References: Higher Topos Theory by Jacob Lurie.
You can first read the prerequisite on simplicial sets.
Categories
Categories are ubiquitous. A morphism reflects some relationship in a category \(\mathcal{C}\). In some cases, these morphisms themselves form a category.
The \((\infty,n)\)-categories are \(\infty\)-categories where all \(k\)-morphisms are invertible for \(k > n\).
- \((\infty,0)\)-categories are \(\infty\)-groupoids, and are generally thought as spaces. You can think of \((\infty,0)\)-categories are generalizations of the concept of spaces because paths and homotopies are invertible.
Why work over \((\infty,1)\)-categories?
- It will be our base line for \((\infty,n)\)-categories, since it can be thought as a category enriched over \((\infty,n-1)\)-categories.
Candidates For \((\infty,1)\)-Categories
Topological Categories
With the idea above one might define \((\infty,1)\)-categories as categories enriched over \((\infty,0)\)-categories, and these can concretely be replaced by topological spaces. This lead us to the first candidate for \((\infty,1)\)-categories: topological categories. These are categories enriched over compactly generated topological spaces, \(\mathcal{C}\mathcal{G}\).
A topological category \(\mathcal{C}\) consists of:
A set of objects \(\mathrm{Ob}(\mathcal{C})\), we will mix the notation and use \(\mathcal{C}\) instead of \(\mathrm{Ob}(\mathcal{C})\).
For each pair of objects \(x,y \in \mathrm{Ob}(\mathcal{C})\), a compactly generated topological space \(\mathcal{C}(x,y)\in \mathcal{C}\mathcal{G}\).
Compactly generated means that the open sets can be seen by all the compact sets, there are no weird open sets.
For \(n\ge 0\) composition maps
\[\mathcal{C}(x_0,x_1) \times \mathcal{C}(x_1,x_2) \times \cdots \times \mathcal{C}(x_{n-1},x_n) \to \mathcal{C}(x_0,x_n)\]
which are continuous.
This definition is simple, but it is the most difficult to work with.
Infinity Categories
To get an idea of how to define \(\infty\)-categories, we can think of the following:
We consider some extreme edge cases that our theory should be able to handle.
If every morphism is invertible, this is a groupoid and everything reduce to homotopy theory.
If there are no \(2\)-morphisms, then we have an ordinary category.
Thus we need something that can behave like category, and topological spaces.
Clearly topological categories, which we get by enriching categories over topological spaces, satisfy this, but we have more direct approaches.
Consider replacing the huge category of topological spaces with a smaller and handleable subcategory. The simplicial sets are a good candidate, which provides a combinatorial framework for homotopy theory.
But for them to behave like space, we need to consider the Kan complexes (since singular simplicial set associated to a topological space is a Kan complex).
\(\infty\)-Categories
Kan Complexes
A simplicial set \(K\) is a Kan complex if for every \(0\le i\le n\), all diagrams of solid arrows below can be filled in with dashed arrows.
Note that this includes outer horns \(i=0,n\), so it includes inverting diagrams like these:
which essentially forces all arrows to be invertible, and we think of Kan complexes are good representatives for groupoids.
Nerve of a Category
Given a category \(\mathcal{C}\), we can form a simplicial set \(N(\mathcal{C})\) called the nerve of \(\mathcal{C}\), defined by
\[N : \mathrm{Cat}\to {\bf Set}_\Delta\] \[N(\mathcal{C})_n = \mathrm{Hom}([n],\mathcal{C}).\]
Theorem. The following are equivalent:
\(K\cong N(\mathcal{C})\) for some small category \(\mathcal{C}\).
\(K\) is an inner-unique-Kan complex, i.e. for any \(0<i<n\), the diagram below can be uniquely filled in.
The point is that a inner-unique-Kan complex behaves almost like an ordinary category. Simplicial sets can be both models for groupoids (spaces) and categories.
In higher category contexts, we don’t really want the uniqueness of composition, think about the composition of paths, there can be many choices, in which there is no canonical way to compose two paths. Just like affine spaces are not vector spaces.
Hence we obtain the following definition for an \(\infty\)-category,
Definition Of \(\infty\)-Categories
An \(\infty\)-category or a weak Kan complex is a simplicial set \(K\) that satisfy an inner-Kan condition, for every \(0<i<n\), the diagram below can be filled in (without assuming uniqueness).
Examples of \(\infty\)-categories include
Any Kan complex.
\(N(\mathcal{C})\) for any small category \(\mathcal{C}\). These objects are identified as ordinary categories for us.
Equivalences Of Topological Categories
The first step to higher category theory is a concept which replaces homotopy equivalence. If \(F:\mathcal{C}\to \mathcal{D}\) is a functor between topological categories, under what conditions is \(F\) an equivalence?
Of course we would not use isomorphism as our concept of equivalence since it is way too strong. Classical category theory says \(F\) is an equivalence iff \(F\) is fully faithful and essentially surjective. In terms of topological categories, this could mean the following definition:
A functor \(F:\mathcal{C}\to \mathcal{D}\) of topological categories is a strong equivalence if
For every \(x,y\in \mathcal{C}\), the map \(F:\mathcal{C}(x,y)\to \mathcal{D}(Fx,Fy)\) is a homeomorphism.
For every \(x\in \mathcal{D}\), there exists \(y\in \mathcal{C}\) such that \(Fy\) is isomorphic to \(x\).
This reduces to equivalences of ordinary categories when we consider the discrete topological categories. But from higher category perspective, we should not require homeomorphism in the first condition, but only homotopy equivalence.
Let \(\mathcal{C}\) be a topological category, the (ordinary) homotopy category \(h\mathcal{C}\) is defined as
\(\mathrm{Ob}(h\mathcal{C}) = \mathrm{Ob}(\mathcal{C})\).
\(\mathrm{Hom}_{h\mathcal{C}}(x,y) = \pi_0(\mathcal{C}(x,y))\).
Composition is induced by the composition in \(\mathcal{C}\) and applying \(\pi_0\).
Consider the topological category of CW complexes whose morphisms are continuous maps equipped with compact-open topology (which states that open sets are \(C(K,U)\), mapping compact sets to open sets)
it’s ordinary homotopy category is denoted by \(\mathcal{H}\) and referred as homotopy category of spaces.
A map \(f:X\to Y\) between topological spaces is said to be a weak homotopy equivalence if it induces a bijection on \(\pi_0\) and for every point \(x\) and \(i\ge 1\) it induces an isomorphism of homotopy groups \(\pi_i(X,x)\to \pi_i(Y,f(x))\).
Spaces in \(\mathcal{C}\mathcal{G}\) are weak equivalent to a CW complex \(X\mapsto [X]\), this gives a functor \(\theta:\mathcal{C}\mathcal{G}\to \mathcal{H}\).
We can convert any topological category into a category enriched over \(\mathcal{H}\), and we will now improve the definition of a homotopy category, denote by \(h\mathcal{C}\) the category defined by
\(\mathrm{Ob}(h\mathcal{C}) = \mathrm{Ob}(\mathcal{C})\).
\(\mathrm{Hom}_{h\mathcal{C}}(x,y) = [\mathcal{C}(x,y)]\).
This sends the compactly generated space \(\mathcal{C}(x,y)\) to a (the homotopy class of) CW complex \([\mathcal{C}(x,y)]\) upto homotopy equivalence.
The composition law on \(h\mathcal{C}\) is obtained from \(\mathcal{C}\) and the functor \(\theta:\mathcal{C}\mathcal{G}\to \mathcal{H}\).
Simplicial Categories
A simplicial category is a category enriched over simplicial sets \({\bf Set}_\Delta\). The category of simplicial categories is denoted by \(\mathrm{Cat}_{\Delta}\).
Simplicial categories can be regarded as simplicial object in the category \(\mathrm{Cat}\),
Given a simplicial category \(\mathcal{C}\), consider the functor \(X:\Delta^{op}\to \mathrm{Cat}\) sending
\[X([n]) = (\mathcal{C}, \mathrm{Hom}_{\mathcal{C}}(x,y)([n])).\]
Conversely, any simplicial object \(X\) in \(\mathrm{Cat}\) whose objects \(\mathrm{Ob}(X([n]))\) are constant can be regarded as a simplicial category.