nLab Adámek's fixed point theorem

Idea

Adámek’s fixed point theorem gives a method to construct an initial algebra of an endofunctor by “iterating” the functor. The fact that this gives a “fixed point” of the endofunctor is given by Lambek's theorem.

Construction

This theorem is attributed to Jiří Adámek but note that the same construction for set-functors already appeared in (Pohlova,1973)).

Theorem (Adámek)

Let $C$ be a category with an initial object $0$ and transfinite composition of length $\omega$, hence colimits of sequences $\omega \to C$ (where $\omega$ is the first infinite ordinal), and suppose $F: C \to C$ preserves colimits of C$\omega$-chains. Then the colimit $\gamma$ of the chain

$$0 \overset{i}{\to} F(0) \overset{F(i)}{\to} \ldots \to F^{(n)}(0) \overset{F^{(n)}(i)}{\to} F^{(n+1)}(0) \to \ldots$$

carries a structure of initial $F$-algebra.

Proof

The $F$-algebra structure $F\gamma \to \gamma$ is inverse to the canonical map $\gamma \to F\gamma$ out of the colimit (which is invertible by the hypothesis on $F$). The proof of initiality may be extracted by dualizing the corresponding proof given at terminal coalgebra.

This approach can be generalized to the transfinite construction of free algebras.

Related entries

• fixed point

• Kleene's fixed point theorem is precisely the de-categorification of this theorem to posets/preorders.

• Knaster-Tarski's fixed point theorem

• Lawvere's fixed point theorem

• Lefschetz fixed point theorem

• Brouwer's fixed point theorem

• Atiyah-Bott fixed point formula

References

• Věra Pohlová. “On sums in generalized algebraic categories.” Czechoslovak Mathematical Journal 23.2 (1973) 235-251 [eudml:12718]

• Jiří Adámek, Free algebras and automata realizations in the language of categories, Commentationes Mathematicae Universitatis Carolinae 15.4 (1974) 589-602 [eudml:16649]

• Jiří Adámek, Věra Trnková, Automata and algebras in categories 37 Springer (1990) [ISBN:9780792300106]