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Dg-categories

Dg-categories

Preface

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

Conventions

kk is a commutative ring.

1. Kernels and Cokernels

Let 𝒞\mathcal{C} be a pointed category. Define the functor R 0:𝒞→Mor 𝒞R_0\colon \mathcal{C} \to \operatorname{Mor}_\mathcal{C} by x↦(0→x)x \mapsto (0\to x). Dually, define L 0:x↦(x→0)L_0 \colon x\mapsto (x \to 0). A cokernel functor is a left adjoint coker\operatorname{coker} of R 0R_0. Dually, a kernel functor is a right adjoint of L 0L_0.

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