The n-Category Café

Great, thanks.

That reminds me: my latest thinking # about higher gauge theory has made it clear to me once again (I had talked about it before #):

The set $U(1)$ is an illusion.

What it really is is the groupoid $\mathbb{R}//\mathbb{Z}$.

sitting in the short exact sequence of 2-groups

$$1 \to \mathbb{R} \to \mathbb{R}//\mathbb{Z} \to \mathbf{B} \mathbb{Z} \to 1 \,.$$

You seem to be conflating ‘categorified mathematical physics’ with ‘phrasing physics category theoretically’. I’m very interested in both, but categorification — taking ideas and pushing them further up the $n$-categorical ladder by replacing equations with isomorphisms — is different from merely taking ideas and phrasing them in terms of category theory. The latter is a prerequisite for the former.

If phrasing existing physics in terms of categories catches on, this could just be the logical extension of the Gruppenpest that hit quantum theory many decades ago. We can do physics without talking about categories, just as people did quantum mechanics without talking about groups… but we understand things more clearly and simply when we use symmetry concepts, and categories are a useful generalization of groups.

If categorified physics catches on, something more interesting may be at work. Namely, a dethronement of the concept of ‘equality’, where the static concept of ‘$x$ is $y$’ is completely replaced by the dynamic ‘$f$ is a process whereby $x$ becomes $y$’.’ And this goes straight to the root of physics. After all, the Greek word physis meant something like ‘becoming’.

Thanks very much for that summary! So, I suppose we’re trying to prove the following:

Conjecture. A topos is equivalent to $Set^{\mathbf{G}}$ for some finite groupoid G iff it has a finite number of connected objects.

Of course, I’m using ‘number’ to mean ‘number of isomorphism classes of’. A connected object is an object $X$ such that Hom$(X,-)$ preserves coproducts, as Todd Trimble explained in his post. He also explained that a projective object is an object $X$ such that Hom$(X,-)$ preserves coequalisers, and that a tiny object is an object which is connected and projective.

To prove the conjecture, we therefore need to show that in a topos with a finite number of connected objects, the tiny objects form a dense subcategory.

I’m stuck at the first hurdle, just showing that there must be any tiny objects at all. In a category of presheaves of a finite groupoid G, my intuition (i.e. what I’ve stolen from other people’s posts) is that the connected objects are the transitive G-actions, and the tiny objects are the simply-transitive actions for one of the groups in the groupoid G, which are just the regular representations for these groups. It seems to me that a morphism $f:A \rightarrow B$ between transitive G-actions must be surjective, and will only be injective if $A \simeq B$ as G-actions. So if $f:A \rightarrow B$, then we know that $|A| \ge |B|$. We can use this to organise our transitive G-actions into a partially-ordered set, with $B \lt A$ iff |$\mathrm{Hom}(B,A)$|$= 0$. The tiny actions are then actually the biggest actions, the ones without any action above them in the poset — so maybe tiny isn’t such a such a good name!

But how can these tiny objects be identified categorically? Perhaps as the colimit of the full subcategory of the connected objects?

I also wanted to say something about spotting the index category C that a presheaf category $Set^{C ^{op}}$ arises from, because I don’t think anyone’s said it yet: it’s always just the smallest full generating subcategory. This is, in fact, just the Yoneda embedding of the index category into the presheaf category, as far as I remember. This is probably equivalent to what Denis-Charles Cisinski said, but it’s worth saying it explicitly.

It sounds like you’d be happy if something like this were true: a group is finite iff it has finitely many isomorphism classes of transitive actions.

I would really like to know if this is true — but luckily, I don’t quite require this! All I need is a way to take a finitary topos T with a finite number of isomorphism classes of connected object, and from it construct a groupoid G such that T$\simeq$$Set^G$. If there’s also an infinite groupoid that happens to have the same finite transitive actions, it doesn’t really matter!

I guess I’ll give a partial answer to this. If $C$ is a finite category, then clearly $C$ is $Fin$-enriched, and there is a Yoneda embedding

$$C \to Fin^{C^{op}}$$

obtained by transforming the hom $C(-, -): C^{op} \times C \to Fin$. The category $Fin^{C^{op}}$ is finitely complete, since $Fin$ is.

Proposition: $Fin^{C^{op}}$ is the free finite-cocompletion of $C$. More precisely, $y_C$ is 2-universal among functors $G: C \to U D$, where $U$ is the forgetful 2-functor (from the 2-category of finitely cocomplete categories $D$ and finitely cocontinuous functors to $Cat$).

Proof. Each $F: C^{op} \to Fin$ can be represented as a finite colimit of representables $C(-, y)$: there is an exact sequence

$$\sum_{y, z \in C} C(-, y) \times C(y, z) \times F(z) \stackrel{\to}{\to} \sum_{z \in C} C(-, z) \times F(z) \to F(-)$$

which expresses $F$ as a coequalizer of maps between finite sums of representables. (One of the maps involves composition in $C$, and the other involves the contravariant action $C(y, z) \times F(z) \to F(y)$ of $C$ on $F$.)

Given a functor $G: C \to D$ to a finitely cocomplete $D$, we define an extension $\hat{G}: Fin^{C^{op}} \to D$ of $G$ along the Yoneda embedding, in the only way which preserves this colimit: namely $\hat{G}(F)$ is the evident coequalizer $G \otimes_C F$ in $D$:

$$\sum_{y, z \in C} G(y) \times C(y, z) \times F(z) \stackrel{\to}{\to} \sum_{z \in C} G(z) \times F(z) \to G \otimes_C F.$$

(The $\times$ is a slight misnomer here; if $S$ is a finite set and $d$ is an object of $D$, then $d \times S$ just means the coproduct of $S$ copies of $d$.) One easily checks that $\hat{G}(-) = G \otimes_C (-)$ is finitely cocontinuous. But since finite cocontinuity forced $\hat{G}(-)$ to be defined this way (up to unique isomorphism), it is the unique finitely cocontinuous extension up to unique isomorphism, as desired.