\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \newcommand{\E} {\mathrm{E}} \)
deepdream of
          a sidewalk
Show Answer
\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \)
Math and science::Algebra::Aluffi

Canonical decomposition

Any function \( f : A \to B \) can be decomposed into a [type of function], followed by an [type of function], followed by an [type of function].

The surjection will be a projection to a partition of \( A \) induced by \( f \). The isomorphism will be from this set to [what set?]. The injection will be [what function?] back to \( B \).

Equivalence relation induced by a function

Let \( f : A \to B \) be a function. \( f \) induces an equivalence relation \( \sim \) on \( A \) as follows: For all \( a', a'' \in A \),

\[ a' \sim a'' \iff f(a') = f(a''). \]

Theorem. Canonical decomposition

Let \( f : A \to B \) be any function, and define \( \sim \) as above. Then \( f \) decomposes as follows:

First we have the canonical projection \( A \to A/\sim \). The last function is the inclusion \( \operatorname{im} f \subseteq B \). The bijection in the middle is defined as:

[\[ \widetilde{f}(?) = \quad ? \]]

This theorem states that the above diagram for \( f \)'s decomposition computes and that \( \widetilde{f} \) is a valid function and is a bijection.


There is ambiguity as to which elements of \( A \) are chosen to represent the equivalence classes in \( A \ \sim \). As a consequence, this theorem should be accompanied by a proof to show that any choice available leads to the same result. Such a proof is said to be a verification of the theorem being well-defined. Aluffi does exactly this, and it is a good example of such proofs.