This paper discusses the formalization of proofs "by diagram chasing", a
standard technique for proving properties in abelian categories. We discuss how
the essence of diagram chases can be captured by a simple many-sorted
first-order theory, and we study the models and decidability of this theory.
The longer-term motivation of this work is the design of a computer-aided
instrument for writing reliable proofs in homological algebra, based on
interactive theorem provers