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Consider the first-order language $L_{\omega\omega}$ of the signature $L:=\{\mathrm{dom}, \mathrm{cod}, \mathrm{comp}\}$, where $\mathrm{dom}$ and $\mathrm{cod}$ are unary function symbols and $\mathrm{comp}$ is a ternary relation symbol. This is intended to be thought of as the language of a single category: $\mathrm{dom}$ resp. $\mathrm{cod}$ are interpreted as functions yielding the domain resp. codomain of a given morphism; $\mathrm{comp}(h, g, f)$ is interpreted as $h=g\circ f$. One can formally write down the axioms of a category (associativity of composition, identity morphisms for composition) as first-order $L$-sentences. If we call the collection of these axioms $T_\text{Cat}$, then an $L$-structure $C$ with $C\models T_\text{Cat}$ is essentially the same as a category. (Okay, one can argue about size issues or whether specific decisions concerning the design of the formal language are natural, for example, whether it would be better to use a two-sorted language with the sorts "objects" and "morphisms" rather than a one-sorted language where everything is a morphism, but let us ignore these issues for now.)

Lawvere famously gave an axiomatization $\mathsf{ETCS}\supseteq T_{\mathrm{Cat}}$ of the category of sets in the language $L_{\omega\omega}$ and showed that a great deal of set theory can be carried out in this theory. I think it is quite remarkable that all the usual concepts of set theory (such as elements, the set of natural numbers, and the cartesian product) can be formulated categorically in the language $L_{\omega\omega}$ of morphisms. Here are some links for further reading for people not familiar with $\mathsf{ETCS}$: nLab, Lawvere's original paper, fully formal presentation of ETCS on the nLab, Tom Leinster's "Rethinking set theory". Lawvere also gave an axiomatization $\mathsf{ETCC}$ of the category of categories (nLab). (To me, this theory seems to be not as established as $\mathsf{ETCS}$ and I don't know to what extent this theory can be used to carry out doing category theory.)

Question: Is it also possible to axiomatize the category of topological spaces (and continuous maps) in the language $L_{\omega\omega}$? Is it then possible to really carry out some topology in this theory? Also, is it possible to axiomatize the category of groups resp. rings in $L_{\omega\omega}$ and then really do some group resp. ring theory? (You can really interpret my question as a question schema: for each theory, you can ask this question.) This would be interesting, because it would show that one can do topology, group theory, ring theory, ... without presupposing some form of set theory. Also, it would show that one can express all (or a great deal of) the theorems of topology, group theory, ring theory, ... just in terms of morphisms, domain, codomain, and composition.

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    $\begingroup$ The question you've asked isn't really what you meant to, I think. ETCS doesn't axiomatize the category of sets fully - there are lots of statements about the category of sets, in the language above, which are independent of ETCS. This is relevant because when you ask "is it possible to axiomatize ---?," it's not clear what you mean by "axiomatize" - if you just mean "write some true statements about," then that's trivially true ($\emptyset$), while if you mean "give a complete axiomatization of" then that's false even for the category of sets. (cont'd) $\endgroup$
    – Noah Schweber
    Commented Mar 3, 2019 at 19:09
  • $\begingroup$ Or at least, $(i)$ ETCS doesn't constitute such an axiomatization and $(ii)$ there is no computable axiomatization at all (the category of sets is complicated enough for Godel's incompleteness theorem to apply to its theory). As to "it would show that one can express all (or a great deal of) the theorems of topology, group theory, ring theory, ... just in terms of morphisms, domain, codomain, and composition," this is already well-known and one of the whole points of category theory in the first place. (cont'd) $\endgroup$
    – Noah Schweber
    Commented Mar 3, 2019 at 19:12
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    $\begingroup$ I don't mean "axiomatizable" in the rigorous mathematical sense. I just wonder whether the theory (for example, topology) is axiomatizable in such a way that a great deal of the theory can be done in the axiomatization. Note that my question is a soft question. Basically, I just wonder: if one studies ETCS, why don't consider similar theories for topology, group theory, ring theory, ... $\endgroup$
    – user103598
    Commented Mar 3, 2019 at 19:13
  • $\begingroup$ As to groups and rings they can be developed as 1 order theories, without sets (but these theories will not be equivalent to the usual group and ring theories, since everything connected to morphisms will be out of description). And your language seems to be too poor for describing what people are interested in these theories. $\endgroup$
    – Sergei Akbarov
    Commented Mar 3, 2019 at 19:13
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    $\begingroup$ @SergeiAkbarov "your language seems to be too poor for describing what people are interested in these theories" I disagree with this - again, one of the whole points of category theory is that the language is rich enough to talk about a huge amount of the stuff we care about. $\endgroup$
    – Noah Schweber
    Commented Mar 3, 2019 at 19:16

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I don't really know what you're after but here is an analogue of ETCS for topological spaces

Dana I. Schlomiuk, An elementary theory of the category of topological spaces, Trans. Amer. Math. Soc. 149 (1970), 259-278, doi:10.1090/S0002-9947-1970-0258914-7

and here's one for (five different) categories of graphs

Demitri Plessas, The Categories of Graphs, PhD thesis, University of Montana (2011) (link)

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  • $\begingroup$ Thank you very much! In these papers, they do exactly what I was looking for! I suppose one can do the same with the category of groups, rings, ... $\endgroup$
    – user103598
    Commented Mar 4, 2019 at 18:11

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