Definition:Inclusion Functor

Definition

Let $\mathbf D$ be a metacategory, and let $\mathbf C$ be a subcategory of $\mathbf D$.

The inclusion functor on $\mathbf C$ is the functor $\Iota_{\mathbf C}: \mathbf C \to \mathbf D$ defined by:

			For all objects $C$ of $\mathbf C$:			$\ds \Iota_{\mathbf C} C $	$\ds := $	$\ds C $
			For all morphisms $f: C_1 \to C_2$ of $\mathbf C$:			$\ds \Iota_{\mathbf C} f $	$\ds := $	$\ds f $

1971: Saunders Mac Lane: Categories for the Working Mathematician: Chapter $\text I$ Categories, Functors and Natural Transformations: $\S \text{3}$ Functors

			For all objects $C$ of $\mathbf C$:			\(\ds \Iota_{\mathbf C} C \)	\(\ds := \)	\(\ds C \)
			For all morphisms $f: C_1 \to C_2$ of $\mathbf C$:			\(\ds \Iota_{\mathbf C} f \)	\(\ds := \)	\(\ds f \)