4.1  Categories and Functors

Definition 4.1.1. A universe \(\mathcal U\) is a set satisfying the following properties:

  1. \(\varnothing \in \mathcal U\);
  2. if \(x\in \mathcal U\) then \(x\subseteq \mathcal U\);
  3. if \(x\in \mathcal U\) then \(\{x\}\in \mathcal U\);
  4. if \(x\in \mathcal U\) then \(\mathcal P(x)\in \mathcal U\);
  5. if \(I\in \mathcal U\) and \(x_i\in \mathcal U\) for all \(i\in I\), then \(\bigcup_{i\in I}x_i \in \mathcal U\);
  6. \(\mathbb N \in \mathcal U\).

Axiom 4.1.2 (Axiom of universes). We add an axiom to the \(\text{ZF}\) theory: For all \(x,\) there exists a universe \(\mathcal U\) such that \(x\in \mathcal U.\)

Definition 4.1.3. Let \(\mathcal U\) be a universe.

  1. A set is called a \(\mathcal U\)-set if it belongs to \(\mathcal U\).
  2. A set is called \(\mathcal U\)-small if it is isomorphic to a set belonging to \(\mathcal U\).

Definition 4.1.4. A category \(\mathcal C\) consists of the following data:

  1. a set \(\operatorname {Ob}(\mathcal C)\) of objects;
  2. for any \(X,Y\in \operatorname {Ob}(\mathcal C),\) a set \(\operatorname {Hom}_{\mathcal C}(X,Y)\) of morphisms from \(X\) to \(Y\);

  3. for any \(X,Y,Z\in \operatorname {Ob}(\mathcal C),\) a map

    \[\operatorname {Hom}_{\mathcal C}(X,Y) \times \operatorname {Hom}_{\mathcal C}(Y,Z) \to \operatorname {Hom}_{\mathcal C}(X,Z)\]

    denoted by \((f,g)\mapsto g\circ f\).

These data satisfy the following properties:

  • The composition operation \(\circ\) is associative.
  • For each \(X\in \operatorname {Ob}(\mathcal C)\), there exists \(\text{id}_X\in \operatorname {Hom}_{\mathcal C}(X,X)\), called the identity, such that \(f\circ \text{id}_X = f\) for all \(f\in \operatorname {Hom}_{\mathcal C}(X,Y)\) and \(\text{id}_X \circ g= g\) for all \(g\in \operatorname {Hom}_{\mathcal C}(Y,X)\).

Definition 4.1.5. A category \(\mathcal C\) is a \(\mathcal U\)-category if \(\operatorname {Hom}_{\mathcal C}(X,Y)\) is \(\mathcal U\)-small for any \(X,Y\in \operatorname {Ob}(\mathcal C).\) The category \(\mathcal C\) is a \(\mathcal U\)-small category if \(\mathcal C\) is a \(\mathcal U\)-category and \(\operatorname {Ob}(\mathcal C)\) is \(\mathcal U\)-small.