Is an "infinite compositions of arrows" meaningful?

Question

For example, deciding whether or not the following is a category seems to depend on the above question (from Awodey's Category Theory, pg. 6):

"What if we take sets as objects and as arrows, those $f : A \rightarrow B$ such that for all $b \in B$, the subset $f$^-1$(b) \subseteq A$ is finite?"

Define for each $n \in N$ the function $f$_n: $N \rightarrow N$ where $f$_n$(x) = max(0, x - n)$.

Then any $f$_n or any finite composition thereof has finite inverse images. Yet the "infinite composition" $... f$₁$f$₂$f$₀ has an infinite inverse image for 0, and so the above does not meet the definition of a category.

If this "infinite composition" is legit, does it follow from the basic definition of a category, or must the definition be made more flexible or precise to accommodate it?

For reference, this is Awodey's definition concerning composition:

"Given arrows $f : A \rightarrow B$ and $g : B \rightarrow C$, [...] there is given an arrow: $g$ o $f : A \rightarrow C$ called the composite of $f$ and $g$."

Thank you for your insight.

Answer 1

This is probably not what the OP is looking for, but there is a notion of "infinite composition of arrows" which often appears for example in categorical homotopy theory:

If $f_0 : X_0 \to X_1$, $f_1 : X_1 \to X_2$, $\ldots$ are morphisms in a category $C$ and the colimit of the diagram $X_0 \to X_1 \to X_2 \to \cdots$ exists (call it $X$) then $X$ is equipped in particular with a canonical map $X_0 \to X$ which is called the transfinite composition of the maps $f_i$. Of course, technically this morphism of $C$ is specified only up to canonical isomorphism because $X$ may be replaced by a (uniquely isomorphic) different colimit of the $X_i$. More generally given an ordinal $\alpha$ which we may view as a category (poset) and a colimit-preserving functor $X_\cdot : \alpha \to C$ (so that for each limit ordinal $\beta \in \alpha$, we have $X_\beta = \operatorname{colim}_{\gamma < \beta} X_\gamma$), we may form the transfinite composition $X_0 \to \operatorname{colim}_\alpha X_\alpha$. One is often concerned with questions such as whether a given class of maps is closed under transfinite compositions (for example, the class of cofibrations in a model category has this property).

Answer 2

(This was slightly too long to fit as a comment.)

As with Qiaochu's and Mariano's comments above, the answer as to whether infinite compositions are a priori meaningful in a category is simply no.

I don't know what it would mean to make this definition more precise -- it seems already, like any acceptable definition, to be completely precise. Moreover, in order to entertain a notion of infinite composition, it seems that one rather needs a less flexible -- i.e., more restrictive -- definition of composition of morphisms in a category. Since as Qiaochu points out, already infinite composition is not always meaningful in the category of sets and functions, such a definition would have to be very restrictive indeed.

It does not seem completely unreasonable to define some sort of category-like structure in which infinite composition is meaningful. Off the top of my head, it seems that some kind of "topology" on the class of objects of the category would be helpful, so that we could speak of a convergent sequence of objects. But it would be much more interesting and fruitful to do this in the context of some particular instance in which one would like to formalize infinite composition, e.g. for certain elements of the symmetric group on an infinite set.