# Category of Pointed Sets as Coslice Category

## Contents

## Theorem

Let $\mathbf{Set}_*$ be the category of pointed sets.

Let $\mathbf{Set}$ be the category of sets.

Let $1 := \left\{{*}\right\}$ be any singleton.

Then:

- $\mathbf{Set}_* \cong 1 \mathbin / \mathbf{Set}$

where $1 \mathbin / \mathbf{Set}$ denotes the coslice of $\mathbf{Set}$ under $1$ and $\cong$ signifies isomorphic categories.

## Proof

Define the functor $F: \mathbf{Set}_* \to 1 \mathbin / \mathbf{Set}$ by:

- $F \left({C, c}\right) := \bar c: 1 \to C$
- $F f := f$

where $\bar c: 1 \to C$ is defined by $\bar c (*) = c$.

Further, define $G: 1 \mathbin / \mathbf{Set} \to \mathbf{Set}_*$ by:

- $G \left({x: 1 \to C}\right) := \left({C, x (*)}\right)$
- $G f := f$

### $F$ is a functor

The definition of $F$ on morphisms is admissible, since for any pointed mapping $f: \left({C, c}\right) \to \left({D, d}\right)$:

\(\displaystyle f \circ \bar c (*)\) | \(=\) | \(\displaystyle f (c)\) | Definition of $\bar c$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle d\) | $f$ is a pointed mapping | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \bar d (*)\) | Definition of $\bar d$ |

Thus by Equality of Mappings, $f \circ \bar c = \bar d$.

So indeed $F f = f$ is a morphism $\bar c \to \bar d$, as desired.

The identity morphisms of both $\mathbf{Set}_*$ and $1 \mathbin / \mathbf{Set}$ are the identity mappings, which $F$ thus preserves.

That $F$ preserves composition is also trivial, since $\mathbf{Set}_*$ and $1 \mathbin / \mathbf{Set}$ both have composition of mappings as their $\circ$.

In conclusion, $F$ is a functor.

### $G$ is a functor

Let $x: 1 \to C$ and $y: 1 \to D$ be objects of the coslice $1 \mathbin / \mathbf{Set}$.

Let $f: C \to D$ be a morphism $x \to y$.

That is, let $f \circ x = y$.

Thus, in particular:

- $f \left({x (*)}\right) = y (*)$

showing that $f$ is a pointed mapping $\left({C, x(*)}\right) \to \left({D, y(*)}\right)$.

Observe that the composition and the identity morphisms of $1 \mathbin / \mathbf{Set}$ and $\mathbf{Set}_*$ are identical.

Because $G$ is the identity on morphisms, it is thus trivially a functor.

### $F$ is an isomorphism

Because $F$ and $G$ are the identity on morphisms, it will suffice to show that:

- $F G (x) = x$ for all objects $x$ of $1 \mathbin / \mathbf{Set}$
- $G F \left({C, c}\right) = \left({C, c}\right)$ for all objects $\left({C, c}\right)$ of $\mathbf{Set}_*$

Explicitly:

\(\displaystyle F G \left({x: 1 \to C}\right)\) | \(=\) | \(\displaystyle F \left({C, x (*)}\right)\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \overline{x (*)}\) |

Now $\overline{x (*)}: 1 \to C$ is defined by $\overline{x (*)} (*) = x (*)$.

Hence $\overline{x (*)} = x$ by Equality of Mappings.

For the other equality:

\(\displaystyle G F \left({C, c}\right)\) | \(=\) | \(\displaystyle G \left({\bar c: 1 \to C}\right)\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left({C, \bar c (*)}\right)\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left({C, c}\right)\) |

where the last equality follows by definition of $\bar c$.

Thus $F$ is shown to be an isomorphism, and hence:

- $\mathbf{Set}_* \cong 1 \mathbin / \mathbf{Set}$

$\blacksquare$

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 1.6$: Example $1.8$