F-алгебра

В теории категорий $F$ -алгебра — это алгебраическая структура, связанная с функтором $F$ . $F$ -алгебры можно использовать в программировании для представления структур данных, таких как списки и деревья.

Определение

$F$ -алгеброй эндофунктора

F:{\mathcal {C}}\longrightarrow {\mathcal {C}}

называется объект $A$ из ${\mathcal {C}}$ вместе с морфизмом в ${\mathcal {C}}$

\alpha :FA\longrightarrow A

Таким образом, $F$ -алгебра — это пара $(A,\alpha )$ .

Гомоморфизмом из $F$ -алгебры $(A,\alpha )$ в $F$ -алгебру $(B,\beta )$ называется морфизм в ${\mathcal {C}}$

f:A\longrightarrow B

для которого верно

f\circ \alpha =\beta \circ Ff:

Для любого заданного эндофунктора $F$ можно рассмотреть категорию, объектами которой являются $F$ -алгебры, а морфизмами — гомоморфизмы между $F$ -алгебрами.

Примеры

Для примера, рассмотрим эндофунктор $F:Set\to Set$ , который отображает множество $X$ в $1+X$ . Здесь $Set$ - категория множеств, $1$ - любое одноэлементное множество, а $+$ — операция копроизведения (дизъюнктное объединение множеств). Тогда множество N неотрицательных целых чисел вместе с функцией $[\mathrm {zero} ,\mathrm {succ} ]:1+\mathbb {N} \to \mathbb {N}$ , которая является копроизведением функций $\mathrm {zero} :1\to \mathbb {N}$ (которая всегда возвращает 0) и $\mathrm {succ} :\mathbb {N} \to \mathbb {N}$ (которая отображает n в n+1), является $F$ -алгеброй.

Литература

Varmo Vene, Categorical programming with inductive and coinductive types
Philip Wadler, Recursive types for free! Университет Глазго, 1990 год, черновик.
Pierce, Benjamin C. «F-Algebras». Basic Category Theory for Computer Scientists. ISBN 0-262-66071-7.

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