• Associativity. We have isomorphisms of sets[1]
    \[ \underbrace{\text{Eq}\webleft (f\circ \text{eq}\webleft (g,h\webright ),g\circ \text{eq}\webleft (g,h\webright )\webright )}_{{}=\text{Eq}\webleft (f\circ \text{eq}\webleft (g,h\webright ),h\circ \text{eq}\webleft (g,h\webright )\webright )}\cong \text{Eq}\webleft (f,g,h\webright ) \cong \underbrace{\text{Eq}\webleft (f\circ \text{eq}\webleft (f,g\webright ),h\circ \text{eq}\webleft (f,g\webright )\webright )}_{{}=\text{Eq}\webleft (g\circ \text{eq}\webleft (f,g\webright ),h\circ \text{eq}\webleft (f,g\webright )\webright )}, \]

    where $\text{Eq}\webleft (f,g,h\webright )$ is the limit of the diagram

    in $\mathsf{Sets}$, being explicitly given by

    \[ \text{Eq}\webleft (f,g,h\webright )\cong \webleft\{ a\in A\ \middle |\ f\webleft (a\webright )=g\webleft (a\webright )=h\webleft (a\webright )\webright\} . \]

Footnotes

[1] That is, the following three ways of forming “the” equaliser of $\webleft (f,g,h\webright )$ agree:
  1. Take the equaliser of $\webleft (f,g,h\webright )$, i.e. the limit of the diagram
    in $\mathsf{Sets}$.
  2. First take the equaliser of $f$ and $g$, forming a diagram
    \[ \text{Eq}\webleft (f,g\webright )\overset {\text{eq}\webleft (f,g\webright )}{\hookrightarrow }A\underset {g}{\overset {f}{\rightrightarrows }}B \]
    and then take the equaliser of the composition
    \[ \text{Eq}\webleft (f,g\webright )\overset {\text{eq}\webleft (f,g\webright )}{\hookrightarrow }A\underset {h}{\overset {f}{\rightrightarrows }}B, \]
    obtaining a subset
    \[ \text{Eq}\webleft (f\circ \text{eq}\webleft (f,g\webright ),h\circ \text{eq}\webleft (f,g\webright )\webright )=\text{Eq}\webleft (g\circ \text{eq}\webleft (f,g\webright ),h\circ \text{eq}\webleft (f,g\webright )\webright ) \]
    of $\text{Eq}\webleft (f,g\webright )$.
  3. First take the equaliser of $g$ and $h$, forming a diagram
    \[ \text{Eq}\webleft (g,h\webright )\overset {\text{eq}\webleft (g,h\webright )}{\hookrightarrow }A\underset {h}{\overset {g}{\rightrightarrows }}B \]
    and then take the equaliser of the composition
    \[ \text{Eq}\webleft (g,h\webright )\overset {\text{eq}\webleft (g,h\webright )}{\hookrightarrow }A\underset {g}{\overset {f}{\rightrightarrows }}B, \]
    obtaining a subset
    \[ \text{Eq}\webleft (f\circ \text{eq}\webleft (g,h\webright ),g\circ \text{eq}\webleft (g,h\webright )\webright )=\text{Eq}\webleft (f\circ \text{eq}\webleft (g,h\webright ),h\circ \text{eq}\webleft (g,h\webright )\webright ) \]
    of $\text{Eq}\webleft (g,h\webright )$.

View Item 1 as PDF

Statistics Cite
2 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: