nLab
pointed set

Contents

Contents

Definition

A pointed set is a pointed object in Set, hence a set SS equipped with a chosen element ss of SS. (Compare inhabited set, where the element is not specified.)

Since we can identify a (set-theoretic) element of SS with a (category-theoretic) global element (a morphism s:*Ss: {\ast} \to S from the terminal object *{\ast}), we see that a pointed set is an object of the under category ptSet\pt \downarrow \Set, or coslice category */Set{\ast}/\Set, of objects under the singleton set *{\ast}.

The category of pointed sets

Definition

Definition

The category Set *\Set_{\ast} of pointed sets is the under category */Set{\ast}/\Set of Set under the singleton set *{\ast}.

So a morphism (S 1,s 1)(S 2,s 2)(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

* s 1 s 2 S 1 S 2. \array{ && {\ast} \\ & ^{s_1}\swarrow && \searrow^{s_2} \\ S_1 &&\to&& S_2 } \,.

The category Set *\Set_{\ast} naturally comes with a forgetful functor p:Set *Setp : \Set_{\ast} \to \Set which forgets the tip of these triangles.

Properties

Proposition

Equipped with the smash product :={\otimes} := {\wedge} of pointed sets, (Set *,)(\Set_{\ast}, {\wedge}) is a closed symmetric monoidal category.

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

See at pointed object for more details.

Interpretation as universal Set-bundle

The morphism Set *Set\Set_{\ast} \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

Set * pt pt* [I,Set] d 0 Set d 1 Set \array{ \Set_{\ast} &\to& \pt \\ \downarrow && \downarrow^{\pt \mapsto {\ast}} \\ [I,\Set] &\stackrel{d_0}{\to}& \Set \\ \downarrow^{d_1} \\ \Set }

in the 1-category Cat, where

  • I={01}I = \{0 \to 1\} is the interval category;

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

  • d i:=[j i,Set]d_i := [j_i, \Set] are the images of the two injections j i:ptIj_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 *Setp : \Set_{\ast} \to \Set.

The way in which Set *Set\Set_{\ast} \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 CSetC \to \Set appearing there as being the right legs of anafunctors. )

Interpretation as 2-subobject-classfier

Observing that usual morphism into the subobject classifier Ω\Omega of the topos Set is the universal truth-value bundle? {}TV\{\top\} \to \TV, and noticing that TV=(1)CatTV = (-1)Cat and Set=0CatSet = 0Cat suggests that Set *SetSet_* \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).

Last revised on July 10, 2020 at 11:34:50. See the history of this page for a list of all contributions to it.