Simplicial objects and higher categories, part I

 

Hi all! Today I’ll introduce simplicial sets, and talk about how they relate to higher categories. Simplicial sets are basically combinatorial models for topological things; in fact, a particular kind of simplicial set (a Kan complex) is essentially equivalent to a topological space! So more general kinds of simplicial sets should model something more general than a topological space. Exploring this will be the topic of this post. The reader should note that these are directly from the (rather terse) notes I took while preparing for the Intel International Science and Engineering Fair last year (so this blog post is currently also serving as the notes for preparing for the Intel Science Talent Institute), so there may be small errors.

We will begin by talking about some combinatorial constructions. Let [n] denote the set \{0,...,n\} equipped with the linear ordering. Define \mathbf{\Delta} to be the category whose objects are the sets [n] with the linear ordering and morphisms the order-preserving maps. A simplicial object of a category \mathscr{C} is a functor \mathbf{\Delta}^\mathrm{op}\to\mathscr{C} (with an analogous definition of a cosimplicial object). When \mathscr{C} is the category of sets, then a simplicial object of \mathscr{C} is called a simplicial set. More explicitly, a simplicial set S consists of the following data:

  1. A set S_n, namely the value of the functor S on [n] for every n\geq 0.
  2. A map \alpha_{!}:S_n\to S_m given by an order-preserving map \alpha:[m]\to [n] (note the direction of the map) of finite sets, that is compatible with composition.

The standard notation when working with simplicial sets is as follows. There is a face map d_i:S_n\to S_{n-1} for 0\leq i\leq n, given by the pullback of the map \alpha:[n-1]\to [n], defined as

\alpha(n)=\begin{cases} n &\mbox{if } n < i \\ n+1 & \mbox{if } n\geq i. \end{cases}

The degeneracy map s_i:S_n\to S_{n+1}, for 0\leq i\leq n, is defined as the pullback of the map \beta:[n+1]\to [n] defined as

\beta(n)=\begin{cases} n &\mbox{if } n \leq i \\ n-1 & \mbox{if } n > i. \end{cases}

It turns out that the morphisms of \mathbf{\Delta} are consisted of the composites of the d_is and s_is subject to the following relations:

s_j d_i=\begin{cases} d_i s_{j-1} &\mbox{if } i< j \\ 1 & \mbox{if } i=j,j+1\\ d_{i-1} s_k & \mbox{otherwise}.\end{cases}

s_j s_i=s_i s_{j+1}\quad \mbox{if } i\leq j

This statement follows from the claim that any morphism [n]\to [m] can be put into a composition d_{i_k}\cdots d_{i_1}s_{j_1}\cdots s_{j_h}, where n+k-h=m and m\geq i_k>\cdots> i_1\geq 0 and 0\leq j_1<\cdots<j_h<n. The proof of the claim is a (simple) exercise, left to the reader.

A map of simplicial sets f:X\to Y is a natural transformation of the functors X and Y. These maps form a category {\mathscr{S}\mathrm{et}}_\Delta of simplicial sets. The first example of a simplicial set one usually encounters is the nerve \mathrm{N}(\mathscr{C}) of a 1-category \mathscr{C}. The set of n-simplices \mathrm{N}(\mathscr{C})_n is the set of maps [n]\to\mathscr{C}. It is possible to characterize those simplicial sets which arise as the nerve of a category, but in order to do this, we must first introduce some terminology.

Let \Delta^{[n]}=\Delta^n denote the representable functor [n]\mapsto\mathrm{Hom}_{{\mathscr{S}\mathrm{et}}_\Delta}([n],[n]). One can naturally identify the set S_n of n-simplices of a simplicial set with \mathrm{Hom}_{{\mathscr{S}\mathrm{et}}_\Delta}(\Delta^n,S). Let \Lambda_j^n\subset\Delta^n denote the j-th horn, defined as follows: an element of (\Lambda_j^n)_i is an order-preserving map f:[i]\to [n] satisfying the condition \{j\}\cup f([m])\neq [n]. In other words, \Lambda^n_j corresponds to subset of \Delta^n in which the jth face and the interior have been removed. We can now characterize those simplicial sets which arise as the nerve of a category:

Proposition: A simplicial set K is isomorphic to the nerve of a small category \mathscr{C} if and only if for each 0<i<n, there is always a unique dotted arrow that makes the following diagram commute:

nerveprop

We will now do some basic geometric constructions, inherently combinatorial. The standard n-simplex \Delta^n is the convex closure of the standard basis of \mathbf{R}^{n+1}. Let X be a topological space. There is a natural functor from \mathscr{T}\mathrm{op}\to{\mathscr{S}\mathrm{et}}_\Delta, where \mathscr{S} is the category of topological spaces, denoted \mathrm{Sing}, whose n-simplices are continuous maps \Delta^n\to X. This simplicial set \mathrm{Sing}(X) is called the singular simplicial complex of the topological space X. The left adjoint to the functor \mathrm{Sing} is called the geometric realization of the simplicial set, and is denoted |X_\bullet|. The geometric realization is the way you imagine a simplicial set, so this construction is particularly natural. The geometric realization of a simplicial set turns out to be a CW-complex! This is taking a very interesting turn.

A simplicial set K is a Kan complex if for every 0\leq i\leq n, there is a lift:

nerveprop

So the nerve of a general category need not be a Kan complex! Also, a Kan complex need not be the nerve of a category, because the lift need not be unique. Let X be a topological space. Then the singular simplicial complex \mathrm{Sing}(X) is a Kan complex. This is because the horn |\Lambda^n_i| can be seen to be a retract of |\Delta^n| in the category of topological spaces. The idea is that Kan complexes are simplicial versions of topological spaces. To make this precise, we’ll have to introduce a basic definition.

Let \mathrm{h}:\mathscr{S}\mathrm{et}_\Delta\to\mathscr{C}\mathrm{at} denote the left adjoint to the nerve functor \mathrm{N}:\mathscr{C}\mathrm{at}\to\mathscr{S}\mathrm{et}_\Delta. This assigns to each simplicial set its homotopy category. Awesome. We have the following results, making precise what we claimed in the first paragraph (and the previous one):

Theorem: The adjoint pair (\mathrm{Sing},|-|) provides an equivalence between the homotopy category of CW complexes and the homotopy category of Kan complexes. Also, a simplicial set is a Kan complex if and only if for all 0<i<n, there is an extension:

nerveprop

and its homotopy category is a groupoid.

As you can see, that particular diagram is quite important in simplicial homotopy theory, the reason explained in the following video, which I may have already linked to in a previous post:

 

It turns out that you can do homotopy theory with simplicial sets! Let’s begin with a definition. Let \mathscr{A} be a class of monomorphisms in \mathscr{S}\mathrm{et}_\Delta. It is said to be saturated if:

  1. \mathscr{A} contains all isomorphisms,
  2. \mathscr{A} is closed under pushouts,
  3. \mathscr{A} is closed under retracts,
  4. \mathscr{A} is closed under coproducts,
  5. \mathscr{A} is closed under \omega-composites.

The intersection of all saturated classes containing a given set of monomorphisms, \Omega, is the saturated class generated by \Omega. A very important saturated class of morphisms is the collection of anodyne maps: The class of anodyne extensions is the saturated class of morphisms generated by the family \{\Lambda^n_k\to \Delta^n|n\geq 1,0\leq k\leq n\}. The importance of anodyne extensions is encoded in the following statement:

Proposition: A map is a Kan fibration if and only if it has the right lifting problem with respect to all anodyne extensions.

Now to define homotopies of maps of simplicial sets. Let f,g:X\to Y be simplicial maps. There is a homotopy from f to g if there is a commutative diagram:

homotopy

In order to gain intuition for this definition, consider the case when X and Y are Kan complexes and then invoking the above theorem. Since \Delta^1 is replaced by the interval [0,1]=I, this definition of homotopy is analogous to ordinary homotopy theory. Similarly, replacing the interval with \mathbf{A}^1, the affine line, takes one into motivic homotopy theory, which is an interesting and exciting field of research.

One cool result is that simplicial homotopy of vertices \Delta^0\to X of X is an equivalence relation if X is a fibrant simplicial set, i.e., a Kan complex. This further solidifies the idea that Kan complexes are essentially the same as topological spaces. This claim can be applied in the following way to show that certain simplicial sets are not Kan complexes. Consider the maps 0,1:\Delta^0\rightrightarrows\Delta^n, for n\geq 1, that classify the vertices 0 and 1, respectively. Then the simplex [0,1]:\Delta^1\to\Delta^n determined by these vertices exist, so there is a simplicial homotopy 0\xrightarrow{\simeq}1. However, it is not possible to find a 1-simplex such that 1\xrightarrow{\simeq}0 (because 0\leq 1), so simplicial homotopy of the vertices of \Delta^n is not an equivalence relation; hence \Delta^n is not fibrant (i.e., it is not a Kan complex).

We also have the following definition of homotopies rel an inclusion. Let X^\prime\subset X and let f,g:X\to Y be simplicial maps. We say f is homotopic to g rel X^\prime if, like above, there is a commutative diagram:

homotopy

such that the following diagram commutes:

relhomotopy

We will use this definition in the next post, to define the homotopy groups of simplicial sets. Then we’ll progress on to \infty-categories!

Have fun,
SKD

 

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