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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.
It is a concrete category, i.e. a category obtained by adding some type of structure onto Set, and requiring that morphisms are functions that respect this added structure. The class of all groups with group homomorphisms as morphisms and function composition as the composition operation forms a large category, Grp. Like Ord, Grp is a concrete ...
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).
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.
Once reserved for athletes, eccentric exercise is becoming increasingly popular in everyday training and physical therapy—especially for people with musculoskeletal conditions like Parkinson’s ...
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From January 2008 to September 2008, if you bought shares in companies when Leslie Rahl joined the board, and sold them when she left, you would have a -81.4 percent return on your investment, compared to a -13.0 percent return from the S&P 500.