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forgetful functor
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(Definition)
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Let $\mathcal{C}$ and $\mathcal{D}$ be categories such that each object $c$ of $\mathcal{C}$ can be regarded an object of $\mathcal{D}$ by suitably ignoring structures $c$ may have as a $\mathcal{C}$ object but not a $\mathcal{D}$ object. A functor $U:\mathcal{C} \to \mathcal{D}$ which operates on objects of $\mathcal{C}$ by ``forgetting'' any
imposed mathematical structure is called a forgetful functor. The following are examples of forgetful functors:
- $U:\mathbf{Grp} \to \mathbf{Set}$ takes groups into their underlying sets and group homomorphisms to set maps.
- $U:\mathbf{Top} \to \mathbf{Set}$ takes topological spaces into their underlying sets and continuous maps to set maps.
- $U:\mathbf{Ab} \to \mathbf{Grp}$ takes abelian groups to groups and acts as identity on arrows.
Forgetful functors are often instrumental in studying adjoint functors.
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"forgetful functor" is owned by RevBobo.
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Cross-references: adjoint functors, identity, abelian groups, continuous maps, topological spaces, maps, group homomorphisms, groups, functor, structures, object, categories
There are 6 references to this entry.
This is version 1 of forgetful functor, born on 2002-05-17.
Object id is 2910, canonical name is ForgetfulFunctor.
Accessed 6272 times total.
Classification:
| AMS MSC: | 18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations) |
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Pending Errata and Addenda
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