisfies an axiom that imposes that, each geometric space of a tangent category must be locally linear. This means, that, despite a space might globally having a complicated geometry, locally it must be simple.

This comes directly from our daily intuition: when I look at the horizon, I see a flat space, even though Ketty Perry and Jeff Bezos says that globally I live on a ball.

Finally, the canonical flip is a local symmetry that tells us that, if I move just a little bit in the direction $u$ and right after in the direction $v$ is the same as going first in the direction $v$ and later in the direction $u$.

Some references

I save you from the formal definition of a tangent category. Instead, I suggest you to read the following resources:

• Check out my notes! Tangent categories: a minicourse

• Cockett, J. R. B. and Cruttwell, G. S. H. (2014). Differential Structure, Tangent Structure, and SDG (Vol 22). Applied Categorical Structures