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(Sub)categories of many well-studied mathematical objects have been characterized purely in terms of their morphisms. Some (famous) examples:

  • Sets and functions, due to Lawvere.
  • Modules over some ring and module homomorphisms, due to Mitchell and Freyd.
  • Sheaves over some site and natural transformations, due to Giraud.
  • Topological spaces and continuous functions, due to Schlomiuk.
  • Boolean algebras and homomorphisms, by combining the above.

In a similar vein:

Is there a characterization of the category of rings and homomorphisms?

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    $\begingroup$ Though I'm primarily interested in rings, this could be made into a big list -- please feel free to add characterizations of other categories of well-known objects! $\endgroup$ – Chris Heunen Oct 23 '14 at 10:22
  • $\begingroup$ For example: the category of complex Hilbert spaces and bounded linear maps, see tac.mta.ca/tac/volumes/22/13/22-13abs.html. $\endgroup$ – Chris Heunen Oct 23 '14 at 10:23
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    $\begingroup$ OK: the category of rings together with its forgetful functor to $\mathbf{Set}$ is terminal among all categories $C$ equipped with a functor $U: C \to \mathbf{Set}$ and a ring structure on the object $U \in [C, \mathbf{Set}]$! That's probably not the kind of thing you want, though... $\endgroup$ – Tom Leinster Oct 23 '14 at 11:33
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    $\begingroup$ @Tom, doesn't your observation work equally well with any algebraic theory in place of rings? It seems to be defining the thing essentially in terms of itself. $\endgroup$ – Paul Taylor Oct 23 '14 at 15:02
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    $\begingroup$ Note that when the "characterization" is in the form of an embedding (Like Freid-Mitchel, or Barr's embedding of a regular category), essentially this embedding should be structure preserving. Such embeddings can indeed be viewed as characterizations of some sort since they allow working with an abstract category as if it were a specific one. $\endgroup$ – Dimitri Chikhladze Oct 23 '14 at 16:23
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Yves Diers has the notion of a Zariski category in [Categories of commutative algebras], which apparently suffices to carry out a lot of commutative algebra in an axiomatic fashion. I reproduce the definition:

A Zariski category is a category $\mathcal{A}$ satisfying the following conditions:

  • $\mathcal{A}$ is cocomplete.
  • $\mathcal{A}$ has a strong generating set whose objects are finitely presentable and flatly codisjunctable.
  • Regular epimorphisms are universal i.e. stable under pullbacks.
  • The terminal object of $\mathcal{A}$ is finitely presentable and has no proper subobject.
  • Binary products of objects are co-universal i.e. stable under pushouts.
  • For any finite sequence of codisjunctable congruences $r_1, \ldots, r_n$ on any object with respect codisjunctors $d_1, \ldots, d_n$, we have $$r_1 \vee^c \cdots \vee^c r_n = \mathrm{id}_{A \times A} \implies d_1 \vee \cdots \vee d_n = \mathrm{id}_A$$ where $\vee^c$ denotes the join in the lattice of congruences on $A$, while $\vee$ denotes the co-union of quotient objects of $A$.

For more details, see the cited book, or the introduction of this article.

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The programme that I propose for answering questions of this kind is described in my paper Foundations for Computable Topology. I had in mind that this might be done for (the opposite of) the category of (commutative) rings, though I confess that my efforts to do so drew a blank.

Thank you for telling me about Schlomiuk's paper. To me now it seems pedestrian and too closely wedded to sets of points, but that is completely unfair because it was written a long time ago. (I would say that it resembles Lawvere's work on the category of sets long before the invention of elementary toposes.)

I notice that Schlomiuk includes an axiom with some similarity to what I call the Phoa Principle, in the form that the Sierpinski space (which he calls $E$ and I call $\Sigma$) has just three endofunctions. It also uses extremal monos, as I have done, coincidentally with the same name but in ignorance of previous work.

The analogue of the Phoa Principle for affine varieties would be that any endofunction of the base ring qua space is a polynomial.

The analogue of my exponential $\Sigma^X$ for rings should be polynomial ring or free symmetric algebra on the underlying abelian group (ie forget the existing multiplication and freely adjoin a new one). However, whilst this works fine for locally compact frames, for rings it goes badly wrong for cardinality reasons.

There is some work by Mike Barr on Banach algebras that might help here, though I couldn't get my ideas to work with it; I will look out the references if you are interested.

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This is a comment rather than an answer.

There is a notion of semi-abelian category which generalizes an abelian category introduced in

Semi-abelian categories, G Janelidze, L Márki, W Tholen - Journal of Pure and Applied Algebra.

The definition can be found on nlab page http://ncatlab.org/nlab/show/semi-abelian+category. It captures properties of categories such as groups, rings without unit etc.

There is a result which characterizes categories of varieties which are semi-abelian. However, as far as I know, there is no embedding theorem proved.

Semi-abelian categories do not include category of rings, which is of course quite different from the category of rings without unit. But, maybe there are some results about this in the semi-abelian category literature, which I am not very familiar with.

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