### 101 things to do with a 2-classifier

#### Posted by David Corfield

Thanks to Tom and Todd, I have an answer to the problem I posed of what does the classifying for 2-categories.

- At level 0, we have the set inclusion $\{1\} \to \{0, 1\}$.
- At level 1, we have the forgetful functor $(Pointed set) \to Set$.
- At level 2, we have the forgetful 2-functor $(Pointed cat)^+ \to Cat$.

$(Pointed cat)^+$ is what I’m calling the 2-category of pointed categories $(C, c)$, but where a map $(C, c) \to (D, d)$ is a functor $F: C \to D$ *together with* a map $F(c) \to d$ in $D$.

If there’s a name already for this 2-category, do please let me know.

On the royal road to categorified logic, let’s see if I can understand the answer in terms of some of the examples I had in mind:

a) Mapping a 2-group, $G$, into Cat.

Here $G$ is taken as a 1 object 2-groupoid. So we’re looking then at permutation 2-representations of $G$. Do people study these?

Above $G$ we should have the action 2-groupoid. Let’s say the single object of $G$ was sent to $C$, so we’re after a 2-groupoid $C // G$. (One might argue that it should be $C /// G$, the *really* weak quotient. Perhaps the notation isn’t too good.)

$C // G$ has as objects the objects of $C$. A map $c \to c^'$ is a 1-morphism $g$ in $G$ together with a map $g \cdot c \to c^'$ in $C$. The 2-morphisms of $C // G$ can wait.

b) Mapping the 2-category Th(2-gp) (theory of 2-groups) into Cat.

Surely someone must have worked out 2-theories as 2-categories. Do you get a nice duality as between Th(gp) and Free groups?

Presumably fibred above Th(2-gp) for a given functor into Cat, you’d have specific maps such as between pairs of objects of that category and a third object, with a comparison map between the monoidal product of the pair and the third object. So essentially encoding the weakness and coherence conditions.

c) Mapping the fundamental 2-groupoid of a space, such as the 2-sphere, into Cat.

So over this 2-groupoid we have another 2-groupoid, with as objects pairs consisting of a point of the 2-sphere and an object in the fibre category. A map between such pairs is a path between the corresponding points on the sphere, together with a map in the fibre over the second point from the endpoint of the lift of the path to the designated object. (No doubt some symbols would have helped here.) Again 2-morphisms are needed.

Is this a well-known construction?

## Re: 101 things to do with a 2-classifier

Something tells me I need a crash course in fibred categories. Maybe Thomas Streicher’s notes.