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 $pt\downarrow Set$, or coslice category $1/Set$, of objects under the singleton $\{•\}$.

The category of pointed sets

Definition

So a morphism $(S_1,s_1)\to (S_2,s_2)$ is a map between sets which maps these chosen elements to each other, i.e., commuting triangles

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

Properties

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

in the 1-category Cat, where

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

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

• $d_i:=[j_i,\mathrm{Set}]$ 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

• Kathryn Hess, Bundle theory for categories .

(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? $\{\top \}\to TV$, and noticing that $\mathrm{TV}=(-1)\mathrm{Cat}$ 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

• David Roberts, comment to: 101 things to do with a 2-classifier

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).