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 theĀ 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.)