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

- hom
_{C}(*X*,*X*) = {id_{X}} for all objects*X* - hom
_{C}(*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

- |hom
_{C}(*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.

### Other articles related to "discrete category":

... If J is a (small)

**discrete category**, then a diagram of type J is essentially just an indexed family of objects in C (indexed by J) ... So, for example, when J is the

**discrete category**with two objects, the resulting limit is just the binary product ... B and the two arrows B → A, B → C, the resulting diagram would simply be the

**discrete category**with the two objects A and C, and the colimit would simply be the ...

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

“The truth is, no matter how trying they become, babies two and under don’t have the ability to make moral choices, so they can’t be “bad.” That *category* only exists in the adult mind.”

—Anne Cassidy (20th century)

“We have good reason to believe that memories of early childhood do not persist in consciousness because of the absence or fragmentary character of language covering this period. Words serve as fixatives for mental images. . . . Even at the end of the second year of life when word tags exist for a number of objects in the child’s life, these words are *discrete* and do not yet bind together the parts of an experience or organize them in a way that can produce a coherent memory.”

—Selma H. Fraiberg (20th century)