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

## The category of pointed sets

### Definition

###### Definition

The category $Set_*$ of pointed sets is the under category $*/Set$ of Set under the singleton set.

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

$\array{ && pt \\ & \swarrow && \searrow \\ S_1 &&\to&& S_2 } \,.$

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

### Properties

###### Proposition

Equipped with the smash product $\otimes := \wedge$ of pointed set, $(Set_*, \wedge)$ is a closed symmetric monoidal category.

The internal hom $Set_*(X,Y)$ is the hom-set in $*/Set$ pointed by the morphism $X \to Y$ that sends everything to the basepoint in $Y$.

#### Interpretation as universal Set-bundle

The morphism $Set_* \to Set$ 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

$\array{ Set_* &\to& pt \\ \downarrow && \downarrow^{pt \mapsto pt} \\ [I,Set] &\stackrel{d_0}{\to}& Set \\ \downarrow^{d_1} \\ Set }$

in the 1-category Cat, where

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

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

• $d_i := [j_i, Set]$ are the images of the two injections $j_i : pt \to I$ 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 $p : Set_* \to Set$.

The way in which $Set_* \to Set$ is the “universal Set-bundle” is discussed pretty explicitly in

(The discussion there becomes more manifestly one of bundles if one regards all morphisms $C \to Set$ 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 $TV = (-1)Cat$ and $Set = 0Cat$ suggests that $Set_* \to Set$ 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 May 19, 2013 18:52:21 by Ben Millwood? (131.111.185.81)