Унивалентный функтор

Унивалентный функтор (строгий функтор) — функтор, который инъективен на каждом множестве морфизмов с фиксированными образом и прообразом. Полный функтор — двойственное понятие — функтор, который сюръективен на каждом множестве морфизмов с фиксированным образом и прообразом.

Соответственно, функтор $F\colon {\mathcal {C}}\longrightarrow {\mathcal {D}}$ между локально малыми категориями ${\mathcal {C}}$ и ${\mathcal {D}}$ :

унивалентнен, если функция $F_{X,Y}$ инъективна,
полон, если $F_{X,Y}$ сюръективна,
вполне унивалентен, если $F_{X,Y}$ биективна

для каждой пары $X,Y$ из ${\mathcal {C}}$ ( $F_{X,Y}\colon \mathrm {Hom} _{\mathcal {C}}(X,Y)\rightarrow \mathrm {Hom} _{\mathcal {D}}(F(X),F(Y))$ — срез функтора на морфизмы $X\to Y$ ).

Унивалентный функтор не обязательно инъективен на объектах категории ${\mathcal {C}}$ , поэтому образ вполне унивалентного функтора не обязан быть категорией, изоморфной ${\mathcal {C}}$ . Аналогично, полный функтор не обязательно сюръективен на объектах. Однако вполне унивалентный функтор инъективен на объектах с точностью до изоморфизма, то есть если $F\colon {\mathcal {C}}\longrightarrow {\mathcal {D}}$ является вполне унивалентным и $F(x)\cong F(y)$ , то $x\cong y$ (в этом случае говорят, что функтор $F$ отражает изоморфизмы).

Любой унивалентный функтор отражает мономорфизмы и эпиморфизмы. Из этого следует, что любой унивалентный функтор из сбалансированной категории отражает изоморфизмы.

Пример унивалентного функтора — забывающий функтор для категории групп $U\colon \mathbf {Grp} \longrightarrow \mathbf {Set}$ : гомоморфизм групп однозначно определяется функцией на множествах-носителях. (Категория с унивалентным функтором в $\mathbf {Set}$ называется конкретной категорией.) Функтор, вкладывающий категорию абелевых групп $\mathbf {Ab}$ в категорию групп $\mathbf {Grp}$ , вполне унивалентный.

Литература

Маклейн С. Глава 1. Категории, функторы и естественные преобразования // Категории для работающего математика = Categories for the working mathematician / Пер. с англ. под ред. В. А. Артамонова. — М.: Физматлит, 2004. — С. 17—42. — 352 с. — ISBN 5-9221-0400-4.

Букур И., Деляну А. Введение в теорию категорий и функторов. — М.: Мир, 1972.

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