pointed set


Category theory


Universal constructions

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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:1Ss: 1 \to S), we see that a pointed set is an object of the under category ptSet\pt \downarrow \Set, or coslice category 1/Set1/\Set, of objects under the singleton {}\{\bullet\}.

The category of pointed sets



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

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

pt S 1 S 2. \array{ && pt \\ & \swarrow && \searrow \\ S_1 &&\to&& S_2 } \,.

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



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

The internal hom Set *(X,Y)Set_*(X,Y) is the hom-set in */Set*/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 *SetSet_* \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 ptpt [I,Set] d 0 Set d 1 Set \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={ab}I = \{a \to b\} is the interval category;

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

  • 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_* \to Set.

The way in which Set *SetSet_* \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 April 20, 2016 at 09:03:53. See the history of this page for a list of all contributions to it.