Editing Groupoid (section)

=== Homological algebra ===
A two term complex
: <math>C_1 ~\overset{d}{\rightarrow}~ C_0</math>
of objects in a [[Concrete category|concrete]] [[Abelian category]] can be used to form a groupoid. It has as objects the set <math>C_0</math> and as arrows the set {{tmath|1= C_1\oplus C_0 }}; the source morphism is just the projection onto <math>C_0</math> while the target morphism is the addition of projection onto <math>C_1</math> composed with <math>d</math> and projection onto {{tmath|1= C_0 }}. That is, given {{tmath|1= c_1 + c_0 \in C_1\oplus C_0 }}, we have
: <math>t(c_1 + c_0) = d(c_1) + c_0.</math>
Of course, if the abelian category is the category of [[coherent sheaves]] on a scheme, then this construction can be used to form a [[presheaf]] of groupoids.