<p>you should have a look at <a href="http://ncatlab.org/nlab/show/forgetful+functor" rel="noreferrer">http://ncatlab.org/nlab/show/forgetful+functor</a>. As the nLab article says, in its common usage there is no precise definition of forgetful functor, though there are some typical examples (such as a functor $U : Alg \to Set$ forgetting some kind of algebraic structure, which will typically have a left adjoint $F : Set \to Alg$ corresponding to the construction of free algebras). And in another (useful) sense, <em>every</em> functor may be regarded as a forgetful functor, and classified according to how much it forgets.  There is no requirement that a functor be unfaithful for it to be called "forgetful", and many common examples of forgetful functors are faithful (such as the forgetful functor $Grp \to Set$ from groups to their underlying sets), while others are not (such as in general the functor $cod : C^\to \to C$ from the <a href="http://ncatlab.org/nlab/show/arrow+category" rel="noreferrer">arrow category</a> of $C$, forgetting everything about an arrow except its codomain).</p>

CC BY-SA 3.0
2014-04-18T12:22:56.840Z
2014-04-18T12:22:56.840Z
Noam Zeilberger
<p>In category theory, a <em>functor</em> between two categories $C,D$ is a map $F$ that assigns to each object (resp. morphism) $x$ of $C$ a corresponding object (resp. morphism) $F(x)$ of $D$ by respecting the incidence relations.</p>

<p>For each pair of objects $x,y$ of $C$, we may then define a map $F_{x,y}$ that takes any morphism $m : x \rightarrow y$ to a morphism $F(m) : F(x) \rightarrow F(y)$. I understand that $F$ is called <em>faithful</em> if every such mapping $F_{x,y}$ is injective, which means intuitively that the relational structure of the category is preserved, although the objects may not be.</p>

<p>There is a related notion of <em>forgetful</em> functor for which I couldn't find a precise definition, so is there anyone willing to help? I mean, is it just the opposite of faithful or is it the combination of unfaithfulness with some other implicit property?</p>

CC BY-SA 3.0
2014-04-17T13:39:15.747Z
See page 148 of Awodey (second edition) for a discussion that distinguishes functors that are forgetful, faithful, and injective on arrows.
András Salamon
<co.combinatorics><soft-question><ct.category-theory>
Faithful functors vs forgetful functors: exact category-theoretic defs?
Super8
# Faithful functors vs forgetful functors: exact category-theoretic defs?

(Asked by: username_0 on 2014-04-17)

In category theory, a *functor* between two categories $C,D$ is a map $F$ that assigns to each object (resp. morphism) $x$ of $C$ a corresponding object (resp. morphism) $F(x)$ of $D$ by respecting the incidence relations.

For each pair of objects $x,y$ of $C$, we may then define a map $F\_{x,y}$ that takes any morphism $m : x \rightarrow y$ to a morphism $F(m) : F(x) \rightarrow F(y)$. I understand that $F$ is called *faithful* if every such mapping $F\_{x,y}$ is injective, which means intuitively that the relational structure of the category is preserved, although the objects may not be.

There is a related notion of *forgetful* functor for which I couldn't find a precise definition, so is there anyone willing to help? I mean, is it just the opposite of faithful or is it the combination of unfaithfulness with some other implicit property?

**Tags:** co.combinatorics, soft-question, ct.category-theory

> **Comments:**
> **username_1**: See page 148 of Awodey (second edition) for a discussion that distinguishes functors that are forgetful, faithful, and injective on arrows.  

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## 1 answers

### Answer by username_2 on 2014-04-18. Score: 5

you should have a look at <http://ncatlab.org/nlab/show/forgetful+functor>. As the nLab article says, in its common usage there is no precise definition of forgetful functor, though there are some typical examples (such as a functor $U : Alg \to Set$ forgetting some kind of algebraic structure, which will typically have a left adjoint $F : Set \to Alg$ corresponding to the construction of free algebras). And in another (useful) sense, *every* functor may be regarded as a forgetful functor, and classified according to how much it forgets. There is no requirement that a functor be unfaithful for it to be called "forgetful", and many common examples of forgetful functors are faithful (such as the forgetful functor $Grp \to Set$ from groups to their underlying sets), while others are not (such as in general the functor $cod : C^\to \to C$ from the [arrow category](http://ncatlab.org/nlab/show/arrow+category) of $C$, forgetting everything about an arrow except its codomain).


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