In mathematics, especially category theory, a discrete category is a category whose only morphisms are the identity morphisms. It is the simplest kind of category. Specifically a category C is discrete if
- homC(X, X) = {idX} for all objects X
- homC(X, Y) = ∅ for all objects X ≠ Y
Since by axioms, there is always the identity morphism between the same object, the above is equivalent to saying
- |homC(X, Y)| is 1 when X = Y and 0 when X is not equal to Y.
Clearly, any class of objects defines a discrete category when augmented with identity maps.
Any subcategory of a discrete category is discrete. Also, a category is discrete if and only if all of its subcategories are full.
The limit of any functor from a discrete category into another category is called a product, while the colimit is called a coproduct.
Famous quotes containing the words discrete and/or category:
“One can describe a landscape in many different words and sentences, but one would not normally cut up a picture of a landscape and rearrange it in different patterns in order to describe it in different ways. Because a photograph is not composed of discrete units strung out in a linear row of meaningful pieces, we do not understand it by looking at one element after another in a set sequence. The photograph is understood in one act of seeing; it is perceived in a gestalt.”
—Joshua Meyrowitz, U.S. educator, media critic. “The Blurring of Public and Private Behaviors,” No Sense of Place: The Impact of Electronic Media on Social Behavior, Oxford University Press (1985)
“I see no reason for calling my work poetry except that there is no other category in which to put it.”
—Marianne Moore (1887–1972)