# Categorical Algebra and its Applications by Francis Borceux download in iPad, ePub, pdf

Every retraction is an epimorphism, and every section is a monomorphism. Functor Functors are structure-preserving maps between categories. Each morphism f has a source object a and target object b.

Morphisms can have any of the following properties. Some authors deviate from the definition just given by identifying each object with its identity morphism.

In a narrow sense, a categorical algebra is an associative algebra, defined for any locally finite category and a commutative ring with unity. Double groupoids are often used to capture information about geometrical objects such as higher-dimensional manifolds or n-dimensional manifolds. They can be thought of as morphisms in the category of all small categories. Noncommutative double groupoids and double algebroids are only the first examples of such higher dimensional structures that are nonabelian.

From the axioms, it can be proved that there is exactly one identity morphism for every object.

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