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My question is a very simple one.

What ways are there to generalize terms such as cardinality (or, more generally, the concept of finiteness) to abstract (and not concretizable) categories?

I have seen two ways to do so: One is to take a terminal object $\mathbf{T}$ and to count the numer of morphisms from $\mathbf{T}$ to an object $\mathbf{A}$ and define this number as the cardinality of $\mathbf{A}$. Clearly, this will coincide with the cardinality as defined in the category of sets as the number of morphisms from $\{1\}$ to another set is the cardinality of the letter. The downside of this approach is that it does not always work well: Even if a category has a terminal object, there is no garuantee that there are morphisms from this object to any other object and, thus, if we define cardinality in this way, it might give very misleading results: For instance, we can have a concrete category in which the number of morphisms from the terminal object does not reflect the finitness of an object that might be obtained by applying the forgetful functor to it.

Another way is to define an object as finite if it has a finite number of endomorphisms. Again, this coincides with the notional of finiteness in the category of sets. However, one can think of a category in which an object $\mathbf{A}$ only has the identity morphism as an endomorphism but an infinite number of morphisms exist between $\mathbf{A}$ and another object $\mathbf{B}$ (for which, again, only the identity morphism is an endomorphism). This creates a scenario in which it is hardly justified to think $\mathbf{A}$ and $\mathbf{B}$ as finite.

Are there other ways to define finitenes? In particular, is there a way to define finitness for a category such that, if this category is concretizable, it can be turned into a concrete category in which only the finite objects (abstractly defined) can have a finite image under the forgetful functor and this can be achieved for any abstractly defined finite object?

EDIT: In the comments, it was asked what categories I have in mind. Here is an answer to this:

The categories I have in mind are only required to have the following: There must exist an object $\mathbf{A}$ such that all finite, non-empty powers of $\mathbf{A}$ are also in the category (the product is of course defined in the usual category-theoretic way and the point of all this is to work with a generalized definition of clones in these categories). Apart from this condition, however, the category can have any possible structure. Although it would be nice to have a general definition of cardinality, all I am really interested in is to distinguish between finite and non-finite objects. The definition should satisfy the three conditions below.

  1. Of course, it should coincide with the usual definition of finiteness if my category is Set.

  2. If $\mathbf{A}$ and another object $\mathbf{B}$ is finite, then the number of morphisms from $\mathbf{B}$ to $\mathbf{A}$ should be finite (it would therefore be enough to have a notion of finiteness "with respect to $\mathbf{A}$", if such a thing makes sense to have).

  3. The definition should give me as many finite objects as the "best" concretezation. By this, I mean the following: If my category is concretizable, then you should not be able to get more finite objects by defining finiteness over the image under the forgetful functor.

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I don't think there is a universal definition of finiteness for arbitrary categories. It might help to know what kind of categories you are interested in. For example, notions of finiteness have been well studied over topoi. Would you be interested in that, or do you have other categories in mind? –  François G. Dorais Sep 20 '10 at 12:00
François is correct that for toposes notions of finiteness have been considered. In particular, you can define finiteness in any of the usual ways you do in set theory; but because the internal language of a topos is intuitionistic these notions need not coincide. This is discussed in detail in Johnstone's "Elephant" (volume 2, D.5). –  Michael A Warren Sep 20 '10 at 14:24
Thank you for your comments. I have edited my question with respect to the question that François asked me. Now, it should be clearer what categories I have in mind. –  Niemi Sep 20 '10 at 16:03
I think that this nLab page partly answers your question: ncatlab.org/nlab/show/compact+object –  François G. Dorais Sep 20 '10 at 16:45
Peter, I think this would not coincide with the usual definition of finiteness in the category of sets. For instance, take $\{1,2\}$. It is obviously finite. However, for any infinite set $B$, there will exist infinitely many mappings from $B$ to $\{1,2\}$. –  Niemi Sep 21 '10 at 14:35
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2 Answers

up vote 5 down vote accepted

If your category has a monoidal structure, you can ask for an object to be dualizable. This is a generalization of finite-dimensionality for vector spaces.

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As François Dorais mentioned above, the notion of "compact object" is quite useful in more general contexts. –  S. Carnahan Sep 29 '10 at 5:57
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In Pareigis' Categories and functors an object $x$ of an abelian category $A$ is called finite if $Hom(x,-) : A \to Ab$ commutes with coproducts. For example, in $R$-Mod, every finitely generated free module is finite. There it is also proved that every cocomplete abelian category, which has a finite projective generator, is equivalent to some $R$-Mod.

There are various notions of finitely generated objects. See for example also Pareigis' book, this question or nlab.

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