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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".
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 ...
Category: Concrete. 56 languages. ... Download as PDF; Printable version; ... This category has the following 15 subcategories, out of 15 total. ...
A linear function u : E 1 → E 2 between vector spaces is entirely determined by its values on a basis of the vector space E 1. 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 ...
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.
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).
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This is a list of building materials.. Many types of building materials are used in the construction industry to create buildings and structures.These categories of materials and products are used by architects and construction project managers to specify the materials and methods used for building projects.