Is the inverse limit always an equalizer?

Question

In EGA, a presheaf $\mathcal{F}$ on a space $X$ with values in a category $\mathcal{C}$ is called a sheaf if it futher satisfies the axiom:

For all open coverings $(U_\alpha)_{\alpha \in I}$ of an open set $U$, with $U_{\alpha} \subset U$, $\mathcal{F}(U)$ is the inverse limit of the system $(\mathcal{F}(U_\alpha))$, $(\mathcal{F}(U_\alpha \cap U_\beta))$ with the obvious restriction maps $(\rho_\alpha)$ and $(\rho_{\alpha \beta})$

In other words, for any object $B$ in $\mathcal{C}$ and $f \in Hom(B,\mathcal{F}(U))$ the map $$f \mapsto (\rho_\alpha \circ f) \in \prod Hom(B,\mathcal{F}(U_\alpha))$$ is a bijection from Hom$(B,\mathcal{F}(U))$ to the collection of $(f_\alpha)$ such that, for any pair $(\alpha, \beta)$ agree upon further restriction: $\rho_{\alpha \beta} \circ f_\alpha = \rho_{\beta \alpha} \circ f_\beta$. (Or in the more usual language given a $B$ with such maps $(f_\alpha)$ there is a unique $f:B \to \mathcal{F}(U)$ such that $f_\alpha=\rho_\alpha\circ f$).

It turns out if $\mathcal{C}$ is one of the usual categories like the category of rings, this definition gives us the usual definition of a sheaf, which we can also show is equivalent to being the equalizer of the sequence $\mathcal{F}(U) \longrightarrow \prod_{\alpha\in I} \mathcal{F}(U_\alpha) {{{}\atop {\Large\longrightarrow}}\atop{{\Large\longrightarrow}\atop {}}}\prod_{\alpha, \beta\in I} \mathcal{F}(U_\alpha \cap U_\beta)$. (Or we can just see that both the definition as a limit and an equalizer satisfy the same universal property).

So say $(A_\alpha), (A_{\alpha \beta}), (\rho_{\alpha \beta})$ is an inverse system. Is it true that by the same reasoning

$$\lim_{\gets} A_\alpha = A\longrightarrow \prod_{\alpha\in I} A_\alpha {{{}\atop {\Large\longrightarrow}}\atop{{\Large\longrightarrow}\atop {}}}\prod_{\alpha, \beta\in I} A_{\alpha \beta}$$

Seems bit strange to me? If I am not making a mistake, doesn't this make one of the constructions superfluous?

Answer 1

This is correct: in any category, given objects $A_\alpha$ and $A_{\alpha\beta}$ with maps $A_\alpha\to A_{\alpha\beta}\leftarrow A_\beta$, assuming the products $\prod A_\alpha$ and $\prod A_{\alpha\beta}$ both exist, then a limit of the diagram formed by all the maps $A_\alpha\to A_{\alpha\beta}\leftarrow A_\beta$ is the same thing as an equalizer of the two canonical maps between the products.

Much more generally, in fact, a limit of any diagram can be constructed as an equalizer of a pair of maps between between products (assuming the products in question exist). This is a basic theorem in category theory; see for instance nLab or Theorem V.2.2 of Mac Lane's Categories for the Working Mathematician. Note that your construction in this case is actually slightly different from the general construction, since your diagram has a particularly simple shape that lets you get away with using some smaller products.