Homology commutes with direct limits

Statement

Each of the homology functors commutes with direct limits, viz the $n^{th}$ homology group of a direct limit of topological spaces is the direct limit of their $n^{th}$ homology groups.

The statement encodes two facts:

• The singular chain complex functor commutes with direct limits; viz the direct limit of the singular chain complexes of a diagram of topological spaces, is the singular chain complex of the direct limit
• The homology functor on a chain complex of Abelian groups, commutes with direct limits.

Related facts

• Cohomology ring functor commutes with direct limits