# The Azimuth Project Localizations of tensor categories and fiber products of schemes

## Main Result

Theorem A. Let $\mathbb{K}$ be a commutative ring and let $X$ and $Y$ be two quasi-compact quasi-separated $\mathbb{K}$-schemes over some quasi-compact quasi-separated $\mathbb{K}$-scheme $S$. Then $\mathsf{Qcoh}(X \times_\mathbb{K} Y)$ is the bicategorical pullback of $\mathsf{Qcoh}(X)$ and $\mathsf{Qcoh}(Y)$.

## Idals

The term ideals is introduced. The name is meant to imply that its like ideals but without “embedding”.

Definition 2.1 Let $\mathcal{C}$ be a tensor category with unit object $\mathcal{O}_\mathcal{C}$. An idal is a morphism $e \colon I \to \mathcal{O}_\mathcal{C}$ such that $\lambda \circ (e \otimes I) = \rho \circ (I \otimes e)$.

This condition is better expressed as a commuting square or as a string diagram.