Proposition 9.5.1.1.6 ▶ Elementary Properties of Functors

Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.

  1. Preservation of Isomorphisms. If $f$ is an isomorphism in $\mathcal{C}$, then $F\webleft (f\webright )$ is an isomorphism in $\mathcal{D}$.1

1When the converse holds, we call $F$ conservative, see Definition 9.6.4.1.1.

Proof of Proposition 9.5.1.1.6.
Item 1: Preservation of Isomorphisms
Indeed, we have
\begin{align*} F\webleft (f\webright )^{-1}\circ F\webleft (f\webright ) & = F\webleft (f^{-1}\circ f\webright )\\ & = F\webleft (\text{id}_{A}\webright )\\ & = \text{id}_{F\webleft (A\webright )} \end{align*}

and

\begin{align*} F\webleft (f\webright )\circ F\webleft (f\webright )^{-1} & = F\webleft (f\circ f^{-1}\webright )\\ & = F\webleft (\text{id}_{B}\webright )\\ & = \text{id}_{F\webleft (B\webright )}, \end{align*}

showing $F\webleft (f\webright )$ to be an isomorphism.

Previous

View Proposition 9.5.1.1.6 as PDF

Statistics Cite
3 tags refer to this tag

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

You can also use the contact form below: