9.5.3 Forgetful Functors

Definition 9.5.3.1.1 ▶ Forgetful Functors

There isn’t a precise definition of a forgetful functor.

Remark 9.5.3.1.2 ▶ Unwinding Definition 9.5.3.1.1

Despite there not being a formal or precise definition of a forgetful functor, the term is often very useful in practice, similarly to the word “canonical”. The idea is that a “forgetful functor” is a functor that forgets structure or properties, and is best explained through examples, such as the ones below (see Example 9.5.3.1.3 and Example 9.5.3.1.4).

Example 9.5.3.1.3 ▶ Forgetful Functors That Forget Structure

Examples of forgetful functors that forget structure include:

  1. Forgetting Group Structures. The functor $\mathsf{Grp}\to \mathsf{Sets}$ sending a group $\webleft (G,\mu _{G},\eta _{G}\webright )$ to its underlying set $G$, forgetting the multiplication and unit maps $\mu _{G}$ and $\eta _{G}$ of $G$.
  2. Forgetting Topologies. The functor $\mathsf{Top}\to \mathsf{Sets}$ sending a topological space $\webleft (X,\mathcal{T}_{X}\webright )$ to its underlying set $X$, forgetting the topology $\mathcal{T}_{X}$.
  3. Forgetting Fibrations. The functor $\mathsf{FibSets}\webleft (K\webright )\to \mathsf{Sets}$ sending a $K$-fibred set $\phi _{X}\colon X\to K$ to the set $X$, forgetting the map $\phi _{X}$ and the base set $K$.

Example 9.5.3.1.4 ▶ Forgetful Functors That Forget Properties

Examples of forgetful functors that forget properties include:

  1. Forgetting Commutativity. The inclusion functor $\iota \colon \mathsf{CMon}\hookrightarrow \mathsf{Mon}$ which forgets the property of being commutative.
  2. Forgetting Inverses. The inclusion functor $\iota \colon \mathsf{Grp}\hookrightarrow \mathsf{Mon}$ which forgets the property of having inverses.

Notation 9.5.3.1.5 ▶ Notation For Forgetful Functors That Forget Structure

Throughout this work, we will denote forgetful functors that forget structure by ${\text{忘}}$, e.g. as in

\[ {\text{忘}}\colon \mathsf{Grp}\to \mathsf{Sets}. \]

The symbol ${\text{忘}}$, pronounced wasureru (see Item 1 of Remark 9.5.3.1.6 below), means to forget, and is a kanji found in the following words in Japanese and Chinese:

  1. 忘れる, transcribed as wasureru, meaning to forget.
  2. 忘却関手, transcribed as boukyaku kanshu, meaning forgetful functor.
  3. 忘记 or 忘記, transcribed as wàngjì, meaning to forget.
  4. 遗忘函子 or 遺忘函子, transcribed as yíwàng hánzǐ, meaning forgetful functor.

Remark 9.5.3.1.6 ▶ Pronunciation of the Words in Notation 9.5.3.1.5

Here we collect the pronunciation of the words in Notation 9.5.3.1.5 for accuracy and completeness.

  1. Pronunciation of 忘れる:

    • Audio:

    • IPA broad transcription: [wäsɯɾe̞ɾɯ].
    • IPA narrow transcription: [ɰᵝäsɨᵝɾe̞ɾɯ̟ᵝ].

  2. Pronunciation of 忘却関手: Pronunciation:

    • Audio:

    • IPA broad transcription: [bo̞ːkʲäkɯ kä̃ɰ̃ɕɯ].
    • IPA narrow transcription: [bo̞ːkʲäkɯ̟ᵝ kä̃ɰ̃ɕɯᵝ].

  3. Pronunciation of 忘记:

    • Audio:

    • Broad IPA transcription: [wɑŋtɕi].
    • Sinological IPA transcription: [wɑŋ⁵¹⁻⁵³t͡ɕi⁵¹].

  4. Pronunciation of 遗忘函子:

    • Audio:

    • Broad IPA transcription: [iwɑŋ xänt͡sz̩ɨ].
    • Sinological IPA transcription: [i³⁵wɑŋ⁵¹ xän³⁵t͡sz̩²¹⁴⁻²¹⁽⁴⁾].

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