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.



Is an "infinite compositions of arrows" meaningful?
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).


