nLab pointed set




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



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.



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.

Pointed objects in the category of pointed sets

The tensor unit of pointed sets is the boolean domain 𝟚\mathbb{2}, and pointed objects in the category of pointed sets are pointed sets AA with a point-preserving function 𝟚A\mathbb{2} \to A.

Natural numbers object

In classical mathematics, the natural numbers object in Set *Set_* is the set of extended natural numbers ¯=+{}\overline{\mathbb{N}} = \mathbb{N} + \{\infty\}, and comes with point-preserving functions z 0:𝟚¯z_0:\mathbb{2} \to \overline{\mathbb{N}} and z s:¯¯z_s:\overline{\mathbb{N}} \to \overline{\mathbb{N}} such that for all pointed sets AA and point-preserving functions f:𝟚Af:\mathbb{2} \to A, g:AAg: A \to A, there is a unique point-preserving function ϕ f,g:¯A\phi_{f, g}:\overline{\mathbb{N}} \to A making the following diagram commute:

𝟚 z 0 ¯ z s ¯ f ϕ f,g ϕ f,g A g A\array{ \mathbb{2} & \stackrel{z_0}{\to} & \overline{\mathbb{N}} & \stackrel{z_s}{\leftarrow} & \overline{\mathbb{N}} \\ & \mathllap{f} \searrow & \downarrow \mathrlap{\phi_{f, g}} & & \downarrow \mathrlap{\phi_{f, g}} \\ & & A & \underset{g}{\leftarrow} & A }

The point-preserving function z 0z_0 represents the function which takes the boolean true to \infty and false to zero, and z sz_s represents the point-preserving function which takes a natural number to its successor and \infty to \infty.

The absorption monoid structure on ¯\overline{\mathbb{N}} is defined by double induction on ¯\overline{\mathbb{N}}, we define

()+():¯×¯¯¯¯(-)+(-):\overline{\mathbb{N}} \times \overline{\mathbb{N}} \to \overline{\mathbb{N}} \wedge \overline{\mathbb{N}} \to \overline{\mathbb{N}}


z 0(p)+z 0(q)=z 0(pq)z s(m)+z 0(q)=z s(m+z 0(q))z_0(p) + z_0(q) = z_0(p \vee q) \qquad z_s(m) + z_0(q) = z_s(m + z_0(q))
z 0(p)+z s(n)=z s(z 0(p)+n)z s(m)+z s(n)=z s(z s(m+n))z_0(p) + z_s(n) = z_s(z_0(p) + n) \qquad z_s(m) + z_s(n) = z_s(z_s(m + n))

for all p,q𝟚p, q \in \mathbb{2} and m,n¯m, n \in \overline{\mathbb{N}}, where pqp \vee q is the disjunction of booleans pp and qq (recall the definition of addition in the natural numbers, inductively defined by 0(p)+0(q)=0(pq)0(p) + 0(q) = 0(p \cdot q), s(m)+0(p)=s(m+0(p))s(m) + 0(p) = s(m + 0(p)), 0(p)+s(n)=s(0(p)+n)0(p) + s(n) = s(0(p) + n), and s(m)+s(n)=s(s(m+n))s(m) + s(n) = s(s(m + n)) for all p,q𝟙p, q \in \mathbb{1} and m,nm, n \in \mathbb{N}). It is a commutative monoid and represents addition in ¯\overline{\mathbb{N}}.

In constructive mathematics, the extended natural numbers ¯\overline{\mathbb{N}} and the disjoint union +{}\mathbb{N} + \{\infty\} are no longer the same; it is +{}\mathbb{N} + \{\infty\} which remains the natural numbers object in Set *Set_*.

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 December 4, 2023 at 20:11:12. See the history of this page for a list of all contributions to it.