# Contents

## Idea

Let $S$ be a noetherian scheme. The h-topology is a Grothendieck topology on the category of schemes, used by Vladimir Voevodsky to construct a triangulated category $DM$ of mixed motives.

## Definition

A morphism $p : X \to Y$ of schemes is called a topological epimorphism? if the underlying topological space of $Y$ is a quotient space of the underlying topological space of $X$, i.e. $p$ is surjective and a subset $A \subset Y$ is open iff its preimage $p^{-1}(A) \subset X$ is open. Such a morphism is further called a universal topological epimorphism if this property is preserved under any base change.

Consider the pretopology on the category of schemes whose covering families are finite families $(p_i : U_i \to X)_i$ such that $p_i$ are of finite type and $\sqcup p_i : \sqcup U_i \to X$ is a universal topological epimorphism?. The Grothendieck topology generated by this pretopology is called the h-topology on the category of schemes.

The h-topology is stronger than the etale and proper topologies.