Examples of 2-Categories
Table of Contents
1. \( 1 \)-Category as a \(2 \)-Category
Let \(\mathcal C \) be a \(1 \)-category. Then \(\mathcal C \) can be regarded as a strict \(2 \)-category \(\mathcal C _{bi} \) where \(\mathcal C _{bi}(x,y) \) is the discrete category (only identities) induced by the set \(\mathcal C(x,y) \) for all objects \(x,y \).
A \(1 \)-category is \( \Set \)-enriched and we have a monoidal functor \(\Set \rightarrow \Cat \). Hence the above is a special case of transfering enrichmenta.
2. Monoidal Categories
Monoidal \(1 \)-categories correspond to \(2 \)-categories with one object.
Let \(\mathcal C \) be a \(2 \)-category. Then \(\mathcal C \rightarrow \CatMon, x \mapsto \mathcal C(x,x) \) is a \(2 \)-functor. (??)
3. \( 2 \)-Category \( \Cat \)
Let \( \Cat \) denote the strict \( 2 \)-category of \( 1 \)-categories where the \( 1 \)-morphisms are functors and the \( 2 \)-morphisms are natural transformations.
In Different Notions of \( 2 \)-Categories we introduced notation for composition of \( 1 \)-morphisms (\( \circ \)), vertical composition of \( 2 \)-morphisms (\( \bullet \)), horizontal composition of \( 2 \)-morphisms (\( \circ \)), left whiskering (\( \triangleleft \)), and right whiskering (\( \triangleright \)). Hence we are now in the position to fix notations for composing functors and natural transformation and compare them with our “old and naive” notation.
- Composition of functors didn’t change.
- Vertical composition: Let \( \alpha \colon F \Rightarrow G \), \( \beta \colon G \Rightarrow H \) be natural transformations (\( F,G,H \colon \mathcal C \rightarrow \mathcal C \) functors). Then we denote the vertical composition by \( \beta \bullet \alpha \colon F \Rightarrow H \). For \( X \in \mathcal C \) it is given by \( (\beta \bullet \alpha)_X = \beta_X \circ \alpha _X \).
Our old notation was just “\( \beta \circ \alpha \)”, but we see that “\( \bullet \)” is better notation to distinguish vertical and horizontal compositon and indicate the relation between normal composition and horizontal composition. Horizontal composition: Let \( \alpha : F \Rightarrow G \), \( \beta \colon F' \Rightarrow G' \) be natural transformations. Then the horizontal composition \( \beta \circ \alpha \) is given by
\begin{equation*} (\beta \circ \alpha) _X = \beta _{GX} \circ F'(\alpha_X) = G'(\alpha _{X}) \circ \beta _{FX}. \end{equation*}We didn’t have an old notation, so it is give to have one now.
Left whiskering: Let \( F \colon \mathcal C \rightarrow \mathcal D \) be a functor and \( \alpha \colon G \Rightarrow H \) a natural transformation. Then the left whiskering \( \alpha \triangleleft F \) is given by
\begin{equation*} (\alpha \triangleleft F) _X = \alpha _{FX}. \end{equation*}Hence left whiskering \( \alpha \triangleleft F \) corresponds to \( \alpha _{F(-)}\) is our old notation.
Right whiskering: Let \( H \colon \mathcal D \rightarrow \mathcal E \) be a functor and \( \alpha \colon F \Rightarrow G \) a natural transformation. Then the right whiskering \( H \triangleright \alpha \) is given by
\begin{equation*} (H \triangleright \alpha)_X = H(\alpha_X). \end{equation*}Hence right whiskering \( H \triangleright \alpha \) corresponds to \( H(\alpha _{-}) \) is our old notation.
4. \(2 \)-Category of Grothendieck fibrations
Strict \( 2 \)-category of Grothendieck fibrations (see Straightening for 1-Categories (Grothendieck Construction)).