LECTURE 11: INTRODUCTION TO HIGHER CATEGORIES

Last time, we met the definition of a quasi-category as a simpicial set satisfying a certain lifting property. While this definition is rather direct and simple, it lacks to provide any interpretation as a model for a higher category. In this lecture, we hope to give a broad picture of higher categories, and to relate more intuitive approaches to that of quasi- categories. If a category is like a set with arrows between the elements of the set, we should inductively think of a 2-category as a category with arrows between the morphisms of C. Thus, what we mean by “higher category” is a category in which one can speak of arrows between arrows, and so on.

1. The idea of enrichment

Let K⊗ be a (unital) monoidal category. A category enriched in K is a category C together such that for all X,Y ∈ C the morphism sets HomC(X,Y ) are objects of K. One must also prescribe the data of identities and compositions in the appropriate way. For instance, the composition rule

cXYZ : HomC(Y,Z) ⊗ HomC(X,Y ) → HomC(X,Z) must be a morphism in the category K.

Example 1.1. There are many examples of enriched categories. For instance, the category dg of dg vector spaces Vectk is enriched in itself. The category of topological spaces also admits a natural enrichment. Indeed, one can speak of “mapping spaces”, which gives Top an enrichment over itself as well.

Note that when we say category, what we really mean is a category enriched in Set. Following this logic, the idea of enrichment gives a natural first guess of what a higher category is.

Definition 1.2. A strict 2-category is a category enriched over the category of categories Cat.

Remark 1.3. When we say the category of categories Cat we mean the (large) category whose objects are categories and morphisms are functors between categories.

Concretely, this definition means that for any two objects X,Y of a 2-category C(2) there is a morphism category HomC(2) (X,Y ). Within this morphism category, we can then talk about “morphisms between morphisms” in a rigorous way. 1 Why have we called such 2-categories strict? This comes from thinking about associa- tivity of the composition rule (which is now a functor)

cXYZ : HomC(2) (Y,Z) ◦ HomC(2) (X,Y ) → HomC(2) (X,Z). By the definition of enrichment, associativity says that we have an equality of functors

cXZW ◦ (cXYZ × id) = cXYW ◦ (id × cYZW ) for all X,Y,Z,W ∈ C(2). It is generally not good practice to ask for two functors to be equal. Indeed, it is more natural to ask for the data of a natural isomorphism

ηXYZW : cXZW ◦ (cXYZ × id) =⇒ cXYW ◦ (id × cYZW ). Asking for such data leads one to the definition of a (weak = non-strict) 2-category.

Remark 1.4. It turns out that weak and strict 2-categories are equivalent, but this should be thought of as a happy accident. When we go higher up the chain, strict n-categories are not at all the same as weaker definitions one can come up with.

When we say “higher category” we are referring to a category in which one has mor- phisms of arbitrary level.

2. Fluffy morphisms

In practice we will still only be concerned with results at the 1-categorical level. Really, we will use the theory of higher categories as a robust model for the concept of “homotopy equivalence”. This means that we will only ever care about higher categories in which all the higher morphisms are quite boring.

Definition 2.1. An (∞, n)-category is a higher category in which all k-morphisms are invertible for k > n. We will simply refer to a (∞, 1)-category as an ∞-category.

Example 2.2. An intuitive example of higher categories comes from topology. Let X be a topological space. Define the category π≤1(X) whose objects are the points of X and morphisms are paths. Note that to every path, we can invert time to obtain an inverse.

Thus, this is a category in which all morphisms are invertible. Hence π≤1(X) is a groupoid called the “fundamental groupoid” of X.

Going further, we can define the 2-category π≤2(X) whose objects and morphisms are the same as π≤1(X). The 2-morphisms are the homotopies between paths. This is what one might call a “2-groupoid”, since it is a 2-category in which all 1, 2-morphisms are invertible.

One can inductively define a higher category π≤∞(X), whose morphsims of all levels are invertible. This is a particular example of a (∞, 0)-category, which we might as well refer to as an ∞-groupoid. We will soon see why one should think of all ∞-groupoids as arising in this way. Thus, “∞-groupoid” is just a fancy way of saying “topological space”. 2 Using the motivating perspective of 2-categories as categories enriched in 1-categories, we arrive at the following model for an (∞, 1)-category: it is a category enriched in (∞, 0)- categories. That is, a topological category.

Definition 2.3. A topological category is a category enriched over the category of topological spaces Top. That is, a category C in which the set of morphisms between any two objects HomC(X,Y ) ∈ Top is a topological space.

We will refer to the category of topological categories by CatTop.

3. Simplicial categories

For many purposes, topological spaces are not as well-behaved as their combinatorial cousin: simplicial sets. For similar reasons, it is often better to consider a simplicial analog of a topological category as a model for an ∞-category.

Definition 3.1. A simplicial category is a category enriched in simplicial sets. That is, a category C in which the set of morphisms between any two objects HomC(X,Y ) ∈ Set∆ is a simplicial set.

Denote the category of simplicial categories by Cat∆. Simplicial categories are related to topological categories in a similar way to how sim- plicial sets are related to topological spaces. Namely, the geometric realization / singular- ization adjunction |−|

Set∆ Top.

Sing determines an adjunction |−|

Cat∆ CatTop.

Sing by applying the original adjunction to the morphism spaces. For instance, if C is a sim- plicial category, we define the topological category |C| in the following way: • its objects are the same as C; • if X,Y ∈ C, define

Hom|C| = |HomC(X,Y )|. Composition is defined by functoriality of realization. The standard realization / singular adjunction between simplicial sets and topological spaces enhances to a Quillen equivalence of model categories. In particular, it determines an equivalence of categories between the homotopy category of simplicial sets and the homotopy category of spaces. 3 We will see there is a similar statement of equivalences at the level of simplicial and topological categories. For this, we need to define the notion of “homotopy” at the cate- gorical level.

4. The homotopy category

Note that there is a functor

π0 : Top → Set which takes a topological space X to its path components π0X.

Definition 4.1. Let C be a topological category. Define its homotopy category ho C to be the (ordinary) category whose objects are the same as that of C and whose morphisms are

Homho C(X,Y ) = π0HomC(X,Y ).

Remark 4.2. The set π0HomC(X,Y ) is precisely the set of homotopy classes of maps X → Y and is sometimes written as [X,Y ].

We use geometric realization to define the concept from simplicial categories.

Definition 4.3. Let C be a simplicial category and |C| the realized topological category. The homotopy category of C is ho C := ho |C|.

By construction, we see that the adjunction between simplicial and topological cat- egories becomes an equivalence at the level of homotopy categories. That is, if C is a simplicial category, then the unit induces a functor of simplicial categories

C → Sing|C| which is an equivalence at the level of homotopy categories. Similarly, if D is a topological category, the counit induces a functor of topological categories

|SingD| → D which is an equivalence on homotopy categories.

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