# Grothendieck monoids of extriangulated categories

arXiv preprint, August 2022. Link: arXiv:2208.02928

Citation: H. Enomoto, S. Saito, Grothendieck monoids of extriangulated categories, arXiv:2208.02928.

## Comment

As a generalization of triangulated and exact categories, we can naturally define the Grothendieck monoid $\mathsf{M}(\mathcal{C})$ of an extriangulated category $\mathcal{C}$. In this paper, we show the following.

• We can classify several subcategories of $\mathcal{C}$ using $\mathsf{M}(\mathcal{C})$:
• Serre subcategories (natural generalization of those in an abelian cat)
• Dense 2-out-of-3 subcategories (2-out-of-3 w.r.t conflations & additively generates $\mathcal{C}$)
• $\mathsf{M}(\mathcal{C})$ behaves well under the localization of $\mathcal{C}$ by a good subcategory $\mathcal{N}$, that is, $\mathsf{M}(\mathcal{C}/\mathcal{N})$ is isomorphic to the monoid quotient $\mathsf{M}(\mathcal{C})/\{[N] \mid N \in \mathcal{N}\}$. This can be applied to:
• The Serre quotient of an abelian category by a Serre subcategory.
• The Verdier quotient of a triangulated category by a thick subcategory.
• The stable category of a Frobenius category.
• For an abelian category $\mathcal{A}$ and its torsionfree class $\mathcal{F}$, we consider a subcategory $\mathcal{C} :=\mathcal{F}[1] * \mathcal{A}$ of the derived category of $\mathcal{A}$. This is extension-closed, so extriangulated. Then we show that $\mathsf{M}(\mathcal{C})$ is the monoid localization of $\mathsf{M}(\mathcal{A})$ with respect to $\{[F] \mid F \in \mathcal{F} \}$. Moreover, this subcategory is natural, since it is characterized by a subcategory $\mathcal{C}$ of the derived category with
• $\mathcal{C}$ is closed under extensions and direct summands.
• $\mathcal{A} \subseteq \mathcal{C} \subseteq \mathcal{A}[1] * \mathcal{A}$ holds. So it is a subcategory with width between $1$ and $2$ intuitively.

Also, we give some questions in the last section.