A pointed set is a set equipped with a chosen element of . (Compare inhabited set, where the element is not specified.)
Since we can identify a (set-theoretic) element of with a (category-theoretic) global element (a morphism ), we see that a pointed set is an object of the under category , or coslice category , of objects under the singleton .
The category of pointed sets
So a morphism is a map between sets which maps these chosen elements to each other, i.e., commuting triangles
The category naturally comes with forgetful functor which forgets the tip of these triangles.
Interpretation as universal Set-bundle
The morphism 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
is the interval category;
is the internal hom category which here is the arrow category of ;
are the images of the two injections 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 .
The way in which is the “universal Set-bundle” is discussed pretty explicitly in
(The discussion there becomes more manifestly one of bundles if one regards all morphisms appearing there as being the right legs of anafunctors. )
Interpretation as 2-subobject-classfier
Observing that usual morphism into the subobject classifier of the topos Set is the universal truth-value bundle? , and noticing that and suggests that 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).