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A concrete category is a pair (C,U) such that . C is a category, and; U : C → Set (the category of sets and functions) is a faithful functor.; The functor U is to be thought of as a forgetful functor, which assigns to every object of C its "underlying set", and to every morphism in C its "underlying function".
The following definition translates this to any category. A concrete category is a category that is equipped with a faithful functor to Set, the category of sets. Let C be a concrete category with a faithful functor U : C → Set. Let X be a set (that is, an object in Set), which will be the basis of the free object to be defined.
Pages in category "Concrete" The following 200 pages are in this category, out of approximately 221 total. This list may not reflect recent changes.
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Set is the prototype of a concrete category; other categories are concrete if they are "built on" Set in some well-defined way. Every two-element set serves as a subobject classifier in Set. The power object of a set A is given by its power set, and the exponential object of the sets A and B is given by the set of all functions from A to B.
In concrete categories, one can thus take a subset of K ′ for K, in which case, the morphism k is the inclusion map. This allows one to talk of K as the kernel, since k is implicitly defined by K. There are non-concrete categories, where one can similarly define a "natural" kernel, such that K defines k implicitly.
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The zero objects in Grp are the trivial groups (consisting of just an identity element). Every morphism f : G → H in Grp has a category-theoretic kernel (given by the ordinary kernel of algebra ker f = { x in G | f ( x ) = e }), and also a category-theoretic cokernel (given by the factor group of H by the normal closure of f ( G ) in H ).