Pro-representability

Question

In his descent II Bourbaki paper, Grothendieck lays out some criteria for a functor $F:C\rightarrow\mathrm{Set}$ to be strictly pro-representable; i.e. isomorphic to an inductive limit of representable functors $Hom(X_i,-)$ where X_i form a projective system with transition morphisms which are epimorphims. Assume $C$ has finite projective limits.

Then Grothendieck forms the comma category $CC=(1\downarrow F)$ and makes the following definitions. Object $A$ of $CC$ dominates object $B$ of $CC$ if there is an arrow $A\rightarrow B$ in $CC$, An object $A$ of $CC$ is minimal if whenever it is dominated by some $B$, such that the morphism in $CC$ is induced by a strict monomorphism $f$ in $C$, $f$ is an isomorphism. He then makes the following claims for $A,B$ in $CC$.

1) If $A$ is minimal and dominates $B$ by an arrow induced by an arrow $v$ in $C$, and $F$ is left exact (commutes with finite projective limits) then $v$ is unique. 2) if $B$ is minimal and dominated by $A$ then $v$ is epi. 3) $F$ is strictly pro-representable iff it is left exact and every object in $CC$ is dominated by a minimal object.

I don't see these things basically because I don't know how to cook up strict monomorphisms to use the minimality hypothesis. For (3) I'm aware of how the proof that any functor is a colimit of representable functors uses the same comma category. But still I need pointers…

Answer 1

If I try to prove 1:

Take objects $A,B$ given by a morphism $f:x\rightarrow y$ in $C$ and objects $u\in F(x)$, $v\in F(y)$ such that $v=F(f)(u)$. Consider another morphism $g:x\rightarrow y$ such that $v=F(g)(u)$. Then take $h:z\rightarrow x$ to be the equalizer of $f,g$. If I understand correctly $h$ is a typical example of a strict monomorphism. And since $F$ is left exact, $F(h):F(z)\rightarrow F(x)$ is the equalizer of the two maps $F(f), F(g): F(x) \rightarrow F(y)$. Of course $u\in F(z)$ since $F(f)(u) = F(g)(u)$. Then the minimality axiom implies that $h$ is an isomorphism, to that $f=g$.

I have no proof for 2 for the moment but this should not be more difficult.

To prove 3:

It's just the usual proof that $F$ is canonically a colimit of representable functors, but using only minimal objects. Then by 2 the transition morphisms will automatically be epimorphism, and you construct a strict pro-object. The usual proof goes as follows: there is a canonical map $$\operatorname{colim}_{(x,u)\in 1\downarrow F}\quad \hom(x,-) \longrightarrow F $$ basically because by Yoneda lemma a morphism $\hom(x,-)\rightarrow F$ is defined uniquely by an object of $F(x)$, corresponding to the image of $1_x \in \hom(x,x)$. This canonical map is always a injective (since you take the colimit over all these couples). In the classical case where you consider all of the comma category it is also surjective for more or less tautological reasons. Here the hypothesis tells you can restrict to minimal couples in the comma category and this will again be surjective.