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](https://ncatlab.org/nlab/show/ETCS), [Lawvere's original paper](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC300477/pdf/pnas00186-0196.pdf), [fully formal presentation of ETCS on the nLab](https://ncatlab.org/nlab/show/fully+formal+ETCS), [Tom Leinster's "Rethinking set theory"](https://arxiv.org/pdf/1212.6543.pdf). Lawvere also gave an axiomatization $\mathsf{ETCC}$ of the category of categories ([nLab](https://ncatlab.org/nlab/show/ETCC)). (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*.



Which branches of mathematics can be done just in terms of morphisms and composition?
I don't really know what you're after but here is an analogue of ETCS for topological spaces



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> 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](https://doi.org/10.1090/S0002-9947-1970-0258914-7)
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and here's one for (five different) categories of graphs



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> Demitri Plessas, *The Categories of Graphs*, PhD thesis, University of Montana (2011) ([link](https://scholarworks.umt.edu/etd/967))
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