Is category theory "conceptual"?

Question

Warning: This question is in part philosophical in nature.

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Several prominent authors in category theory (CT) have claimed that CT is 'conceptual'. It is my impression that this sentiment is widely shared in the categorical community and is used often to explain the value of CT to outsiders. However, it is quite unclear to me whether this claim is ever really argued for, either mathematically, meta-mathematically or philosophically. More simply put: what do category theorists mean by 'concept' when they claim their discipline is 'conceptual'?

Here are some examples (all italics mine):

• Lawvere & Schanuel in "Conceptual Mathematics", write:

  There are in these pages general concepts that cut across the artificial boundaries dividing arithmetic, logic, algebra, geometry, calculus, etc.

  And later:

  Our goal in this book is to explore the consequences of a new and fundamental insight about the nature of mathematics which has led to better methods for understanding and using mathematical concepts.

  Also:

  The goal of this book has been to show how the notion of maps leads to the most natural account of the fundamental notions of mathematics [...].

• In "Adjointness and Quantifiers" (1969), Lawvere writes:

  More recently, the search for universals has also taken a conceptual turn in the form of Category Theory [..]" and later he mentions "the conceptual sphere of category theory.

• ncatlab states as a main goal of CT:

  Conceptual unification. A major driving force behind the development of category theory is its ability to abstract and unify concepts. General statements about categories apply to each specific concrete category of mathematical structures.

• Jean-Pierre Marquis writes about adjoints:

  ...the fact that they can all be described using the same language illustrates the profound unity of mathematical concepts and mathematical thinking."

Clearly the word "concept" is used in an emphatic sense here. A concept is not just any old expression, but an extremely fruitful one with a high level of generality (and yet mathematically speaking fully operational and not fuzzy). I also think that there is some connection to naturality. For example, adjoint functors are often said to be 'conceptual inverses'.

To clarify this question more I will distinguish between an internal and external use of "concept":

Internal: For example, to understand set theory, one will have to understand the 'concept' of power set at some point. This is an internal use.

External: Category theorists, however, claim that their concepts (e.g. limit, adjoint) have an external value, inasmuch as they apply to situations outside of pure category theory. (E.g. the Cartesian product in set is an instance of the product in pure CT.)

My thoughts and questions, unfortunately fuzzy and philosophical:

1. Is the word 'conceptual' meant in an absolute or relative way? For example, is set theory 'conceptual' LESS conceptual than category theory, or perhaps not conceptual at all?

   2. Is the following an (elementary) example of conceptual generalization: the set-theoretic concept of singleton set is generalized into the CT-concept of terminal object?
2. How does this relate to approaches to concepts in more classical formal logic? For example, axiomatic set theory can be said to 'give' the concept of set membership axiomatically. In contrast, category theory could claim that the category of sets is simply one which fulfills certain conceptual determinations, which are formulated in the more universal language of category theory. (I am alluding to ETCS here).
3. Is the use of 'concept' by category theorists not in a way circular? If one asks: why is category theory conceptual? one perhaps gets the answer: because relevant mathematical concepts are all formulated in category theory. If this is so, the idea of 'conceptual mathematics' is more of a motto for a research program than a claim about the nature of CT.

Answer 1

Warning: this answer subsumes various answers I gave on FB (and I think this sadly uncovers my secret identity on the big blue f). It is reported only to involve in the discussion as much people as possible before this thread is closed for being subjective and whatever else you want to say :-) (no polemic intended, but I find a nice intention what spurred the OP to open the thread)

My initial answer:

I personally see conceptualisation as a method to do mathematics. One of the features of category theory is that instead of proving a single theorem you enucleate a pattern that describes several similar situations or results (one of the most blatant achievements in this direction is functorial semantics). In fact, the way in which these patterns interact becomes itself an object of mathematical study, thanks to conceptualisation. Think for example to de/categorification, or to the study of coherence for monoidal / higher categorial structures (a feature of mathematical objects, i.e. the fact that they only compose or identify up to some additional information living "one type higher", can be the subject of a rather intricate "theory of higher patterns"). Category theory as a conceptual look at mathematics is, in fact, a systematic review of the rules under which mathematical objects relate, done using mathematical language itself.

To this, the OP replied, and I re-replied:

What made me ask the question in the first place is that in the work of Lawvere, one never finds an independent explanation of what 'conceptual' means (save for allusions to Hegel here and there). At the beginning (also of the 'conceptual mathematics' textbook) one thinks one will be explained what makes category theory conceptual. But this metamathematical or metalogical notion of 'concept' or 'conceptual' is never given any precise meaning (to my knowledge).

This is true. I've never noticed it so thank you for pointing out. I assume that the metamathematical meaning of "conceptual" is what I always had in mind, and that also taking conceptual and categorical as synonyms trivially solves the issue. But I agree that there is no reason to do so, in principle. I was reinforced in this belief by the feeling that what Lawvere had in mind with his book with Schanuel was an introductory text to categorical logical thinking, adapted to the "layman", and that they took category theory as a synonym of some sort of conceptualization for mathematics. The fact that somehow "conceptualization" is a synonym for "studying things categorically" is better appreciated in Lawvere's three pillars:

1. functorial semantics is a conceptualization of universal algebra, because informally a theory is defined as the widget from which you generate concrete models of structures.
2. the definition of an elementary topos is a conceptualization of the notion of sheaf, because it generalizes the geometric intuition for a matching family of "functions" into the powerful description via a subobject classifier and a Lawvere-Tierney topology
3. the definition of a cohesive topos is a conceptualization of several classical constructions of differential geometry because it clarifies what are the universal properties of most of the gadgets defined there (not without a certain taste for hegelian nomenclature).

In the end, if you accept that category theory is the way in which mathematics implements (a form of) structuralism, then you must accept that the categorical eye to mathematical problems is one among many; so, conceptual (categorical) must be opposed to practical approach.

I believe that the categorical thinking is "conceptual" in the sense that it opposes to other ways to do Mathematics which are not (like for example the "I don't care about general theorems, I want to kick the fu**ing integral in the mouth" approach of many people in mathematical analysis, probability and suchlike, or, more blatantly, the "I would really like to know how much machine-time this algorithm is going to cost me" approach of applied mathematicians). These are very legitimate ways to do mathematics, but they are not at all "conceptual", in that they concentrate on a single entity and its "esoteric" (ἐσώτερος, interior) patterns. Conceptualization on the contrary moves entities from the outside, not without a rather funny taste for proving statements as corollaries of gigantic structural theorems (why the inverse image function f^{-1} : PY -> PX preserves unions and finite intersections? Because it's the left part of a geometric morphism between presheaf categories!)

I personally like vey much the motto that when you do category theory you don't want to dis/prove a theorem: you want to understand why it is trivially true or trivially false (this mixes two quotes, one by Peter Freyd and one another from Richard Garner). But this approach is not something you find in every corner of mathematics, nor it is something you can hope to use to understand some subtle questions in combinatorics, or to solve the traveling salesman problem, or the Riemann hypothesis.

In a way, the parts of mathematics that exhibit a "dialectic tension" (this is really a term stolen from Lawvere), like algebra and geometry are so naturally seen in a categorical lens because it is that very dialectic that you are trying to describe mathematically (you want to state the fact that a space is determined by its algebra of continuous functions, and the only way to do that is via a functor), whereas those parts that live on their own (like subtle cardinality argument in finite group theory) are not influenced by this paradigm at all. This is again very legitimate: I'd be a really bad category theorist, if I said that all mathematics can be reduced to category theory. Instead, I am a category theorist, so I rather say that all mathematics should be reduced to category theory :-)

Another user raised an interesting point, finally:

"But this approach is not something you find in every corner of mathematics, nor it is something you can hope to use to understand some subtle questions in combinatorics, or to solve the traveling salesman problem, or the Riemann hypothesis." Why do you think so? I believe category theory used to be something to only understand trivialities, but that it's not the case anymore (I think Peter May said something like this or maybe it was other homotopy theorist). Of course, triviality is something very subjective. Maybe you mean something like: solely category theory will not be enough to solve the Riemann hypothesis since category theory only organize data coherently in order to the limited human brain process it?

To which my answer, and I promise it's the last, was

Of course I'd be the first interested in seeing a proof of a famous conjecture relying completely on categorical language. But in the end, what does "completely" mean here? Is the proof of Weil conjectures in the etale topos a categorical proof? You mention Peter May; is the definition of an operad "categorical"? Is the Doplicher-Roberts reconstruction theorem a theorem in category theory even if it was proved by two physicists (Sergio Doplicher by the way is fluent in four languages -ancient sanskrit is one of them-)?

Most mathematicians would say that these results motivated an interest for category theory but that they somehow pre-exist categorical language. They do not follow from CT, instead they motivate a systematic approach to circumvent, or to guide you through, the complexity of involved proofs. They surely obtain a theorem (all three are very specific theorems, indeed), but my point is that it is not the reason why we worship them.

Roughly, the purpose of category theory is to understand, using mathematics, why mathematics behaves in the way we observe. The incidental is really important here, as without it we would fall into logic or philosophy. Instead, we have theorems, but they regard mathematical structures as a whole, and more than ever they are about the meaning of a theorem, not its content. This is what I mean by "that's not the reason why we worship them": I do not like operads (only) because they prove that algebras for the little cube are whatever-whatever. Mostly, I like them because I can characterize "operations" in a much broader sense functorial semantics does using only monads; they pop out everywhere, and as a particular case, when I do algebraic topology, I find that a particular example of a really specific operad describes infinite loop spaces. But to my eye this is not category theory anymore, because it regards a specific entity, not the overall abstract semantics and the general rules of manipulation that the specific example covers. To put it simply, it is astoundingly beautiful mathematics, it only goes against the fundamental principle of category theory, answering the question of why things are the way they are, and not a different one.

Sorry for this incredible, shameful wall of text.