Category theory without categories

I was bewildered by my first exposure to category theory. My first semester in graduate school I had a textbook with definitions like “A gadget is an object G such that whenever you have this unfamiliar constellation of dots and arrows, you’re allowed to draw another arrow from here to there.” What? Why?!

I revisited category theory occasionally after college, going through cycles of curiosity followed by revulsion. It took several cycles before I could put my finger on why I found category theory so foreign.

There are numerous obstacles to appreciating category theory, but the biggest may be diagrammatic reasoning. More specifically, having to learn diagrammatic reasoning at the same time as facing other challenges.

Why should diagrammatic reasoning be difficult? Isn’t the purpose of diagrams to make things clearer?

Usually diagrams are supplementary. They illustrate things that are described verbally. But in category theory, the diagrams are primary, not supplementary. Instead, you have definitions and theorems stated in the language of diagrams [1].

In practice, category theory uses a style of presentation you’re unlikely to have seen anywhere else. But this is not essential. You could do category theory without drawing diagrams, though nobody does. And, importantly for this post, you can use category theory-like diagrams without reference to categories. That’s what William Lawvere does in his book Conceptual Mathematics. The book uses Karate Kid-like pedagogy: the student gains fluency with a practice before being told its significance.

When you see a triangle made of two arrows, what’s the significance of the existence of an arrow that makes the diagram into a triangle? What difference does it make when the missing arrow goes on one side of the diagram or the other as in the two diagrams below?

The answer is not obvious if you’re unfamiliar with this way of thinking, and yet the problem has nothing to do with category theory per se. You could ask the question in the context of finite sets and functions. And that’s what Lawvere’s book does, acquainting the reader with diagrammatic reasoning before getting into category theory as such.


[1] Or more to the heart of the matter, you have definitions in terms of functions (“morphisms”) that are represented by diagrams. The difficulty does not come from the diagrams but rather from formulating things in terms of the existence and uniqueness of functions rather than more tangible arguments about sets and elements. This indirect formulation may seem unnecessary, or even sadistic, but it is generalizes further.

4 thoughts on “Category theory without categories

  1. I have been a silent reader. But this topic incites interest in me.
    I have not yet studied – category theory. I would love to have your outlook on this.
    * What is the primary distinction b/w set theory and category theory?
    I believe the concept of prime focus was “set” in set theory, whereas in category theory, its “morphisms”, which seems to be (but is not) a sweet generalisation of “relations”. It could be “set” is defined by the objects it holds. whereas in “categories”, claiming them as a unified-lumped object, is defined by the “morphism” that led to them.
    Philosophically, “sets” focus on definitive aspect in description of entities (mathematical or not) with its operation “belonging”; whereas categories focus on behavioural aspect of entities, with morphims / functors / transformations.

  2. A favorite quote from the first edition of Lawvere & Schanuel:

    “We all begin gathering mathematical ideas in early childhood, when we discover that our two hands match, and later when we learn that other children also have grandmothers, so that this is an abstract relationship that a child might bear to an older person, and then that ‘uncle’ and ‘cousin’ are of this type also; when we tire of losing at tic-tac-toe and work it all out, never to lose again; when we first try to decide why things look bigger as they get nearer, or whether there is an end to counting.

    “As the reader goes through it, this book may add some treasures to the collection, but that is not its goal. Rather we hope to show how to put the vast storehouse in order, and to find the appropriate tool when it is needed, so that the new ideas and methods collected and developed as one goes through life can find their appropriate places as well. … ”

    … and from mathematician John Isbell (1930-2005), commenting on his colleague Lawvere (1937-2023):

    “… Talking with Bill, I often feel like a fly buzzing around a cow. (It seems to me I can liken Bill to a cow, if I’m just a fly myself.) On any easy question, I’ll probably see the answer first. But his thoughts seem to move on a level where I don’t function. I can barely see down there. …”

  3. The most important thing about categories is that they are everywhere in mathematics

    *The category of sets and functions
    *The category of sets and relations
    *The category of a transitively closed oriented reflexive graph, in particular the transitive closure of an arbitrary oriented graph (with reflexive = closed under paths of length 0)
    *The category with 1 object and the morphisms the natural numbers)

    *The category of vector spaces and linear maps

    *The category of topological spaces and continuous maps
    *The category of topological spaces and homotopy classes of continuous maps
    *The category of open subsets of R^n’s and smooth maps.
    *The category of smooth manifolds and smooth maps
    *The category of groups and homomorphisms

    *The category of representations of a group G and intertwining operators
    *The category of Banach spaces and bounded linear maps
    *The category of rings and ring homomorphisms
    *The category of modules over a ring R and R-linear maps
    *The category of complexes of R-modules and homotopy classes of maps of complexes (this is a more advanced example)
    any time you have “things” with “thing structure preserving transformations” that you can compose and which have an identity, you have a category.

    If you think of a category as a transitively closed oriented graph, functors are just graph maps and indeed “oriented graphs and graph maps” is another example of a category from which it is only a mild generalisation to categories whose objects are certain (small) categories and the morphisms are functors

    Once you look at the mathematical world in this way it becomes quite natural to see what can naturally be expressed. Can you say “surjective” in function in this language (yes) and does it make sense for other categories (yes, a so called epimorphism).
    Q: Can one say the axiom of choice holds in this language?
    A: yes for every surjective function f:X ->> Y, there is choice function, s:Y ->X such that fºs = 1_Y, a _section_. This is a typical example of a diagram (in crappy ascii art):

    X = X
    |. ^
    f | | \exists s
    V |
    Y = Y

    Q: Does this formulation make sense for other categories?
    A: yes, exactly as written
    Q: Is there such a splitting “choice” section in other categories:
    A: in general, most definitely not. E.g the map
    S^3 -> S^2 which sends a vector (z_1, z_2) in C^2 with |z_1|^2 + |z_2|^2 = 1 to the line it spans, is a continuous surjection (epimorphism in the category of topological spaces), but there is no continuous section! This is a rather fundamental fact and one natural way to prove it is using algebraic topology which is essentially constructing functors from the category of topological spaces to the category of groups and then using algebra to conclude about topological spaces and continuous function, but this comment has become much too long already.

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