11.1.4 Precomposition and Postcomposition

    Let $\mathcal{C}$ be a category and let $A,B,C\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$.

    Let $f\colon A\to B$ and $g\colon B\to C$ be morphisms of $\mathcal{C}$.

    1. 1.

      The precomposition function associated to $f$ is the function

      \[ f^{*} \colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (B,C\webright ) \to \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,C\webright ) \]

      defined by

      \[ f^{*}\webleft (\phi \webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\phi \circ f \]

      for each $\phi \in \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (B,C\webright )$.

    2. 2.

      The postcomposition function associated to $g$ is the function

      \[ g_{*} \colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,B\webright ) \to \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,C\webright ) \]

      defined by

      \[ g_{*}\webleft (\phi \webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g\circ \phi \]

      for each $\phi \in \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,B\webright )$.

    Let $A,B,C,D\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$ and let $f\colon A\to B$ and $g\colon B\to C$ be morphisms of $\mathcal{C}$.

  • 1.

    Interaction Between Precomposition and Postcomposition. We have

  • 2.

    Interaction With Composition I. We have

  • 3.

    Interaction With Composition II. We have

  • 4.

    Interaction With Composition III. We have

  • 5.

    Interaction With Identities. We have

    \begin{align*} \webleft (\operatorname {\mathrm{id}}_{A}\webright )^{*} & = \operatorname {\mathrm{id}}_{\operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,B\webright )},\\ \webleft (\operatorname {\mathrm{id}}_{B}\webright )_{*} & = \operatorname {\mathrm{id}}_{\operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,B\webright )}. \end{align*}
  • Item 1: Interaction Between Precomposition and Postcomposition
    Clear.

    Item 2: Interaction With Composition I
    Clear.

    Item 3: Interaction With Composition II
    Clear.

    Item 4: Interaction With Composition III
    Clear.

    Item 5: Interaction With Identities
    Clear.


Noticed something off, or have any comments? Feel free to reach out!


You can also use the contact form below: