Categories and Functors

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

$\varnothing \in \mathcal U,$
$\forall x\ (x\in \mathcal U \rightarrow x \subseteq \mathcal U),$
$\forall x\ (x\in \mathcal U \rightarrow \lbrace x\rbrace \in \mathcal U),$
$\forall x\ (x\in \mathcal U \rightarrow \mathcal P(x)\in \mathcal U),$
$\forall I\in \mathcal U\ \forall i\in I\ \left( x_i\in \mathcal U \rightarrow \bigcup_{i\in I}x_i\in \mathcal U\right),$
$\mathbb N\in \mathcal U.$

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.

A set $$x$$ is a $\mathcal U$ -set if $x\in \mathcal U.$
A set $$x$$ is a $\mathcal U$ -small if it is isomorphic to a set $y\in \mathcal U.$

Definition 4.1.4. A category $\mathscr C$ consists of:

a set $\operatorname {Ob}(\mathscr C),$
for any $X,Y\in \operatorname {Ob}(\mathscr C),$ a set $\operatorname {Hom}_{\mathscr C}(X,Y).$
for any $X,Y,Z\in \operatorname {Ob}(\mathscr C),$ a map
\[\operatorname {Hom}_{\mathscr C}(X,Y) \times \operatorname {Hom}_{\mathscr C}(Y,Z) \to \operatorname {Hom}_{\mathscr C}(X,Z)\]
called the composition and denoted by $(f,g)\mapsto g\circ f,$ satisfying the following properties:
- (associativity) $\forall f\in \operatorname {Hom}_{\mathscr C}(X,Y)\ g\in \operatorname {Hom}_{\mathscr C}(Y,Z)\ h\in \operatorname {Hom}_{\mathscr C}(Z,W)\ \left((h\circ g)\circ f)=h\circ (g\circ f)\right),$
- (identity) $\forall X\in \operatorname {Ob}(\mathscr C)\ \exists \text{id}_X\in \operatorname {Hom}_{\mathscr C}(X,X)\ \forall f\in \operatorname {Hom}_{\mathscr C}(X,Y)\ \forall g\in \operatorname {Hom}_{\mathscr C}(Y,X)\ \left(\left(f\circ \text{id}_X = f\right) \land \left( \text{id}_X \circ g = g\right)\right).$

Definition 4.1.5. Let $\mathscr C$ be a category. An element of $\operatorname {Ob}(\mathscr C)$ is an object of $\mathscr C.$ For $X,Y\in \operatorname {Ob}(\mathscr C),$ an element of $\operatorname {Hom}_{\mathscr C}(X,Y)$ is a morphism in $\mathscr C.$ The morphism $\text{id}_X$ is the identity of $$X.$$
$\mathscr C$ is a $\mathcal U$ -category if $\operatorname {Hom}_{\mathscr C}(X,Y)$ is $\mathcal U$ -small for any $X,Y\in \operatorname {Ob}(\mathscr C).$
$\mathscr C$ is a $\mathcal U$ -small category if $\mathscr C$ is a $\mathcal U$ -category and $\operatorname {Ob}(\mathscr C)$ is $\mathcal U$ -small.

References

Kashiwara M. & Schapira P. (2006). Categories and sheaves. Springer Berlin Heidelberg. https://doi.org/10.1007/3-540-27950-4

Semantics

4.1. Categories and Functors

References