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Category Theory is fundamental, offering a basis to reason about things like set theory. With it's minimal restrictions (mostly the associativity of morphism composition) it seems difficult to find anything that cannot be studied using it. Even non associative operations like the operator of quasigroups can be studied by cleverly hoisting it into an associative structure via separating left and right multiplication. Even logic itself is studied via topos.

I've been trying to understand it's limits. The best I have found is a quote from Jean-Yves Girard:

The limitations of categories–insofar as we can judge them from the sole logical viewpoint–lies in their spiritualism, their extreme spiritualism : everything is up to isomorphism. In particular categories cannot explain locative logical constructions such as intersection types–or if you prefer, the categorical viewpoint compelled us to consider these artifacts as non-logical. In the same way, category-theory cannot explain the prenex form of ludics, which are based on equalities and which are definitely impossible to explain by means of isomorphisms.

The "locative logical constructions" and "ludics" he speaks of appear to be a logical construction of his own creation described in his 200 page opus, of which I have not fully dissected. This seems like an unsatisfying limit, trusting an author that his construction is unique enough to defy study.

What kinds of systems cannot be studied fully using Category Theory due to its inherent limitations?

Edit: In response to KCd's query about the word "study," I'm referring to limitations such as how an ancient Greek mathematician using rational numbers will eventually find a triangle whose ratio of its sides cannot be fully captured. Or how a set theorist applying ZFC must avoid some constructions such as "the set of all sets." On the other hand, a programmer will never find a computable function which cannot be computed via a Turing machine, although they may never actually choose to do so, preferring more convenient forms (such as "Java"). In the domain of programming, Turing Machines are universal. I use an informal word, "study," rather than something more formal out of fear of my own ignorance. I don't know what I don't know.

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What does "studied fully" really mean? The tone of your post sounds like someone who is a bit overly enthusiastic about a particular way of viewing things before using that viewpoint to prove something hard.

To use category theory in an area of math that is not category theory, you need to understand that area of math very well first. You won't prove nontrivial theorems just by mucking around with categorical generalities. At some point you need to really do something in that area of math. For that matter, the categories that have been introduced by mathematicians are largely based on a solid understanding of some other area of math (algebraic topology, algebraic geometry, number theory, and so on).

While category theory has penetrated quite far in algebra and topology, it isn't needed for all of it (see an answer here for an example from algebra) and many areas of analysis (e.g., PDEs with their hard estimates) do not currently rely on category theory. Perhaps Clausen and Scholze will change the overall relation between analysis and category theory with their condensed mathematics, but analysts are less enthusiastic users of category theory as a tool to prove theorems (not just restate things in a categorical language, even if that can be conceptually clarifying on its own).

That a certain point of view can contain others does not necessarily mean it ought to be the right way to study everything. For example, the foundation of essentially all of modern mathematics is set theory, but that does not mean people who work in the representations of Lie groups are actually using set theory as the primary (or even secondary) technical tool to prove their theorems. Each group is a semigroup, but that doesn't mean semigroups are more important in mathematics than groups; see a discussion about groups vs. semigroups here.

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  • $\begingroup$ Overly enthusiastic sounds right. Or rather, I am following a teacher that could be described as overly enthusiastic (I appreciate that!), and I'm curious to see the downsides. I find classes so extraordinarily all-encompassing, and associativity of composition so fundamental to how we approach language, that I have trouble formulating questions that don't fit the Category Theory mold (even if it's not the most convenient tool on the tool belt). But having seen what Godel did to Principia Mathematica, I'm hesitant to assume my naive understanding is complete enough to see its limits clearly. $\endgroup$
    – Cort Ammon
    Nov 24, 2021 at 15:06
  • $\begingroup$ I added an edit to the question on that vein. You are right to question the words "study fully." I was striving for something that didn't have a trivial definition that could be mapped into category theory too easily, while not being impossibly vague for the stack exchange format. My interest is less "is this the right tool" and more "is this a tool that can always be applied to do the job (in theory)." It's always possible to kill a bumblebee with an anti-aircraft gun, although you may do more damage to the surroundings than the bee. $\endgroup$
    – Cort Ammon
    Nov 24, 2021 at 15:08
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Question: "What kinds of systems cannot be studied fully using Category Theory due to its inherent limitations?"

Answer: It is better to write down a list of fields "using" category theory "in an essential way" and then choose a field that is not on this list.

If you study objects such as real manifolds, complex manifolds or algebraic varieties you will in many cases study various types of cohomology groups, homology groups, Chow groups, algebraic K-theory and these groups are in many cases defined using the language of "derived functors" - this is a language originating in the 1950s and the work of many people. Hence if you study a field where cohomology and homology is important you will end up using "category theory".

Many of todays long standing conjectures in pure mathematics are related to problems with calculating these groups - conjectures such as the Hodge conjecture, the Standard conjecture (on Weil cohomology theories) and various conjectures in arithmetic (related to algebraic K-theory and special values of L-functions). The study of algebraic K-theory $K_n(X)$ and the Higher Chow groups include the Chern character:

The Chern character is an isomorphism between higher algebraic K-groups and higher Chow groups

$$ ch^m_i: \operatorname{K}_m(X)_{\mathbb{Q}}^{(i)} \cong \operatorname{CH}^i(X,m)_{\mathbb{Q}}.$$

The higher algebraic K-groups are defined in terms of the category of vector bundles on $X$, the higher Chow groups are defined in terms of the set of all closed integral subschemes of $X$. Hence Chow theory is "more set theoretical", algebraic K-theory is "more category theoretical". Hence you can study these conjectures using "less category theoretical machinery" - then you study the Chow groups. Else you study algebraic K-theory.

Example: The Chern character above does not exist in "complete generality", hence if you want to use "less category theory" and use higher Chow groups to study "conjectures on special values of L-functions", you must first prove existence of a Chern character as above in greater generality. But in this process of generalization you will have to study algebraic K-theory and then it is inevitable: You must study some category theory in order to understand algebraic K-theory.

Why do we use divisors in algebraic geometry?

But: When you study algebraic geometry, much of the litterature is written in a "functorial language" (in the EGA, SGA book series and the stacks project), hence you will usually need some category theory when reading these books/notes.

Example: The Yoneda lemma is much used in algebra and algebraic geometry in defining objects such as the Hilbert scheme, Quot scheme and various other "moduli spaces". The Lemma is not a "deep lemma". It says that you may for any locally small category $C$ define the "functor of points" $h$

$$h: C \rightarrow Funct(C,Sets),$$

and that the functor $h$ "embeds" $C$ as a subcategory of $Funct(C,Sets)$. The Hilbert scheme is constructed using this Lemma. Does this mean you "use category theory" in an essential way in your work when you study the Hilbert scheme?

Example: If $X$ is a scheme and $E$ is a finite rank locally trivial sheaf on $X$, you may define the projective space bundle $\mathbb{P}(E^*)$ in two ways: You may define it using representable functors and the Yoneda lemma, or you may define it using a local trivialization of $E$ and glueing of schemes. In the case of the grassmannian functor you construct an open cover of representable sub functors and "glue functors". Hence in this situation you have a choice. You can choose a more direct approach and avoid using the functor of points and category theory. In many cases you have this type of choice: A direct approach and an "indirect approach" using the Yoneda lemma. In these cases where you do have such a choice: Does this mean you "use category theory" in your work? A "problem" with the Yoneda lemma in algebraic geometry is that you end up studying non small categories - categories where the objects form a class (and not a set). Hence you use non standard set theory. I believe there are quite many people that are using this construction without being aware of this fact.

Example: Whenever you study objects and have a notion of "isomorphism", this notion depends on which category you are considering. This does not neccessarily mean you "use category theory" in your work. What it means is that you must be careful which category you are working in, if you change the category your isoclasses change. As an example: Consider the case where you study complex quasi projective manifolds up to isomorphism. If you add the hypothesis that your manifolds are algebraic, then it follows you are studying smooth complex projective varieties. The same for coherent sheaves. If $E$ is a coherent sheaf on a complex quasi projective manifold $X$, and if $X$ happens to be projective it follows $E$ "is" projective (or: It is "equivalent" to a coherent algebraic sheaf $E^{alg}$ on $X$):

Motivating (iso)morphism of varieties

This does not necessarily mean you "use category theory" when you study coherent sheaves on complex quasi projective manifolds. There is for any complex quasi projective algebraic manifold $X$ an associated complex analytic manifold $X^{s}$ (see the above link) and for any coherent algebraic sheaf $E$ on $X$ there is a corresponding coherent analytic sheaf $E^s$ on $X^s$, and this correspondence is functorial. When you use this correspondence - does this mean you "use category theory" and that "category theory is essential" in your work? In this particular case it means you study smooth complex projective varieties, but where you include methods from complex analysis as a tool.

Example: If someone studies "Weil cohomology theories" and "algebraic cycles", this study involves much category theoretical machinery. Does this mean this person studies "category theory"?

https://en.wikipedia.org/wiki/Yoneda_lemma#Formal_statement

https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Algebraic_geometry

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