Contents

Yoneda lemma

# Contents

## Idea

On the one hand, a map into a classifying space classifies some kind of object, it is a classifying morphism, meaning that it characterizes the object (only) up to equivalence.

On the other hand, the more refined concept of a moduli stack is such that morphisms into it characterize the object itself. Hence it makes sense to say that such maps into moduli stacks not just classify, but modulate the given object.

Indeed, that is the idea that originally gave rise to the name moduli.
Because, more specifically, a moduli stack $\mathbf{F}$ of a certain kind of objects is such that morphisms $X \stackrel{\chi}{\longrightarrow} \mathbf{F}$ into it determine a bundle

$\array{ P \\ \downarrow \\ X }$

of $\mathbf{F}$-like objects over $X$. It is this bundle which is “being modulated (by $\chi$) as one moves around in $X$”, much as in the language of electronics a waveform is being modulated as one moves around in time.

## Examples

### Metric structure

To see that the difference really matters, consider the map of classifying spaces

$B O(n) \longrightarrow B GL(n)$

for the orthogonal group and the general linear group. These classify, respectively, $O(n)$-principal bundles and $GL(n)$-principal bundles. While as geometric (e.g. topological or smooth) bundles these are different, their equivalence classes are the same. Accordingly the above map is in fact a homotopy equivalence and accordingly for $\Sigma$ any $n$-dimensional manifold whose tangent bundle is classified by $\iota \tau_\Sigma \colon \Sigma \longrightarrow B GL(n)$, then the space of lifts $\iota \hat \tau_\Sigma$ in

$\array{ && B O(n) \\ & {}^{\mathllap{\iota \hat \tau_\Sigma}} \nearrow & \downarrow \\ \Sigma &\stackrel{\iota \tau_\Sigma}{\longrightarrow}& B GL(n) }$

is contractible. Hence classifying maps see no difference here.

However, there is an important difference which the modulating morphisms do see. Write

$\mathbf{B}O(n) \longrightarrow \mathbf{B}GL(n)$

for the corresponding morphism of smooth moduli stacks (see at looping and delooping for more on this). Then a lift $e$ of the modulating map $\tau_\Sigma \colon \Sigma \longrightarrow \mathbf{B}GL(n)$ to one that modulates an actual orthogonal bundle

$\array{ && \mathbf{B} O(n) \\ & {}^{\mathllap{e}} \nearrow & \downarrow \\ \Sigma &\stackrel{\tau_\Sigma}{\longrightarrow}& \mathbf{B} GL(n) }$

is genuine data: this is a choice of orthogonal structure/vielbein on $\Sigma$ and hence a Riemannian metric on $\Sigma$.

### $G$-Structure for maximal compact subgroup inclusions

An analogous situation is obtained for any inclusion of a maximal compact subgroup into a given Lie group and the corresponding notion of G-structure. All $G$-structures arising this way are invisible to classifying maps, but are seen by modulating maps. See at twisted differential c-structure for a list of further examples

The terminology “modulating” in the context of “moduli stacks” would seem to be inevitable but is not used much in practice. One place where it is used consistently is