**Category Theory**

A variant of this approach replaces relations with operations, the basis of universal algebra. In this variant the axioms often take the form of equations, or implications between equations.

A more abstract variant is category theory, which abstracts sets as objects and the operations thereon as morphisms between those objects. At this level of abstraction mathematical objects reduce to mere vertices of a graph whose edges as the morphisms abstract the ways in which those objects can transform and whose structure is encoded in the composition law for morphisms. Categories may arise as the models of some axiomatic theory and the homomorphisms between them (in which case they are usually concrete, meaning equipped with a faithful forgetful functor to the category **Set** or more generally to a suitable topos), or they may be constructed from other more primitive categories, or they may be studied as abstract objects in their own right without regard for their provenance.

Read more about this topic: Mathematical Object

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### Famous quotes containing the words theory and/or category:

“Could Shakespeare give a *theory* of Shakespeare?”

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