# Main Wiki Dg-categories

## Dg-categories

### Preface

This page an experimental summary of Gustavo Jasso’s lecture on Dg-categories.

### Conventions

$k$ is a commutative ring.

### 1. Kernels and Cokernels

Let $\mathcal{C}$ be a pointed category. Define the functor $R_0\colon \mathcal{C} \to \operatorname{Mor}_\mathcal{C}$ by $x \mapsto (0\to x)$. Dually, define $L_0 \colon x\mapsto (x \to 0)$. A cokernel functor is a left adjoint $\operatorname{coker}$ of $R_0$. Dually, a kernel functor is a right adjoint of $L_0$.

There exists a cokernel functor if and only if every morphism in $\mathcal{C}$ has a cokernel.