Just as a monoid consists of an underlying set with a binary operation 'on top of it' which is closed, associative and with an identity, a category consists of an underlying digraph with an arrow composition operation 'on top of it' which is transitively closed, associative, and with an identity at each object. In fact, a category's composition operation, when restricted to a single one of its objects, turns that object's set of arrows (which would all be loops) into a monoid. 6'7 height. Synonyms [ ] • ( group to which items are assigned ):,,,,,,,,, • See also Hyponyms [ ].

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Schematic representation of a category with objects X, Y, Z and morphisms f, g, g ∘ f. (The category's three identity morphisms 1 X, 1 Y and 1 Z, if explicitly represented, would appear as three arrows, from the letters X, Y, and Z to themselves, respectively.) Category theory formalizes and its concepts in terms of a labeled called a, whose nodes are called objects, and whose labelled directed edges are called arrows (or ). A has two basic properties: the ability to the arrows, and the existence of an arrow for each object. The language of category theory has been used to formalize concepts of other high-level such as,,.

Informally, category theory is a general theory of. Several terms used in category theory, including the term 'morphism', are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself. And introduced the concepts of categories,, and in 1942–45 in their study of, with the goal of understanding the processes that preserve mathematical structure. Category theory has practical applications in, for example the usage of. It may also be used as an axiomatic foundation for mathematics, as an alternative to and other proposed foundations.

Contents • • • • • • • • • • • • • • • • • • • • • • • • Basic concepts [ ] Categories represent abstractions of other mathematical concepts. Many areas of mathematics can be formalised by category theory as. Hence category theory uses abstraction to make it possible to state and prove many intricate and subtle mathematical results in these fields in a much simpler way. A basic example of a category is the, where the objects are sets and the arrows are functions from one set to another.