The n-Category Café

I’ll decide, just for definiteness, to work once and for all over complex finite dimensional vector spaces, which I take to live in the category $$\mathrm{Vect} \,.$$ A 2-vector space for me is a $\mathrm{Vect}$-module category. I am restricting attention to those 2-vector space which admit a basis which means that they look like $$\mathrm{Mod}_A$$ for some (finite dimensional, complex in my setup) algebra $A$. (That’s because $\mathrm{Mod}_A = \mathrm{Hom}(\Sigma A, \mathrm{Vect})$ just like an ordinary vector space $V$ with basis $S$ is $V \simeq \mathrm{Hom}(S, \mathbb{C})$).

One point I might emphasize is that this means we are looking at something more general that Kapranov-Voevodsky 2-vector spaces, which are those 2-vector spaces of the form $$\mathrm{Mod}_{\mathbb{C}^{\oplus n}}$$ only. The following simple application is supposed to illustrate what happens when we don’t restrict to the leftmost part of the chain of inclusions $$\mathrm{KV}2\mathrm{Vect} \hookrightarrow \mathrm{Bim} := 2\mathrm{Vect}_{b} \hookrightarrow 2\mathrm{Vect} \,.$$

So lets look at line 2-bundles (with connection and parallel transport). This are 2-vector bundles $$\array{ P_2(X) \\ \downarrow^{\mathrm{tra}} \\ 2\mathrm{Vect}_{b} }$$ with local structre (see The First Edge of the Cube for the general formalism used here) given by the canonical 2-representation (html) of shifted $U(1)$:

$$\rho : \Sigma \Sigma U(1) \to 2\mathrm{Vect}_b$$

$$\rho : \array{ & \nearrow \searrow^{\mathrm{Id}} \\ \bullet &\Downarrow^c& \bullet \\ & \searrow \nearrow_{\mathrm{Id}} } \;\;\; \mapsto \;\;\; \array{ & \nearrow \searrow^{\mathbb{C}} \\ \mathbb{C} &\Downarrow^{\cdot c}& \mathbb{C} \\ & \searrow \nearrow_{\mathbb{C}} } \,.$$

This simply meas that the typical fiber looks like $\mathbb{C}$ thought of as a placeholder for $\mathrm{Mod}_{\mathbb{C}} = \mathrm{Vect}$, i.e. like the canonical 1-dimensional 2-vector space, and that the local transitions of this fiber are given by the canonical action of $\Sigma U(1)$ on this.

But this then also means that the generic fiber of our 2-vector bundle is a gadget equivalent to our typical fiber. Since the equivalence is equivalence internal to $2\mathrm{Vect}_b := \mathrm{Bim}$, this means it is Morita equivalence of algebras.

Notice that the ground field is Morita equivalent to all algebras of the form $\mathrm{End}(V)$, with the weak equivalence induced by the bimodules

$$\mathbb{C} \stackrel{V}{\to} \mathrm{End}(V)$$

and

$$\mathrm{End}(V) \stackrel{V^*}{\to} \mathbb{C}$$

satisfying

$$\mathbb{C} \stackrel{V}{\to} \mathrm{End}(V) \stackrel{V^*}{\to} \mathbb{C} \simeq \mathbb{C} \stackrel{\mathbb{C}}{\to} \mathbb{C}$$

and

$$\mathrm{End}(V) \stackrel{V^*}{\to} \mathbb{C} \stackrel{V}{\to} \mathrm{End}(V) \simeq \mathrm{End}(V) \stackrel{\mathrm{End}(V)}{\to} \mathrm{End}(V)$$

(all of this taking place in $\mathrm{Bim}$).

So we find that the existence of a smooth local $\rho$-trivialization of our 2-vector bundle

$$\array{ P_2(X) \\ \downarrow^{\mathrm{tra}} \\ 2\mathrm{Vect}_{b} }$$

namely a completion to a square

$$\array{ P_2(Y) &\stackrel{\pi}{\to}& P_2(X) \\ \downarrow^{\mathrm{triv}} &\Downarrow^t& \downarrow^{\mathrm{tra}} \\ \Sigma \Sigma U(1) &\stackrel{\rho}{\to}& 2\mathrm{Vect}_{b} }$$

is (possibly recall the discussion at Higher Clifford Algebras at this point)

a vector bundle $$V \to Y$$ over the “cover” space $Y$ (some surjective submersion $\pi : Y \to X$) arising as the components of the local trivialization transformation $t$

$$t^{-1} : (y \in Y) \mapsto \;\; \mathbb{C} \stackrel{V_x}{\to} (\mathrm{End}(V) \simeq \mathrm{tra}(x))$$

which identitfies each algebra coming from the fibers of our 2-vector bundle, after being pulled back to $Y$, with the endomorphism algebra of a fixed vector space $V$.

From this local trivialization, we obtain now, by the general $n$-transport nonsense, the transition

$$g := \pi_2^* t^{-1} \circ \pi_1^* t$$

Over points $y \in Y^{[2]}$, its components are the vector spaces

$$g_y = \mathbb{C} \stackrel{L_y}{\to} \mathbb{C} := \mathbb{C} \stackrel{V_{\pi_2(y)}}{\to} \mathrm{tra}(\pi(\pi_1(y))) \stackrel{V^*_{\pi_1(y)}}{\to} \,.$$

These are 1-dimensional and equipped with a canonical product operation

$$\pi_{12}^* L \otimes \pi_{23}^* L \simeq \pi_{13}^* L$$

coming from forming the obvious filled triangle from the transition functions, as described in section: Parallel $n$-transport in String- and Chern-Simons $n$-transport.

In a flattened-out notation circumventing having to draw these triangles here in MathML, the operation consists of cancelling the center piece

$$\mathrm{End}(V) \stackrel{V^*}{\to} \mathbb{C} \stackrel{V}{\to} \mathrm{End}(V) \simeq \mathrm{End}(V) \stackrel{\mathrm{End}(V)}{\to} \mathrm{End}(V)$$

in

$$\begin{aligned} \pi_{12}^* L \otimes \pi_{23}^* L &= \mathbb{C} \stackrel{V_{\pi_1(y)}}{\to} \mathrm{tra}(\pi(y)) \stackrel{V^*_{\pi_2(y)}}{\to} \mathbb{C} \stackrel{V_{\pi_2(y)}}{\to} \mathrm{tra}(\pi(y)) \stackrel{V^*_{\pi_3(y)}}{\to} \mathbb{C} \\ & \simeq \mathbb{C} \stackrel{V_{\pi_1(y)}}{\to} \mathrm{tra}(\pi(y)) \stackrel{V^*_{\pi_3(y)}}{\to} \mathbb{C} \end{aligned}$$

So we see: the descent data on $Y$ for the local trivialization of the line 2-bundle we started with is nothing but a line bundle gerbe.

Moreover, we see that the local trivialization $$t^{-1} : (y \in Y) \mapsto \;\; \mathbb{C} \stackrel{V_x}{\to} (\mathrm{End}(V) \simeq \mathrm{tra}(x))$$ we started with is the corresponding gerbe module (this is the part we miss when working not with the full $2\mathrm{Vect}_b$): a vector bundle on $Y$ which descends to a twisted vector bundle on $X$ whose twist s precisely our bundle gerbe.

(From what I have said so far it is not clear that it is necessary for $t$ to define a smooth vector bundle on $Y$, but is sufficient for $L$ to be smooth. So if $V$ wasn’t in the first place, we can replace it after the fact with one that is.)

The whole discussion straightforwardly incorporates the 2-transport, too. See 2-Vector Transport and Line Bundle Gerbes. That’s the toy example of rank-1 2-vector bundles in terms of all of $2\mathrm{Vect}_b$.

And now I’ll go to bed. Need to get up real early tomorrow.