Contents

category theory

# Contents

## Idea

### In terms of 2-Segal spaces

Given a 2-Segal space $X_\bullet$ such that the spans

$X_1 \times X_1 \stackrel{(\partial_2, \partial_0)}{\leftarrow} X_2 \stackrel{\partial_1}{\rightarrow} X_1$

and

$pt \leftarrow X_0 \stackrel{s_0}{\to} X_1$

admit pull-push integral transforms in some given cohomology theory $h$. Then the Hall algebra of $X$ with coefficients in $H$ is the associative algebra structure on $h(X_1)$ induced by these pull-push operations.

This is the perspective of Dyckerhoff-Kapranov 12, def. 8.1.8.

### Motivic Hall algebra

Specifically for a given algebraic stack $X$ and with

$X \leftarrow X^{(2)} \rightarrow X\times X$

denoting the moduli stack of 2-flags of coherent sheaves on $X$, then the corresponding pull-push multiplication on the motivic Grothendieck ring $K(X)$ is called the motivic Hall algebra of $X$ (due to Dominic Joyce reviewed e.g. in Bridgeland 10, 4.2). Discussion of motivic Hall algebras of Calabi-Yau 3-folds is in (Kontsevich-Soibelman 08).

### In terms of constructible sheaves

The Hall algebra of an abelian category is the Grothendieck group of constructible sheaves/perverse sheaves on the moduli stack of objects in the category. The Hall algebra is an algebra because the constructible derived category of the moduli stack of objects in an abelian category is monoidal in a canonical way.

This perspective is taken from (Webster11). See there for more details.

## References

A good survey is given in

The characterization via 2-Segal spaces is due to

Canonical references on Hall algebras include the following.

Motivic Hall algebras:

Last revised on September 30, 2018 at 03:07:36. See the history of this page for a list of all contributions to it.