Definition:Subcategory

Definition

Definition 1

Let $\mathbf C$ be a metacategory.

A subcategory of $\mathbf C$ is a metacategory with:

			Objects:					Any collection of objects from $\mathbf C$
			Morphisms:					Any collection of morphisms from $\mathbf C$
			Composition:					Inherited from $\mathbf C$
			Identity morphisms:					Inherited from $\mathbf C$

Colloquially, a subcategory of $\mathbf C$ is a "part of $\mathbf C$ that is a category in its own right".

Definition 2

Let $\mathbf C$ be a metacategory.

A subcategory of $\mathbf C$ is a monic functor $F: \mathbf D \to \mathbf C$ to $\mathbf C$

Also see

• Full Subcategory