##  [Cohomology Theory](/cohomology-theory) 

  ##  [Cohomology Theory](https://mathlogic.quantumdictionary.io/cohomology-theory-0) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A contravariant assignment (usually a functor) from a category of spaces (topological, differentiable, or algebraic) to graded abelian groups, modules, or rings that captures global obstruction classes and supports natural product structures and long exact sequences.

 

 

 

 

 





 

 



 ##  [Cohomology Theory](https://puremath.quantumdictionary.io/cohomology-theory-1) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A contravariant algebraic invariant that assigns graded groups or rings to spaces (or spectra) and satisfies axioms such as homotopy invariance and exactness in classical forms; cohomology theories record global obstructions, classify structures, and carry multiplicative or cohomological operations.