A category is itself a type of mathematical structure, so we can look for "processes" which preserve this structure in some sense; such a process is called a functor.
This process can be extended for all natural numbers n, and these are called n-categories. There is even a notion of ω-category corresponding to the ordinal number ω.
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Anonymous2013-08-31 19:24
Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
As in real analysis, in complex analysis the symbol \infty, called "infinity", denotes an unsigned infinite limit. x ightarrow \infty means that the magnitude |x| of x grows beyond any assigned value. A point labeled \infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere.
The next wave of excitement in set theory came around 1900, when it was discovered that Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself, and not a member of itself.
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Anonymous2013-08-31 22:24
Determinacy refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from the start in the sense that one player must have a winning strategy. The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property.
For any set A, the power set of A (with the empty set removed) has a choice function.
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Anonymous2013-08-31 23:54
In class theories such as Von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory, there is a possible axiom called the axiom of global choice which is stronger than the axiom of choice for sets because it also applies to proper classes. And the axiom of global choice follows from the axiom of limitation of size.
Now, consider stronger forms of the negation of AC. For example, if we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than ¬AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets.
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Anonymous2013-09-01 1:25
Identity: For every object x, there exists a morphism 1x : x → x called the identity morphism for x, such that for every morphism f : a → b, we have 1b ∘ f = f = f ∘ 1a.
If a morphism f has domain X and codomain Y, we write f : X → Y. Thus a morphism is represented by an arrow from its domain to its codomain. The collection of all morphisms from X to Y is denoted homC(X,Y) or simply hom(X, Y) and called the hom-set between X and Y. Some authors write MorC(X,Y) or Mor(X, Y). Note that the term hom-set is a bit of a misnomer as the collection of morphisms is not required to be a set.