category theory

# Contents

## Idea

### In terms of 2-Segal spaces

Given a 2-Segal space ${X}_{•}$ such that the spans

${X}_{1}×{X}_{1}\stackrel{\left({\partial }_{2},{\partial }_{0}\right)}{←}{X}_{2}\stackrel{{\partial }_{1}}{\to }{X}_{1}$X_1 \times X_1 \stackrel{(\partial_2, \partial_0)}{\leftarrow} X_2 \stackrel{\partial_1}{\rightarrow} X_1

and

$\mathrm{pt}←{X}_{0}\stackrel{{s}_{0}}{\to }{X}_{1}$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 $h\left({X}_{1}\right)$ induced by these pull-push operations.

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

### 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.

Revised on May 14, 2013 11:49:29 by Urs Schreiber (82.169.65.155)