Category of chain complexes
Encyclopedia
In mathematics, chain complexes arise naturally in topology and geometry. For example, homology and cohomology theories all make use of chain complexes. To define a chain complex, fix an abelian category, say the category of modules over a commutative ring. Let Mi be a series of objects in this category with maps 
such that the composition of two consecutive maps is zero. Then the collection of Mi along with the morphisms f is called a chain complex
. There is a natural notion of a morphism between chain complexes called a chain map. Given two complexes M* and N*, a chain map between the two is a series of homomorphisms from Mi to Nj such that the entire diagram commutes. Chain complexes with chain maps form a category.
such that the composition of two consecutive maps is zero. Then the collection of Mi along with the morphisms f is called a chain complexChain complex
In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
. There is a natural notion of a morphism between chain complexes called a chain map. Given two complexes M* and N*, a chain map between the two is a series of homomorphisms from Mi to Nj such that the entire diagram commutes. Chain complexes with chain maps form a category.