The Clowder Project

The punctual category^[1] is the category $\mathsf{pt}$ where

• Objects. We have
  \[ \text{Obj}\webleft (\mathsf{pt}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \star \webright\} . \]

• Morphisms. The unique $\textup{Hom}$-set of $\mathsf{pt}$ is defined by
  \[ \textup{Hom}_{\mathsf{pt}}\webleft (\star ,\star \webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \text{id}_{\star }\webright\} . \]

• Identities. The unit map
  \[ \mathbb {1}^{\mathsf{pt}}_{\star } \colon \text{pt}\to \textup{Hom}_{\mathsf{pt}}\webleft (\star ,\star \webright ) \]

  of $\mathsf{pt}$ at $\star $ is defined by

  \[ \text{id}^{\mathsf{pt}}_{\star } \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{\star }. \]

• Composition. The composition map
  \[ \circ ^{\mathsf{pt}}_{\star ,\star ,\star } \colon \textup{Hom}_{\mathsf{pt}}\webleft (\star ,\star \webright ) \times \textup{Hom}_{\mathsf{pt}}\webleft (\star ,\star \webright ) \to \textup{Hom}_{\mathsf{pt}}\webleft (\star ,\star \webright ) \]

  of $\mathsf{pt}$ at $\webleft (\star ,\star ,\star \webright )$ is given by the bijection $\text{pt}\times \text{pt}\cong \text{pt}$.

We have an isomorphism of categories^[2]

The empty category is the category $\emptyset _{\mathsf{cat}}$ where

• Objects. We have
  \[ \text{Obj}\webleft (\emptyset _{\mathsf{cat}}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\emptyset . \]

• Morphisms. We have
  \[ \textup{Mor}\webleft (\emptyset _{\mathsf{cat}}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\emptyset . \]

• Identities and Composition. Having no objects, $\emptyset _{\mathsf{cat}}$ has no unit nor composition maps.

The $n$th ordinal category is the category $\mathbb {n}$ where^[3]

• Objects. We have
  \[ \text{Obj}\webleft (\mathbb {n}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft [0\webright ],\ldots ,\webleft [n\webright ]\webright\} . \]

• Morphisms. For each $\webleft [i\webright ],\webleft [j\webright ]\in \text{Obj}\webleft (\mathbb {n}\webright )$, we have
  \[ \textup{Hom}_{\mathbb {n}}\webleft (\webleft [i\webright ],\webleft [j\webright ]\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \webleft\{ \text{id}_{\webleft [i\webright ]}\webright\} & \text{if $\webleft [i\webright ]=\webleft [j\webright ]$,}\\ \webleft\{ \webleft [i\webright ]\to \webleft [j\webright ]\webright\} & \text{if $\webleft [j\webright ]<\webleft [i\webright ]$,}\\ \emptyset & \text{if $\webleft [j\webright ]>\webleft [i\webright ]$.} \end{cases} \]

• Identities. For each $\webleft [i\webright ]\in \text{Obj}\webleft (\mathbb {n}\webright )$, the unit map
  \[ \mathbb {1}^{\mathbb {n}}_{\webleft [i\webright ]} \colon \text{pt}\to \textup{Hom}_{\mathbb {n}}\webleft (\webleft [i\webright ],\webleft [i\webright ]\webright ) \]

  of $\mathbb {n}$ at $\webleft [i\webright ]$ is defined by

  \[ \text{id}^{\mathbb {n}}_{\webleft [i\webright ]} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{\webleft [i\webright ]}. \]

• Composition. For each $\webleft [i\webright ],\webleft [j\webright ],\webleft [k\webright ]\in \text{Obj}\webleft (\mathbb {n}\webright )$, the composition map
  \[ \circ ^{\mathbb {n}}_{\webleft [i\webright ],\webleft [j\webright ],\webleft [k\webright ]} \colon \textup{Hom}_{\mathbb {n}}\webleft (\webleft [j\webright ],\webleft [k\webright ]\webright ) \times \textup{Hom}_{\mathbb {n}}\webleft (\webleft [i\webright ],\webleft [j\webright ]\webright ) \to \textup{Hom}_{\mathbb {n}}\webleft (\webleft [i\webright ],\webleft [k\webright ]\webright ) \]

  of $\mathbb {n}$ at $\webleft (\webleft [i\webright ],\webleft [j\webright ],\webleft [k\webright ]\webright )$ is defined by

  \[ \begin{gathered} \text{id}_{\webleft [i\webright ]}\circ \text{id}_{\webleft [i\webright ]} = \text{id}_{\webleft [i\webright ]},\\ \webleft (\webleft [j\webright ]\to \webleft [k\webright ]\webright )\circ \webleft (\webleft [i\webright ]\to \webleft [j\webright ]\webright ) = \webleft (\webleft [i\webright ]\to \webleft [k\webright ]\webright ).\end{gathered} \]

Here we list some of the other categories appearing throughout this work.

1. The category $\mathsf{Sets}_{*}$ of pointed sets of Chapter 3: Pointed Sets, Definition 3.1.3.1.1.
2. The category $\mathsf{Rel}$ of sets and relations of Chapter 5: Relations, Definition 5.2.1.1.1.
3. The category $\mathsf{Span}\webleft (A,B\webright )$ of spans from a set $A$ to a set $B$ of .
4. The category $\mathsf{ISets}\webleft (K\webright )$ of $K$-indexed sets of .
5. The category $\mathsf{ISets}$ of indexed sets of .
6. The category $\mathsf{FibSets}\webleft (K\webright )$ of $K$-fibred sets of .
7. The category $\mathsf{FibSets}$ of fibred sets of .
8. Categories of functors $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$ as in Definition 8.9.1.1.1.
9. The category of categories $\mathsf{Cats}$ of Definition 8.9.2.1.1.
10. The category of groupoids $\mathsf{Grpd}$ of Definition 8.9.4.1.1.