Category whose only morphisms are the identity morphisms

In mathematics, in the field of category theory, a **discrete category** is a category whose only morphisms are the identity morphisms:

- 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, we can express the above as condition on the cardinality of the hom-set

- | hom
_{C}(*X*, *Y*) | is 1 when *X* = *Y* and 0 when *X* is not equal to *Y*.

Some authors prefer a weaker notion, where a discrete category merely needs to be equivalent to such a category.

##
Simple facts

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. Thus, for example, the discrete category with just two objects can be used as a diagram or diagonal functor to define a product or coproduct of two objects. Alternately, for a general category **C** and the discrete category **2**, one can consider the functor category **C**^{2}. The diagrams of **2** in this category are pairs of objects, and the limit of the diagram is the product.

The functor from **Set** to **Cat** that sends a set to the corresponding discrete category is left adjoint to the functor sending a small category to its set of objects. (For the right adjoint, see indiscrete category.)