January 17, 2008

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?

Posted at January 17, 2008 10:14 AM UTC

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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.

Posted by: David Corfield on January 17, 2008 2:17 PM | Permalink | Reply to this

Re: 101 things to do with a 2-classifier

Is that morphism $F(c)\rightarrow d$ supposed to be an isomorphism?
Posted by: Tim Silverman on January 17, 2008 7:19 PM | Permalink | Reply to this

Re: 101 things to do with a 2-classifier

For the present purposes, these morphisms $F(c) \to d$ are general ones, not isomorphisms.

Going down one dimension, the fiber over an object $b$ of $B$ of a discrete (op)fibration $E \to B$, as in the universal case $Pointed set \to Set$, is by definition the subcategory of $E$ whose 1-cells map to $1_b$.

Analogously, the (locally discrete) fiber over a $0$-cell $C$ of the $2$-categorical (op)fibration

$(Pointed cat)^+ \to Cat$

is by definition the sub-$2$-category whose $2$-cells map to a given identity 2-cell $1_{1_C}$. If one imposes the isomorphism convention, then this fiber is not $C$ as we want, but rather the underlying groupoid of $C$.

But there may be other reasons for considering that possibility.

Posted by: Todd Trimble on January 17, 2008 9:42 PM | Permalink | Reply to this

Re: 101 things to do with a 2-classifier

Your PointedCat^+ thing is the category you get when applying the Grothendieck construction to the identity functor Cat–>Cat (ie the homotopy colimit of that functor).

Posted by: Richard Lewis on January 19, 2008 6:33 PM | Permalink | Reply to this

Re: 101 things to do with a 2-classifier

It may be relevant that $(PointedCat)$ is the tangent bicategory $T_{pt}(Cat)$ of $(Cat)$ at the category $pt$. Since $(Cat)$ is a pointed bicategory, this is the natural option for the universal $(Cat)$-bundle.

Going back down some dimensions, it may be obvious to everybody else, but I haven’t seen it said out loud: the set $\mathbf{2}$ that is the subobject classifying is $\{pt,\emptyset\}$ and the only pointed thing inside $\mathbf{2}$ is $pt$. There are lots of two-element sets, but this one makes the pattern clearer.

Sorry for stating the obvious, but then the choice of $(PointedSets)$ for the 2-classifier doesn’t seem as mysterious as it did to me at first.

Posted by: David Roberts on January 20, 2008 11:44 AM | Permalink | Reply to this

Re: 101 things to do with a 2-classifier

So I kept talking about how the sequence

$X \hookrightarrow X//G \to \mathbf{B} G$

reminded me of the universal $G$-bundle in its groupoid incarnation

$G \to G//G \to \mathbf{B}G$

The thing is that there is the universal $Set$-bundle $\array{ s^{-1} pt \\ \downarrow \\ T_pt Set \\ \downarrow \\ Set }$

and that the action groupoid sequence is the pullback of that along the representation functor

$\array{ X &\to& s^{-1} pt \\ \downarrow && \downarrow \\ X//G &\to& T_pt Set \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& Set } \,.$

Hm, that fits in beautifully with what I am trying to do:

In Section s of bundles … I pointed out that if you have a $G$ bundle given by a cocycle

$g : Y^\bullet \to \mathbf{B} G$

and if you choose a representation

$\rho : \mathbf{B} G \to Set$

of $G$, then the space of sections of the $\rho$-associated bundle is precisely the space of lifts

$\array{ && X // G \\ &{}^\sigma\nearrow& \downarrow \\ Y^\bullet &\stackrel{\rho}{\to}& \mathbf{B}G } \,.$

So the situation is

$\array{ &&X &\to& s^{-1} pt \\ &&\downarrow && \downarrow \\ &&X//G &\to& T_pt Set \\ &{}^\sigma \nearrow&\downarrow && \downarrow \\ Y^\bullet &\stackrel{g}{\to}&\mathbf{B}G &\stackrel{\rho}{\to}& Set } \,.$

Beautiful. It’s so obvious now that I see it, but I was blind to that until now.

Posted by: Urs Schreiber on March 26, 2008 5:14 PM | Permalink | Reply to this

Re: 101 things to do with a 2-classifier

David, thanks so much for making me see this.

Finally I am able to nicely unify the different points of view on sections of $n$-bundles which I have been talking about.

In case anyone remembers, I grew fond of the fact that a section is a morphism into the corresponding transport functor/cocycle (for instance here or here.)

In the attempt to put this into a larger perspective, I started characterizing them in terms of tangent categories (for instance here) and ever since bugged John about how there seems to be a relation to groupoidification.

Then, more recently, I started thinking here about that concept of section as a lift of the cocycle through the action groupoid. The reason is that in this formulation, in contrast to the other two above, all that ever appears are groupoids, and hence this formulation, if everything is Lie, can be differentiated and considered entirely in the world of $L_\infty$-algebras (described here).

But it’s annoying to have $2\frac{1}{2}$ different cool abstract definitions of what a section of an $n$-bundle is. Of course in special examples one could see that all definitions agreed, but I had no clue what abstract mechanism related them.

Now I have. Thanks to your remark above. The solution is given in the following diagram (for 0-Cat, but the generalization is essentially obvious):

Posted by: Urs Schreiber on March 26, 2008 9:34 PM | Permalink | Reply to this

Re: 101 things to do with a 2-classifier

I have to point out further how very cool this is:

remember all that business about tangent categories: $T_{pt} Set$ is is defined to be a subcollection of maps of the standard interval $I := \{\bullet \to \circ\}$ into $Set$. Now we see how it all comes together with the theory of sections:

Our transformation into the cocycle is an element of

$Hom(Y^\bullet \otimes I, Set) \simeq Hom(Y^\bullet , Hom(I, Set)) \,.$

That’s how the $Hom(I,Set)$ appears. Then the constraint that this is actually a transformation starting at the terminal cocycle yields the constraint appearing in the definition of tangent categories: the $\{\bullet\} \hookrightarrow \{\bullet \to \circ\}$ has to be fixed.

It’s pretty remarkable what is gained by realizing universal bundles not as spaces, but as $n$-groupoids. Not only is $\mathbf{E} G$ then the right home for the curvature (when shifted further with one $\mathbf{B}$), but also for the sections (when not shifted.) Sounds weird on first sight. But works rather neatly.

Posted by: Urs Schreiber on March 26, 2008 9:48 PM | Permalink | Reply to this

Re: 101 things to do with a 2-classifier

I am wondering about the pushout of $\array{ \mathbf{B}G &\stackrel{\rho}{\to}& Set \\ \downarrow \\ \mathbf{B E}G }$

in 2Cat. Here the top left item is the group $G$ regarded as a one-object groupoid. The bottom item is the one-object 2-groupoid with $G$ as its 1-morphisms and a unique 2-morphism between any two 1-morphisms.

Does a strict pushout exist? Or a slightly weak version?

Posted by: Urs Schreiber on March 27, 2008 7:48 PM | Permalink | Reply to this

Re: 101 things to do with a 2-classifier

Something we could do then to get categorified logic going is to reconsider a simple theory in predicate logic, such as that tale of poodles and owners, told here and here.

Dogs and their owners came in a very simple theory which took the form of a category, featuring prominently two objects and an arrow, interpreted as the owning function from dogs to people.

So it would be good to think of a theory where the basic action takes place in a 2-category. Perhaps if we had a couple of 1-morphisms going from one object to another and a 2-morphism between them, all of which gets interpreted in Cat. Any good candidates?

Posted by: David Corfield on January 20, 2008 5:45 PM | Permalink | Reply to this
Read the post What has happened so far
Weblog: The n-Category Café
Excerpt: A review of one of the main topics discussed at the Cafe: Sigma-models as the pull-push quantization of nonabelian differential cocycles.
Tracked: March 27, 2008 2:09 PM