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.
More concretely, a simplicial set $S$ is a collection of sets $S_n$ for $n \in \mathbb{N}$, so that elements in $S_n$ are to be thought of as $n$-simplices, equipped with a rule that says:
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.
A simplicial set is a presheaf on the simplex category $\Delta$, that is, a functor $X : \Delta^{op} \to Sets$, equivalently a simplicial object in the category Set of sets.
With the standard morphisms of presheaves as morphisms, simplicial sets form the category sSet (also called $SSet$ or $sSet$).
More explicitly this means the following.
A simplicial set $S \in sSet$ is
for each $n \in \mathbb{N}$ a set $S_n \in Set$ – the set of $n$-simplices;
for each injective map $\delta_i : \overline{n-1} \to \overline{n}$ of totally ordered sets $\bar n \coloneqq \{ 0 \lt 1 \lt \cdots \lt n \}$
a function $d_i : S_{n} \to S_{n-1}$ – the $i$th face map on $n$-simplices;
for each surjective map $\sigma_i : \overline{n+1} \to \bar n$ of totally ordered sets
a function $\sigma_i : S_{n} \to S_{n+1}$ – the $i$th degeneracy map on $n$-simplices;
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 $S_n$ assigned by a simplicial set to the standard $n$-simplex $[n]$ is the set of $n$-simplices in this space, hence the way of mapping a standard $n$-simplex into this spaces.
For $S$ a simplicial set, the face map
is dual to the unique injection $\delta^i : [n-1] \rightarrow [n]$ in the category $\Delta$ whose image omits the element $i \in [n]$.
Similarly, the degeneracy map
is dual to the unique surjection $\sigma^i : [n+1] \rightarrow [n]$ in $\Delta$ such that $i \in [n]$ has two elements in its preimage.
The maps $\delta^i$ and $\sigma^i$ satisfy certain obvious relations – the simplicial identities – dual to those spelled out at simplex category.
(based on cubical set)
The face maps go from sets $S_{n+1}$ of $(n+1)$-dimensional simplices to the corresponding set $S_{n}$ of $n$-dimensional simplices and can be thought of as sending each simplex in the simplicial set to one of its faces, for instance for $n=1$ the set $S_2$ of 2-simplices would be sent in three different ways by three different face maps to the set of $1$-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 $S_n$ of $n$-simplices to sets $S_{n+1}$ of $(n+1)$-simplices by regarding an $n$-simplex as a degenerate or “thin” $(n+1)$-simplex in the various different ways that this is possible. For instance, again for $n=1$, a degeneracy map may act by sending
Notice the $Id$-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 $[n]$ denote the object of $\Delta$ corresponding to the totally ordered set $\{ 0, 1, 2,\ldots, n\}$. Then the represented presheaf $\Delta(-, [n])$, typically written as $\Delta[n]$ is an example of a simplicial set. In particular we have $\Delta[n]_m=Hom_\Delta([m],[n])$ and hence $\Delta[n]_m$ is a finite set with $\binom{n+m+1}{n}$ elements.
By the Yoneda lemma, the $n$-simplices of a simplicial set $X$ are in natural bijective correspondence to maps $\Delta[n] \rightarrow X$ of simplicial sets.
If $C$ is a small category, the nerve of $C$ is a simplicial set which we denote $NC$. If we intepret the poset $[n]$ defined above as a category, we define the $n$-simplices of $NC$ to be the set of functors $[n] \rightarrow C$. Equivalently, the $0$-simplices of $NC$ are the objects of $C$, the $1$-simplices are the morphisms, and the $n$-simplices are strings of $n$ composable arrows in $C$. Face maps are given by composition (or omission, in the case of $d_0$ and $d_n$) and degeneracy maps are given by inserting identity arrows.
Recall from simplex category or geometric realization the standard functor $\Delta \to Top$ which sends $[n] \in \Delta$ to the standard topological $n$-simplex $\Delta^n$. This functor induces for every topological space $X$ the simplicial set
called the simplicial singular complex of $X$. This simplicial set is always a Kan complex and may be regarded as the fundamental ∞-groupoid of $X$.
Following up on the idea of ‘’thinness’’, a singular simplex $f: \Delta^n \to X$ may be called thin if it factors through a retraction $r: \Delta^n \to \Lambda^{n-1}_i$ to some horn of $\Delta^n$, then the well known Kan condition on $S X$ can be strengthened to say that every horn in $S X$ 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 $E$ is a topos containing such an interval $I$, then we obtain a functor $\Delta \to E$ sending $[n]$ to the subobject
The corresponding geometric realization/nerve adjunction $E \leftrightarrows Set^{\Delta^{op}}$ is the geometric morphism which classifies $I$.
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.
Similarly, also the category $Set^{\Delta}$ of cosimplicial sets is a classifying topos: for inhabited linear orders. See at classifying topos the section For (inhabited) linear orders.
(…) homotopy theory (…) Kan complex (…) quasi-category (…)
For the moment see at dendroidal set the section Relation to simplicial sets
simplicial set
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