Question 1
How much of group/ring/lattice/... theory can be expressed in purely categorical terms (only using the notions object, morphism, identity morphism, and composition), that is, as properties of the category of groups/rings/lattices/...?
I know that the question is vague. Let me be a bit more precise.
For example, consider group theory. Whenever one proves a theorem $X$ about groups, one can ask: can $X$ be expressed purely in categorical terms? For instance, consider the following theorem $X$: "for each homomorphism $\varphi\colon G\to H$, there are homomorphisms
$$G\stackrel{\varphi_1}{\longrightarrow} G/\ker(\varphi)\stackrel{\varphi_2}{\longrightarrow}\text{im}(\varphi)\stackrel{\varphi_3}{\longrightarrow} H,$$
where $\varphi_1$ is surjective, $\varphi_2$ is an isomorphism, and $\varphi_3$ is injective." This theorem can be expressed purely in categorical terms, since each concept occurring in the theorem happens to have a categorical description: surjective --> epimorphism, injective --> monomorphism, quotient --> coequalizer, kernel, image. In contrast to, for example, Lagrange's theorem: it contains the notion of the "order" of a group, which I guess cannot be defined in terms of morphisms and composition.
In fact, one can (more) precisely define what I mean by "in purely categorical terms": consider the first-order language $L$ consisting of two sorts, objects and morphisms, two function symbols $\text{dom}$ and $\text{cod}$ (from morphisms to objects), and a ternary relation symbol between morphisms $f, g, h$ expressing that $h = g\circ f$. Then for each first-order (or higher-order) sentence $\varphi$ in the language $L$ and each category $\mathcal C$, one can ask whether $\varphi$ is satisfied in $\mathcal C$, $$\mathcal C\models \varphi.$$ In this sense, one has defined a notion of a "property" $\varphi$ of a category. For example, one can write down an $L$-sentence $\varphi_1$ expressing the associativity of composition. Then each category $\mathcal C$ satisfies $\varphi_1$. Also, one can write down an $L$-sentence $\varphi_2$ expressing that "for all objects $A, B$, there is an object $A\times B$ satisfying the universal property of the product". Then a category $\mathcal C$ satisfies $\varphi_2$ if and only if $\mathcal C$ has products.
Lawvere demonstrated that essentially all theorems of set theory can be expressed in this limited language (surprise: we don't need a binary relation $\in$) and derived from a small number of axioms (written down in the language), see ETCS.
Coming back to my question, I wonder how many theorems $X$ of group/ring/lattice/... theory can be expressed in this categorical language, in the sense that we can find an $L$-sentence $\varphi_X$ "expressing" $X$ (that is, $\mathcal C\models\varphi_X$, where $\mathcal C$ is the category of groups/rings/lattices/..., should be equivalent to $X$).
As another random example, can the spectral theorem of linear algebra, stating that an endomorphism $f$ is normal and triangularizable if and only if there is an orthonormal basis $B $consisting of eigenvectors of $f$, be expressed as a property of the category of finite-dimensional vector spaces?
Question 2
A related question that is coming to my mind is the following: sometimes in mathematics, it happens that two categories, say $\mathcal C$ and $\mathcal D$, studied in two different branches of mathematics, say $A$ and $B$, respectively, are shown to be equivalent. (For example, consider $A$ to be the study of commutative C*-algebras and $B$ the study of compact Hausdorff spaces.) How much does this say about the similarity of $A$ and $B$ as mathematical branches? If many questions and theorems can be expressed in the categorical language, this means that $A$ and $B$ are, as branches, essentially the same, since then each statement occurring in $A$ can be translated into the branch $B$ and vice versa. Otherwise (for example if some property of, say, Hausdorff spaces doesn't have a categorical analogue), this could mean that there can still be questions in the branch $B$ that cannot be translated into the branch $A$, therefore each of the two branches having its own right to exist.
from Hot Weekly Questions - Mathematics Stack Exchange
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