Simplicial sets generalize the idea of simplicial complexes: a simplicial set is like a combinatorial space built up out of gluing abstract simplices to each other. Equivalently, it is an object equipped with a rule for how to consistently map the objects of the simplex category into it.
One of the main uses of simplicial sets is as combinatorial models for the (weak) homotopy type of topological spaces. They can also be taken as models for ∞-groupoids. This is encoded in the model structure on simplicial sets. For more reasons why simplicial sets see MathOverflow here.
The quick abstract definition of a simplicial set goes as follows:
A simplicial set is a presheaf on the simplex category , that is, a functor from the opposite category of the simplex category to the category Set of sets; equivalently this a simplicial object in Set.
Explicitly this means the following.
A simplicial set is
a function – the th face map on -simplices ( and );
for each surjective map of totally ordered sets
a function – the th degeneracy map on -simplices ( and );
such that these functions satisfy the simplicial identities.
The definition is to be understood from the point of view of space and quantity: a simplicial set is a space characterized by the fact that and how it may be probed by mapping standard simplices into it: the set assigned by a simplicial set to the standard -simplex is the set of -simplices in this space, hence the way of mapping a standard -simplex into this spaces.
For a simplicial set, the face map
is dual to the unique injection in the category whose image omits the element .
Similarly, the degeneracy map
is dual to the unique surjection in such that has two elements in its preimage.
(based on cubical set)
The face maps go from sets of -dimensional simplices to the corresponding set of -dimensional simplices and can be thought of as sending each simplex in the simplicial set to one of its faces, for instance for the set of 2-simplices would be sent in three different ways by three different face maps to the set of -simplices, for instance one of the face maps would send
another one would send
On the other hand, the degeneracy maps go the other way round and send sets of -simplices to sets of -simplices by regarding an -simplex as a degenerate or “thin” -simplex in the various different ways that this is possible. For instance, again for , a degeneracy map may act by sending
Notice the -labels, which indicate that the edges and faces labeled by them are “thin” in much the same way as an identity morphism is thin. They depend on lower dimensional features, (however notice however that a simplicial set by itself is not equipped with any notion of composition of simplices, nor really, therefore, of identities. See quasicategory for a kind of simplicial set which does have such notions and simplicial T-complex for more on the intuitions behind this idea of ‘’thinness’’).
Let denote the object of corresponding to the totally ordered set . Then the represented presheaf , typically written as is an example of a simplicial set. In particular we have and hence is a finite set with elements.
By the Yoneda lemma, the -simplices of a simplicial set are in natural bijective correspondence to maps of simplicial sets.
If is a small category, the nerve of is a simplicial set which we denote . If we intepret the poset defined above as a category, we define the -simplices of to be the set of functors . Equivalently, the -simplices of are the objects of , the -simplices are the morphisms, and the -simplices are strings of composable arrows in . Face maps are given by composition (or omission, in the case of and ) and degeneracy maps are given by inserting identity arrows.
Following up on the idea of ‘’thinness’’, a singular simplex may be called thin if it factors through a retraction to some horn of , then the well known Kan condition on can be strengthened to say that every horn in has a thin filler. This also helps to give some intuitive underpinning to the idea of thin element in this simplicial context.
For the moment see bar construction.
The category of simplicial sets is a presheaf category, and so in particular a Grothendieck topos. In fact, it is the classifying topos of the theory of “intervals”, meaning totally ordered sets equipped with distinct top and bottom elements.
Specifically, if is a topos containing such an interval , then we obtain a functor sending to the subobject
The usual geometric realization into topological spaces cannot be obtained in this way precisely, since Top is not a topos. However, there are Top-like categories which are toposes, such as Johnstone's topological topos.
A pedagogical introduction to simplicial sets is
A very clear and explicit exposition on the basics of simplicial sets is
Another clear exposition is in the classic
A useful (if old) survey article is:
More advanced treatments include
Some more facts about homotopical aspects of simplicial sets are discussed in section 2 of