| This resource is not discoverable in the library. If the resource was recently submitted, it may require some time before it becomes discoverable through this search service. | | Resource Title: | subobject classifier
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| Description: | ... Consider a set A and a subset ... . B can be thought of as a property of A: there is a function ... , such that ... iff ... . This function can be seen to be uniquely determined by the subset B, and conversely. If we denote ... the set of all subsets of A (the power set of A), and ... the set of all functions from A to ... , then ... . In fact, we have established a commutative diagram ... where ... and i are inclusion functions and k is the unique constant function. Any function ... gives rise to a unique set B making the above diagram commute. ... In category theory, a ... subobject classifier is the generalization of the above example, where A is an object of a certain given category ... and B is a subobject of A, ... is replaced by a terminal object, and 2 is replaced by what is known as a ... subobject classifier , or a ... truth object . If we think of the category ... Set , 2 ... elements of a given set as to whether they belong to a certain subset or not, via a characteristic function. If the value of the function is 1, then the element is in that subset, otherwise it is not. Formally, let ... be a category with a terminal object 1. A ... subobject classifier is an object Omega in ... such that, for any monomorphism ... , there exists a ... unique morphism chi_B such that ... is a pullback diagram. chi_B is called the ... characteristic morphism of f and top is a ... truth morphism . In a category with a terminal object 1, a subobject classifier may or may not exist. If it does, it is unique up to isomorphism. Suppose C has a terminal object 1, has pullbacks, and has a subobject Omega. Then for any object X in ... , any morphism ... gives rise to a unique monomorphism ... via the pull back of f and top: ... Since Omega is a subobject classifier, g determines f uniquely as well. So what we have is a one-to-one correspondence ... between the subobject functor and hom functor. It can be verified that the bijection is actually a natural isomorphism, so that Sub is a representable functor. Conversely, it may be shown that if Sub is representable, then C has a subobject classifier. |
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