8.1.1 Foundations

A category $\smash {\webleft (\mathcal{C},\circ ^{\mathcal{C}},\mathbb {1}^{\mathcal{C}}\webright )}$ consists of:

  • Objects. A class $\text{Obj}\webleft (\mathcal{C}\webright )$ of objects.
  • Morphisms. For each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, a class $\textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )$, called the class of morphisms of $\mathcal{C}$ from $A$ to $B$.
  • Identities. For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, a map of sets

    \[ \mathbb {1}^{\mathcal{C}}_{A}\colon \text{pt}\to \textup{Hom}_{\mathcal{C}}\webleft (A,A\webright ), \]

    called the unit map of $\mathcal{C}$ at $A$, determining a morphism

    \[ \text{id}_{A} \colon A \to A \]

    of $\mathcal{C}$, called the identity morphism of $A$.

  • Composition. For each $A,B,C\in \text{Obj}\webleft (\mathcal{C}\webright )$, a map of sets

    \[ \circ ^{\mathcal{C}}_{A,B,C} \colon \textup{Hom}_{\mathcal{C}}\webleft (B,C\webright )\times \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright ) \to \textup{Hom}_{\mathcal{C}}\webleft (A,C\webright ), \]

    called the composition map of $\mathcal{C}$ at $\webleft (A,B,C\webright )$.

such that the following conditions are satisfied:

  1. Associativity. The diagram
    commutes, i.e. for each composable triple $\webleft (f,g,h\webright )$ of morphisms of $\mathcal{C}$, we have
    \[ \webleft (f\circ g\webright )\circ h = f\circ \webleft (g\circ h\webright ). \]
  2. Left Unitality. The diagram

    commutes, i.e. for each morphism $f\colon A\to B$ of $\mathcal{C}$, we have

    \[ \text{id}_{B}\circ f=f. \]
  3. Right Unitality. The diagram

    commutes, i.e. for each morphism $f\colon A\to B$ of $\mathcal{C}$, we have

    \[ f\circ \text{id}_{A}=f. \]

Let $\mathcal{C}$ be a category.

  1. We also write $\mathcal{C}\webleft (A,B\webright )$ for $\textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )$.
  2. We write $\textup{Mor}\webleft (\mathcal{C}\webright )$ for the class of all morphisms of $\mathcal{C}$.

Let $\kappa $ be a regular cardinal. A category $\mathcal{C}$ is

  1. Locally small if, for each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, the class $\textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )$ is a set.
  2. Locally essentially small if, for each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, the class
    \[ \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )/\webleft\{ \text{isomorphisms}\webright\} \]

    is a set.

  3. Small if $\mathcal{C}$ is locally small and $\text{Obj}\webleft (\mathcal{C}\webright )$ is a set.
  4. $\kappa $-Small if $\mathcal{C}$ is locally small, $\text{Obj}\webleft (\mathcal{C}\webright )$ is a set, and we have $\# {\text{Obj}\webleft (\mathcal{C}\webright )}<\kappa $.


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