LECTURE 8: SIMPLICIAL SETS, I

1. Definitions

Let ∆ be the following . • Objects: totally ordered sets

[n] = {0 < 1 < ··· < n}

where n ≥ 0 is any non-negative integer. • : order preserving maps of sets θ :[n] → [m]. That is, i < j =⇒ θ(i) < θ(j).

Remark 1.1. If we think about the objects of ∆ as categories themselves

[n] = {0 → 1 → · · · → n} then the morphisms in ∆ are precisely the .

We refer to ∆ as the finite ordinal category.

Definition 1.2. A simplicial set is a

X : ∆op → Set.

In general, if C is any category, a simplicial object in C is a functor

X : ∆op → C.

If X is a simplicial set, we let X[n] denote the image of the object [n] ∈ ∆. The category of simplicial sets is the

op Set∆ = Fun(∆ , Set). Unwinding the definition, a of simplicial sets f : X → Y is equivalent to prescribing a map of sets f[n]: X[n] → Y [n] for each n satisfying some conditions that we spell out below.

Example 1.3. Singular set. Let Top be the category of topological spaces, and let

n n+1 X |∆ | = {(t0, . . . , tn) ∈ (R≥0) | ti = 1} i≥0 be the standard n-simplex. For any order preserving map θ :[n] → [m] there is a natural map of spaces n m θ∗ : |∆ | → |∆ |. 1 This defines a functor ∆ → Top, [n] 7→ |∆n|. Further, for any Y ∈ Top, we have the simplicial set

Sing(Y ) : ∆op → Set n [n] 7→ HomTop(|∆ |,Y ) called the singular simplicial set associated to Y .

Example 1.4. Standard simplex. For each n let

n op ∆ = Hom∆(−, [n]) : ∆ → Set.

This is called the standard n-simplex. Note that this is simply the image of [n] under the Yoneda embedding op ∆ → Fun(∆ , Set) = Set∆. In particular n HomSet∆ (∆ ,X) = X[n] for each n.

In pictures, we can think about ∆1 as simply the two vertex diagram

∆1 : 0 → 1 and ∆2 as being prescribed by the following diagram 1

∆2 : 0 2. For instance, ∆2([1]) = ∆([1], [2]) = {0 → 1, 1 → 2, 1 → 2} reads off all the 1-simplices inside ∆2.

1.1. A combinatorial description. So far, the definition of a simplicial set might seem a little abstract. One can prescribe the data of a simplicial set more explicitly as a collection of sets

{X[n]}n≥0 together with the following data of face and degeneracy maps. The category ∆ is generated by two classes of morphisms: (1) Face maps. For each n ≥ 1 and 1 ≤ j ≤ n define the maps

dj :[n − 1] → [n] ( i , i < j i 7→ . i + 1 , i ≥ j 2 (2) Degeneracy maps. For each n ≥ 1 and 1 ≤ j ≤ n define the maps

sj :[n + 1] → [n] ( i , i ≤ j i 7→ . i − 1 , i > j Generated means that any order preserving morphism [n] → [m] can be written as a composition of these two classes of maps. The category is not, however, freely generated by these maps. There are a number of relations involving sj’s and di’s such as djdi = didj−1 and sjdi = disj−1 whenever both sides are defined. We won’t list all the relations here, but refer to Chapter 1 of [?] for a textbook reference. Consequently we can define a simplicial set by the following data:

• a collection of sets X = {X[n]}n≥0, where we refer to X[n] as the n-simplices; ∗ ∗ • a collection of morphisms {dj : X[n] → X[n−1]} (the face maps) and {sj : X[n] → X[n + 1]} (the degeneracy maps) satisfying some compatibilities. We will often define a simplicial set in this hands-on way.

2. Geometric Realization

We have already constructed a functor Sing(−): Top → Set∆ given by the singular set. One utility of simplicial sets is that they give us combinatorial rules for gluing spaces together in terms of simplices. Geometric realization makes this precise. I’m going to write the abstract definition, then interpret it in a way that hopefully makes clear what is going on. Given any category C and object X ∈ C one can consider the overcategory (or slice category) C/X whose objects are maps Y → X and whose morphisms are commuting triangles

Y Y 0

X There is a notion of an undercategory.

Given a simplicial set X we can then consider the overcategory Set∆/X . Think about this as the category of maps of simplical sets into X. We want to think about a smaller category, namely the category of maps of just the n-simplices into X, that we denote ∆/X . This category fits into a

∆/X Set∆/X

∆(−) ∆ Set∆ 3 The lower horizontal map is the Yoneda embedding sending [n] 7→ ∆n. The right vertical map is the map that forgets the map into X defining the object of the overcategory and just remembers the source object. 1

Concretely, the objects ∆/X consists of maps, for each n ∆n → X.

These are precisely the n-simplices X[n] of X. The morphisms are commuting triangles of simplicial sets ∆n ∆m

X Note that any simplicial set defines a composition of functors

∆(−) ∆/X → ∆ −−−→ Set∆. The following lemma consists of throwing definitions around

Lemma 2.1. There is an isomorphism of simplicial sets   ∆(−) =∼ colim ∆/X → ∆ −−−→ Set∆ −→ X

Roughly, this lemma just says that X is glued together from its n-simplices.

Remark 2.2. An important analogous phenomena to keep in mind is the gluing condition for sheaves. Suppose Y is a topological space and F is a on Y . Let Open(Y ) be the poset of open sets of Y . For an open set U ⊂ Y , we can present the value F(U) as the following ∼  op op F  F(U) = lim Open(Y )/U → Open(Y ) −→ Set In the limit, the first functor is the forgetful functor and the second functor is defined by the sheaf F. Perhaps even more analogous is the case of a cosheaf G on Y . Then, the cosheaf condition is equivalent to ∼  G  G(U) = colim Open(Y )/U → Open(Y ) −→ Set .

Using this perspective, the following definition is properly motivated.

Definition 2.3. Let X be a simplicial set. Define the geometric realization of X to be the topological space

 |∆(−)|  |X| := colim ∆/X → ∆ −−−−→ Top .

Here, |∆(−)| is the functor from Example 1.3 which sends [n] to the geometric n-simplex n n |∆ | ⊂ R .

1This is a pullback diagram in the category of categories. 4 Remark 2.4. We have already seen how colimits can be used to glue together a space. Morally, this is what the colimit definition above is encoding. This becomes more apparent when one uses the following alternative description of the realization |X|. For each n, we consider the topolgical space X[n] × |∆n|. Here, we are viewing X[n] with the discrete topology. We write a generic element in this space as (x; t0, . . . , tn). We can present the realization as a quotient of the form   G n |X| =  X[n] × |∆ | / ∼ n≥0 where the equivalence relation is of the form

(di(x); t0, . . . , tn) ∼ (x; t0, . . . , ti−1, 0, ti, . . . , tn).

Almost tautologically, one has the following.

Proposition 2.5. The functor | − | is left adjoint to Sing(−):

|−| a Set∆ Top.

Sing Proof. We use the categorical fact that colimits / limits commute with the Yoneda em- bedding in the following way

Hom(colim (−),Y ) = lim Hom(−,Y ).

(−) Also, note that the singular space was defined via the formula Sing(Y ) = HomTop(|∆ |,Y ). Using these two facts, for any space Y we obtain isomorphisms

(−) HomTop(|X|,Y ) = HomTop(colim ∆ ,Y ) ∼ (−) = lim HomTop(|∆ |,Y ) ∼ (−) = lim HomSet∆ (∆ , Sing(Y )) ∼ (−) = HomSet∆ (colim ∆ , Sing(Y ))

= HomSet∆ (X, Sing(Y )).



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