category theory

# Contents

## Definition

A pointed set is a set $S$ equipped with a chosen element $s$ of $S$. (Compare inhabited set, where the element is not specified.)

Since we can identify a (set-theoretic) element of $S$ with a (category-theoretic) global element (a morphism $s:1\to S$), we see that a pointed set is an object of the under category $pt↓Set$, or coslice category $1/Set$, of objects under the singleton $\left\{•\right\}$.

## The category of pointed sets

### Definition

###### Definition

The category ${\mathrm{Set}}_{*}$ of pointed sets is the under category $*/\mathrm{Set}$ of Set under the singleton set.

So a morphism $\left({S}_{1},{s}_{1}\right)\to \left({S}_{2},{s}_{2}\right)$ is a map between sets which maps these chosen elements to each other, i.e., commuting triangles

$\begin{array}{ccc}& & \mathrm{pt}\\ & ↙& & ↘\\ {S}_{1}& & \to & & {S}_{2}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && pt \\ & \swarrow && \searrow \\ S_1 &&\to&& S_2 } \,.

The category ${\mathrm{Set}}_{*}$ naturally comes with forgetful functor $p:{\mathrm{Set}}_{*}\to \mathrm{Set}$ which forgets the tip of these triangles.

### Properties

###### Proposition

Equipped with the smash product $\otimes :=\wedge$ of pointed set, $\left({\mathrm{Set}}_{*},weedge\right)$ is a closed symmetric monoidal category.

The internal hom ${\mathrm{Set}}_{*}\left(X,Y\right)$ is the hom-set in $*/\mathrm{Set}$ pointed by the morphism $X\to Y$ that sends everything to the basepoint in $Y$.

#### Interpretation as universal Set-bundle

The morphism ${\mathrm{Set}}_{*}\to \mathrm{Set}$ is an example of a generalized universal bundle: the universal Set-bundle. The entire structure here can be understood as arising from the (strict) pullback diagram

$\begin{array}{ccc}{\mathrm{Set}}_{*}& \to & \mathrm{pt}\\ ↓& & {↓}^{\mathrm{pt}↦\mathrm{pt}}\\ \left[I,\mathrm{Set}\right]& \stackrel{{d}_{0}}{\to }& \mathrm{Set}\\ {↓}^{{d}_{1}}\\ \mathrm{Set}\end{array}$\array{ Set_* &\to& pt \\ \downarrow && \downarrow^{pt \mapsto pt} \\ [I,Set] &\stackrel{d_0}{\to}& Set \\ \downarrow^{d_1} \\ Set }

in the 1-category Cat, where

• $I=\left\{a\to b\right\}$ is the interval category;

• $\left[I,\mathrm{Set}\right]=\mathrm{Arr}\left(\mathrm{Set}\right)$ is the internal hom category which here is the arrow category of $\mathrm{Set}$;

• ${d}_{i}:=\left[{j}_{i},\mathrm{Set}\right]$ are the images of the two injections ${j}_{i}:\mathrm{pt}\to I$ of the point to the left and the right end of the interval, respectively – so these functors evaluate on the left and right end of the interval, respectively;

• the square is a pullback;

• the total vertical functor is the forgetful functor $p:{\mathrm{Set}}_{*}\to \mathrm{Set}$.

The way in which ${\mathrm{Set}}_{*}\to \mathrm{Set}$ is the “universal Set-bundle” is discussed pretty explicitly in

(The discussion there becomes more manifestly one of bundles if one regards all morphisms $C\to \mathrm{Set}$ appearing there as being the right legs of anafunctors. )

#### Interpretation as 2-subobject-classfier

Observing that usual morphism into the subobject classifier $\Omega$ of the topos Set is the universal truth-value bundle? $\left\{\top \right\}\to TV$, and noticing that $\mathrm{TV}=\left(-1\right)\mathrm{Cat}$ and $\mathrm{Set}=0\mathrm{Cat}$ suggests that ${\mathrm{Set}}_{*}\to \mathrm{Set}$ is a categorified subobject classifier: indeed, it is the subobject classifier in the 2-topos Cat.

For discussion of this point see

• David Corfield: 101 things to do with a 2-classifier (blog)

It was David Roberts who pointed out in

the relation between these higher classifiers and higher generalized universal bundles, motivated by the observations on principal universal 1- and 2-bundles in

• David Roberts, Urs Schreiber, The inner automorphism 3-group of a strict 2-group, Journal of Homotopy and Related Structures, Vol. 3(2008), No. 1, pp. 193-244, (arXiv).

Revised on December 2, 2010 09:49:41 by Urs Schreiber (87.212.203.135)