Discrete category

---

title: "Discrete category" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["categories-in-category-theory"] description: "Category whose only morphisms are the identity morphisms" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Discrete_category" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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

:| homC(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.)

References

• Robert Goldblatt (1984). Topoi, the Categorial Analysis of Logic (Studies in logic and the foundations of mathematics, 98). North-Holland. Reprinted 2006 by Dover Publications, and available online at Robert Goldblatt's homepage.

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::