Category theory; homological algebra Definition

Forgetful functor

Definition

Let $\sigma$ be an algebraic signature and let $K$ be a class of $\sigma$ -algebras. Let $\mathcal{K}$ be the category whose objects are the members of $K$ and whose morphisms are the homomorphisms between them. The forgetful functor $U \colon \mathcal{K} \to \mathbf{Set}$ is the functor that sends each algebra $A$ to its underlying set. It sends each homomorphism $f \colon A \to B$ to the same map $f$ between the underlying sets.

More generally, let $\sigma' \subseteq \sigma$ . The functor that sends a $\sigma$ -algebra to its reduct to the operations of $\sigma'$ , and each homomorphism to itself, is also called forgetful.

Remarks

$U$ is a functor because the identity homomorphism is the identity map and the composite of homomorphisms is their composite as maps. It is faithful: two homomorphisms $A \to B$ with the same underlying map are equal.

Example: the forgetful functor $\mathbf{Ring} \to \mathbf{Ab}$ keeps the addition of a ring and drops its multiplication and unit. A ring homomorphism is additive, so it is sent to a group homomorphism.

Non-example: consider the functor $\mathbf{Grp} \to \mathbf{Set}$ that sends a group $G$ to the set of its subgroups, and a homomorphism $f$ to $H \mapsto f(H)$ . It is a functor, but not a forgetful functor. It does not send $G$ to its underlying set.

The term is also used informally for functors such as $\mathbf{Top} \to \mathbf{Set}$ , which forget a topology rather than operations.

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Category concept
Functor concept
Homomorphism concept
Forgetful functor definition

Tag	`26RWFX4H2H`
URI	`m2://26RWFX4H2H`
Kind	definition
Writing	W0 stub
MSC	18A22, 08C05
Defines	Forgetful functor
Uses	Category, Functor, Homomorphism

Forgetful functor

Definition

Remarks

Reading path

Further reading

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