Noun

(en noun)

An abstract sphere of influence, real or imagined.

* 1907 , Tada Kanai, translated by Arthur Lloyd, Seven Buddhist Sermons , ""

Why should we despise anything in the realm of Buddha?

* 2006 , Christian Neef, " Diary of a Collapsing Superpower", Spiegel Magazine , November 22,

At home in Moscow, Mikhail Sergeyevitch Gorbachev, who had launched a campaign to rejuvenate the Soviet realm  

* {{quote-magazine, date=2013-05-17

, author=George Monbiot, authorlink=George Monbiot , title=Money just makes the rich suffer , volume=188, issue=23, page=19 , magazine=(The Guardian Weekly) citation , passage=In order to grant the rich these pleasures, the social contract is reconfigured. […]  The public realm is privatised, the regulations restraining the ultra-wealthy and the companies they control are abandoned, and Edwardian levels of inequality are almost fetishised.}}

The domain of a certain abstraction.

* 1922 , Judson Eber Conant,The Church The Schools and Evolution , "",

One thing more which the scientific man does is to accord primacy to that realm of truth which is primary in importance.

(formal, or, legal) A territory or state, as ruled by a specific power, and particularly those territories ruled by a king.

* 1874 , ,

And, of this island realm , he and his companion were the undisputed sovereigns.

* 1913 , Leslie Alexander Toke, Catholic Encyclopedia , "",

Then seeing his life was threatened he fled the realm and crossed over to Flanders,

Noun

(categories)

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

*

The traditional way of describing the similarities and differences between constituents is to say that they belong to categories'' of various types. Thus, words like ''boy'', ''girl'', ''man'', ''woman'', etc. are traditionally said to belong to the category''' of Nouns, whereas words like ''a'', ''the'', ''this'', and ''that'' are traditionally said to belong to the ' category of Determiners.

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.

(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.

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.