The SEP article on category theory says:

Categories, functors, natural transformations, limits and colimits appeared almost out of nowhere in a paper by Eilenberg & Mac Lane (1945) entitled “General Theory of Natural Equivalences.” We say “almost,” because their earlier paper (1942) contains specific functors and natural transformations at work, limited to groups. A desire to clarify and abstract their 1942 results led Eilenberg & Mac Lane to devise category theory. The central notion at the time, as their title indicates, was that of natural transformation. In order to give a general definition of the latter, they defined functor, borrowing the term from Carnap, and in order to define functor, they borrowed the word ‘category’ from the philosophy of Aristotle, Kant, and C. S. Peirce, but redefining it mathematically. [emphasis added]

But this seems like something of a throwaway historical grounding if one considers that Kant's (or Aristotle's) categories of ontology seem far removed from modern categories of sets, setoids, groups, groupoids, etc.

However, homomorphisms are important to modern category theory. So looking through the (Meiklejohn translation of the) first Critique:

That is to say, the three categories [of quantity], in which the unity in the production of the quantum must be homogeneous throughout, are transformed solely with a view to the connection of heterogeneous parts of cognition in one act of consciousness, by means of the quality of the cognition, which is the principle of that connection.

... But this very synthetical unity remains, even when I abstract the form of space, and has its seat in the understanding, and is in fact the category of the synthesis of the homogeneous in an intuition; that is to say, the category of quantity, to which the aforesaid synthesis of apprehension, that is, the perception, must be completely conformable.

... In all subsumptions of an object under a conception, the representation of the object must be homogeneous with the conception; in other words, the conception must contain that which is represented in the object to be subsumed under it. For this is the meaning of the expression: "An object is contained under a conception." Thus the empirical conception of a plate is homogeneous with the pure geometrical conception of a circle, inasmuch as the roundness which is cogitated in the former is intuited in the latter.

And so on and on; there are 18 results for "Find: homogeneous" in the first Critique. (Emphasis added in the above.) Now, per the Greek roots of -geneous and -morphism as "species/kind" and "form/shape," are there uses of homogeneity and homomorphism that invoke roughly equivalent concepts? Not identical, but similar enough for category-talk of homogeneity in Kant to be what Eilenberg and Mac Lane had in mind regarding the significance of homomorphisms in their category-talk, so far as these authors sought to ground their metatheory historically?

For what it's worth, the nLab discussion of this topic makes no overt mention of a possible homogeneity-homomorphism overlap as such. The source material they quote involves some fairly direct claims that modern category-theory talk inherits Aristotle and Kant's (and even Hegel's!) category-talk.

Note: my question is, to be honest, very similar (you might say homomorphic to!) this earlier PhilosophtSE question. Conifold there remarks that:

However, the idea that objects are merely relational placeholders in a structure was certainly alien to both Aristotle and Kant.

I would be cautious about using a phrase as implicitly strong as "certainly alien," here, though, seeing as Kant did say in the first Critique (the quote after the ellipsis is from here):

On the other hand, the internal determinations of a substantia phaenomenon in space are nothing but relations, and it is itself nothing more than a complex of mere relations. ... That I, when abstraction is made of these relations, have nothing more to think, does not destroy the conception of a thing as phenomenon, nor the conception of an object in abstracto, but it does away with the possibility of an object that is determinable according to mere conceptions, that is, of a noumenon. It is certainly startling to hear that a thing consists solely of relations; but this thing is simply a phenomenon, and cannot be cogitated by means of the mere categories: it does itself consist in the mere relation of something in general to the senses. In the same way, we cannot cogitate relations of things in abstracto, if we commence with conceptions alone, in any other manner than that one is the cause of determinations in the other; for that is itself the conception of the understanding or category of relation. But, as in this case we make abstraction of all intuition, we lose altogether the mode in which the manifold determines to each of its parts its place, that is, the form of sensibility (space); and yet this mode antecedes all empirical causality.

Kant's relevant phrasing is even harder for me to follow in the Transcendental Aesthetic, since he seems to have the formal intuition of space/spatial form of intuition as relational in some basic capacity, but as anteceding relationality in another (also basic) one. (Since he subtly distinguishes formal intuition from forms-of-intuition, perhaps space considered as a formal intuition is more/less relational than space as a form-of-intuition?) Regardless, I'm also having trouble reading Kant as saying that categories themselves are primarily/merely relational; he seems (to me) to say, "No," on one level, but, "Not quite," on another... So, I'm not sure any of this is especially pertinent to Conifold's observation.

EDIT:: TL;DR version: are the following (roughly) equivalent?

  1. (Kant) A is homogeneous with B.
  2. (Category theory) A is homomorphic to B.


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