Definition
In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example the category of topological spaces and the category of groups, and trivially also the category of sets itself. On the other hand, the homotopy category of topological spaces is not concretizable, i.e. it does not admit a faithful functor to the category of sets.
Related concepts
Banach spacesCardinal numberCategory (category theory)Category of groupsCategory of relationsCategory of setsCategory of topological spacesClass (set theory)CodomainComplete latticeComplete latticesContraction (operator theory)Equivalence of categoriesFaithful functorForgetful functorFunctorGarrett BirkhoffGroup action (mathematics)HomotopyHomotopy category of topological spacesIdentity functorMathematical structureMathematicsMorphismNatural equivalenceNatural transformationPartial functionPermutation groupPeter FreydPosetProper classProperty (philosophy)Relation (mathematics)Saunders Mac LaneSet (mathematics)Signature (logic)Structure (mathematical logic)SubfunctorTopological spaceTopos theoryTotal functionUnit ballYoneda embedding
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