# Notion of infinity in categories

Please excuse me if the question is too vague or uninteresting.

Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$. Motivated by the equivalence of Dedekind-finiteness and finiteness for sets (assuming Choice), let us define that an object $A$ is $\mathcal{C}$-infinite if there is an object $A'$ in $\mathcal{C}$ corresponding to a proper subobject of $A$ and an isomorphism $f: A \to A'$ in $\mathcal{C}$.

In the category Set, we have the property that ($B$ is Set-infinite and $B \subseteq A) \implies A$ is Set-infinite. The analogous property does not hold (with this definition of infinity) over arbitrary categories. For instance, the additive group $\mathbb{Q}$ has $\mathbb{Z}$ as a proper subgroup, is finite in the category Grp (since any homomorphism $f: \mathbb{Q} \to \mathbb{Q}$ must actually be an isomorphism), but $\mathbb{Z}$ is Grp-infinite by $g:\mathbb{Z} \to 2\mathbb{Z}$ with $z \mapsto 2z$.

In which categories would a version of this transitivity property hold? What about other properties depending on a notion of infinity? Or determining whether a given object in a category is infinite or not?

A result somewhat related to this notion is that a Banach space isomorphic to all its infinite-dimensional closed subspaces is isomorphic to a separable Hilbert space. Any such Banach space is obviously Ban-infinite.

I don't know much Category Theory or Logic, but this has been troubling me for a while (I'm an undergraduate student). Thanks in advance for your contributions.

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Is the notion you give equivalent to the notion "There is a monomorphism $A\to A$ which is not epic", and if not why is the one you specify preferable to the one I suggest here? – Asaf Karagila Aug 18 '14 at 6:07
Even in very well-behaved categories (like toposes) there are many (inequivalent) ways to say when objects are finite. See e.g. ncatlab.org/nlab/show/finite+set and ncatlab.org/nlab/show/finite+object for definitions, interrelations, and properties of these various definitions. – Chris Heunen Aug 18 '14 at 9:42
Alternatively, in monoidal categories like that of Hilbert spaces there is a way to characterize finiteness (finite dimensionality) via reflexivity, see ncatlab.org/nlab/show/dualizable+object: the finite-dimensional Hilbert spaces are precisely the dualizable objects. (Not precisely what you're asking, but might be worth pointing out.) – Chris Heunen Aug 18 '14 at 9:46
@Chris: This reflects nicely from axiom of choice related definition of finiteness. John Truss had a nice paper about seven definitions of finiteness, and the relations between them in the absence of choice; and there are other definitions as well. – Asaf Karagila Aug 18 '14 at 10:07
Hm, yes, I was pointed out there is this notion of being a cohopfian" object that would correspond to $\mathcal{C}$-finiteness... it seems this has been investigated in categories of rings, modules, groups and topological spaces. – José Siqueira Aug 19 '14 at 10:33

This is an enlarged comment. See section $8.2$ of $[1]$.

Transitivity Property: Let $B$ be $\mathscr{C}$-infinite, and let $B\hookrightarrow A$. Then $A$ is $\mathscr{C}$-infinite. This is not satisfied for all $\mathscr{C}$; e.g., let $\mathscr{C}$ be the category of presheaves on $(\mathbb{N},\leq)$ (Counterexample $8.2.4$, $[1]$).

The following theorem gives a sufficient condition (there might be many more) for the transitivity property to hold.

Theorem 1 (Proposition $8.2.3$, $[1]$): Let $A$ be an object of a topos $\mathscr{C}$ such that there is a complemented subobject $B$ of $A$ that is $\mathscr{C}$-infinite. Then $A$ is itself $\mathscr{C}$-infinite.

Proof (taken from $[1]$): Let $C\hookrightarrow B$ be a subobject of $B$ not containing the global element $b:\mathbf{1}\to B$ and an isomorphism $C\to B$. As $\models b\in B$, it follows that $\models\neg(b\in \overline{B})$; hence $\models\neg(b\in S\cup\overline{B})$. Thus $S\coprod\overline{B}$ is a proper subobject of $A$. There is an isomorphism $A\to S\coprod \overline{B}$. Hence $A$ is $\mathscr{C}$-infinite. $\square$

### Bibliography

$[1]$ Francis Borceux, Handbook of Categorical Algebra: Volume $3$, Sheaf Theory. $522$ pp. Cambridge University Press. Published $1994$.

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I think the OP is asking about what conditions on a category we can impose to force the transitivity property to hold, not what conditions on objects we can impose. – Sam Hopkins Aug 18 '14 at 2:27
@SamHopkins, if you impose a condition on all objects of a category, then it'll become a condition imposed on the category. – Michal R. Przybylek Aug 18 '14 at 12:28