symmetric monoidal (∞,1)-category of spectra
Adámek’s fixed point theorem gives a method to construct an initial algebra of an endofunctor by “iterating” the functor.
This theorem is attributed to Jiří Adámek but note that the same construction for set-functors already appeared in (Pohlova,1973)).
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
carries a structure of initial $F$-algebra.
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.
Kleene's fixed point theorem is precisely the de-categorification of this theorem to posets/preorders.
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]
Last revised on November 30, 2023 at 15:48:18. See the history of this page for a list of all contributions to it.