A few weeks ago, I finally struck up the courage to take some baby steps reading Lurie’s Higher topos theory. In a series of posts mostly written for my own benefit, I will untangle some of the basic definitions and provide some easy examples. The first one is one I was already somewhat familiar with: simplicial sets.
Definition. For each , write for the poset . The full subcategory of on these is denoted , the simplex category. Concretely, it has objects for all , and morphisms
A simplicial set is a functor . This can be described rather concretely using the objects and the face and degeneracy maps between them; see e.g. Tag 0169. The category of simplicial sets is usually denoted , , or (in analogy with cosimplicial sets ).
The representable simplicial set is usually denoted or . Then the Yoneda lemma shows that the functor given by is represented by , i.e.
Definition. The geometric realisation functor is defined as follows: for , the geometric realisation is the standard -simplex
(If no confusion arises, it may also be denoted .) This is functorial in : for a map (equivalently, by the Yoneda lemma, a map ) we get a continuous map by
For an arbitrary simplicial set , write
where the transition map corresponding to a map over is defined via
This is functorial in , and when it coindices with the previous definition because the identity is terminal in the index category.
Remark. In a fancier language, is the left Kan extension of the functor along the Yoneda embedding . (Those of you familiar with presheaves on spaces will recognise the similarity with the definition of for a continuous map of topological spaces, which is another example of a left Kan extension.)
Remark. It is a formal consequence of the definitions that geometric realisation preserves arbitrary colimits (“colimits commute with colimits”). This also follows because it is a left adjoint to the singular set functor, but we won’t explore this here.
Wisdom. The most geometric way to think about a simplicial set is through its geometric realisation.
For example, we can define the horn in as the union of the images of the maps coming from the face maps for . Since geometric realisation preserves colimits (alternatively, stare at the definitions), we see that the geometric realisation of is obtained in the same way from the maps , so it is the -simplex with its interior and face opposite the vertex removed.
The geometric realisation is a good first approximation for thinking about a simplicial set. However, when thinking about -categories (e.g. in the next few posts), this is actually not the way you want to think about a simplicial set. Indeed, homotopy of simplicial sets (equivalently their geometric realisations) is stronger than equivalence of -categories. (More details later, hopefully.)