Inverse limit without category theory

Question

In Hirsch's book Differential Topology, he by and large does not use any category theory, with the exception of one passage on pg. 52 which I am trying to understand. It is as follows (paraphrased):

Let $X$ be a set and $\mathcal{A}$ a family of subsets, such that $\mathcal{A}$ contains $X$ and is closed under unions. Suppose we have a contravariant functor from the partially ordered (by inclusion) set $\mathcal{A}$ to the category of sets. That is, to every $A\in\mathcal{A}$ there is associated a set $\mathcal{F}(A)$, and to every pair of sets $A,B\in\mathcal{A}$ with $A\subset B$ there is assigned a map of sets $\mathcal{F}_{AB}:\mathcal{F}(B)\rightarrow \mathcal{F}(A)$, denoted by $x\mapsto x|A$, where $x\in\mathcal{F}(B)$ such that $\mathcal{F}_{AB}\mathcal{F}_{BC}=\mathcal{F}_{AC}$ whenever $A\subset B\subset C$, and $\mathcal{F}_{AA}=$ identity map of $\mathcal{F}(A)$.

...

A structure functor is continuous if the following holds. If $\{Y_\alpha\}$ is any simply ordered family of elements of $\mathcal{A}$, and $\cup Y_\alpha=Y$, then the inverse limit of the maps $\mathcal{F}_{Y_{\alpha}Y}:\mathcal{F}(Y)\rightarrow \mathcal{F}(Y_\alpha)$ is a map $\mathcal{F}(Y)\rightarrow \text{inv lim} \mathcal{F}(Y_\alpha)$.

(It seems kind of strange that he explains what a functor is, but assumes the reader knows what an inverse limit is, but whatever.)

So I am trying to understand the last paragraph--in particular, what an inverse limit is, and what a continuous structure functor is--without knowing anything really about category theory beyond the definition of a functor given in the first paragraph. Is there a way to understand the inverse limit in this specific instance without knowing category theory more generally?

(If the answer is simply that I need to go read up on category theory, that's fine, and I would appreciate if someone would tell me so.)

Answer 1

In topological spaces we can realise inverse limits of spaces as subspaces of products. So as a means to construct spaces we don't really need category theory. But I don't know the text you're referring to, and this might use these concepts more extensively than just as a way to construct a space.

Answer 2

Since $\mathcal{F}(A)$ is a set for every set $A$, we have a a functor $\mathcal{F}:Sets\to Sets$.

Therefore, we can use the explicit construction of the inverse limits for the category of sets as a subset of a product set. As the wikipedia secton that I have linked says, this construction works for a number of categories such as topological spaces, rings and many others.

The inverse limit satisfies some properties that characterize it up to unique isomorphism, so in particular there might be instances in which a different definition of the inverse limit set is more convinient as long as it satisfies those properties. For example, in the situation of your book, when the chain of maps is a chain of inclusions, the inverse limit is characterized by the union of the sets in the chain. Therefore $Y=\cup Y_\alpha$ is precisely the inverse limit of the $Y_\alpha$'s with inclusions.

Your book defines a continuous functor as one that satisfies $\mathcal{F}(Y)=\lim \mathcal{F}(Y_\alpha)$. More generally, in analogy with continuous functions in topology, a continuous functor is one that preserves limits. This is the case here, since $Y=\lim Y_\alpha$, we have $\mathcal{F}(\lim Y_\alpha)=\lim\mathcal{F}(Y_\alpha)$. I don't think the author really adds any mathematical meaning by saying "structure functor" other than just saying that functors preserve structure (in this case, it sends sets to sets and maps of sets to maps of sets). I might be wrong if he uses this expression before to mean something in particular.

And of course I suggest you to read some category theory, at least the basic definitions of categories and functors, but for this particular question I think you can skip it for now.