nLab discrete category

Idea

The term discrete category (or discrete groupoid) is used in two different ways:

1. A category is discrete if it is equivalent to (or, sometimes, isomorphic to) a category whose only morphisms are identities; i.e., if small, to a set. In other words, this is a 0-truncated object of Cat; see also discrete object in a 2-category. This meaning is discussed below in Categorical meaning.

2. A category with additional “topological” or “cohesive” structure is discrete if that topological structure consists only of discrete spaces. This meaning is discussed below in Topological meaning.

See also the related discussion at discrete space.

Categorical meaning

A category is discrete if it is both a groupoid and a preorder. That is, every morphism should be invertible, any two parallel morphisms should be equal. The idea is that in a discrete category, no two distinct (nonisomorphic) objects are connectable by any path (morphism), and an object connects to itself only through its identity morphism.

Often one also assumes that a discrete category is skeletal; a category is both discrete and skeletal if and only if it contains only identity morphisms. However, this definition is evil, because it states that objects (the source and target of the identity morphism in question) are equal; it is cleaner to separate the discreteness from the skeletality.

A (small) discrete category may be identified with its set of isomorphism classes. Conversely, given a collection $S$ of objects, the discrete category over $S$ is the category with $S$ as its collection of objects and only identity morphisms.

Topological meaning

If $C$ is a category enriched or internal to topological spaces, then there is another completely different meaning of discrete: that the topology on the arrows (and the objects, in the internal case) is the discrete topology. In this sense a discrete category is an internal category in discrete spaces sitting inside a more general kind of spaces. This usage can be applied more generally for internal or enriched categories in any context with a notion of discrete space.

This is especially confusing if one extends the use of “discrete category over $S$” to the case of internal categories, when $S$ is an object of some ambient category. With this usage, if $S$ is a topological space, then the “discrete internal category over $S$” in Top will not be discrete in the topological sense: it still remembers the topology on that space.

Because of potential confusion, in cases when topology or cohesion is present it is perhaps better to use 0-truncated for categorically-discrete categories and groupoids, reserving discrete for those which are topologically-discrete.

Adjoints

The discrete category construction extends to a fully faithful functor $Set \to Cat$. It has a left adjoint given by taking connected components and a right adjoint given by the underlying set of objects functor (which itself has a right adjoint given by the indiscrete category construction).

Related concepts

• discrete space

• discrete group

• discrete groupoid, discrete object in a 2-category

• discrete ∞-groupoid