The difference between Category and Monoid

When used as nouns, category means a group, often named or numbered, to which items are assigned based on similarity or defined criteria, whereas monoid means a set which is closed under an associative binary operation, and which contains an element which is an identity for the operation.

Monoid is also adjective with the meaning: containing only one kind of metrical foot.

check bellow for the other definitions of Category and Monoid

  1. Category as a noun:

    A group, often named or numbered, to which items are assigned based on similarity or defined criteria.

    Examples:

    "This steep and dangerous climb belongs to the most difficult category."

    "I wouldn't put this book in the same category as the author's first novel."

  2. Category as a noun (mathematics):

    A collection of objects, together with a transitively closed collection of composable arrows between them, such that every object has an identity arrow, and such that arrow composition is associative.

    Examples:

    "One well-known category has sets as objects and functions as arrows."

    "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."

  1. Monoid as a noun (mathematics):

    A set which is closed under an associative binary operation, and which contains an element which is an identity for the operation.

  1. Monoid as an adjective (prosody):

    Containing only one kind of metrical foot.

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