3 Schlessinger’s theorem

Let \(F\) be a functor from \(\mathbf{C}_\Lambda \) to \(Sets\) such that \(F(k)\) is a singleton.

Theorem 43

✓

If \(F\) satisfies \((H_1)\), \((H_2)\) and \((H_3)\), then \(F\) admits a hull.

Theorem 44

If \(F\) admits a hull, then \(F\) satisfies \((H_1)\), \((H_2)\) and \((H_3)\).

Theorem 45

✓

If \(F\) satisfies \((H_1)\), \((H_2)\), \((H_3)\) and \((H_4)\), then \(F\) is pro-representable.

Theorem 46

✓

If \(F\) is pro-representable, then \(F\) satisfies \((H_1)\), \((H_2)\), \((H_3)\) and \((H_4)\).

Proof ▶

\(h_R\) satisfies all four conditions as noted in lemma 42, so this is trivial (using lemma 20 for \((H_3)\)).

The main difficulty is the backward direction of \((1)\).