INVERSE IMAGE FUNCTOR Meaning and
Definition
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An inverse image functor is a concept used in category theory to describe a particular type of mapping between categories. In category theory, a functor is a structure-preserving mapping between categories. Specifically, an inverse image functor is a functor that maps objects and morphisms from a target category to a source category.
Given a functor F: C → D between categories C and D, the inverse image functor (also known as the pullback functor) denoted by F*: D → C is a mapping that associates to each object Y in D an object F*Y in C, and to each morphism g: Y → Z in D a morphism F*g: F*Y → F*Z in C. This functor essentially "pulls back" objects and morphisms from the target category D to the source category C.
The inverse image functor preserves certain properties of the original functor F. For example, if F is a continuous functor between topological spaces, then F* is a functor that preserves compactness and connectedness. Similarly, if F is a functor between abelian categories, then F* is an exact functor that preserves exact sequences.
By defining inverse image functors, category theory provides a powerful framework to study relationships between categories and their mappings. It allows for the understanding and analysis of various mathematical structures in a more abstract and general way, leading to insights and connections across different areas of mathematics.
