Contents

Definition

Given an adjunction $⟨F,G,\eta ,\epsilon ⟩:X\to A$, the canonical presentation of an object $a\in obj(A)$ is the fork

(this is indeed a fork, by the naturality of $\epsilon$).

Properties

• In general, this fork need not be a coequalizer, but if $G$ is monadic, then we do get a coequalizer. To see this, note that the above pair is $G$-split: When applying $G$ to the fork, we get the split coequalizer

  $$T^{2}x\underset{Th}{\overset{{\mu }_x}{\rightrightarrows }}Tx\stackrel{h}{\to }x$$T^2 x\underoverset{T h}{\mu_x}{\rightrightarrows}T x\overset{h}{\rightarrow}{x}

  for the monad $𝕋:=⟨T=GF,\eta ,\mu =G\epsilon F⟩$ corresponding to the given adjunction and for the $𝕋$-algebra $⟨x,h⟩=⟨Ga,G{\epsilon }_a⟩$. Hence, by the monadicity theorem, $G$ (in particular) reflects coequalizers for our pair.

• The two parallel arrows $FG{\epsilon }_a$ and ${\epsilon }_{FGa}$ appearing in the canonical presentation have a common section, namely, $F{\eta }_{Ga}$ (by the triangle identities). Hence, whenever $G$ is monadic, the resulting coequalizer is a reflexive coequalizer.

Example

If $A=\mathrm{Grp}$, $X=\mathrm{Set}$ and $G$ is the forgetful functor, then for a group $a$, $FGa$ is just the free group on the elements of $a$, and $\epsilon$ is the projection, taking a ”formal product” of elements of $a$ to the actual product in $a$ (since by a triangle identity we have $G{\epsilon }_a(⟨t⟩)=t$ where $t\in \mathrm{Ga}$ and $⟨t⟩={\eta }_{Ga}(t)=$ the reduced word with one letter $t$).

Since the coequalizer of $\cdot \underset{f}{\overset{g}{\rightrightarrows }}\cdot$ in $\mathrm{Grp}$ is the familiar quotient by the normal subgroup generated by elements like $f(t)g(t{)}^{-1}$, the canonical presentation (a coequalizer in this case) is indeed a presentation of $a$ in terms of generators and relations.

References

• Categories Work, p. 153.

• John Baez, ”Universal algebra and diagrammatic reasoning,” slides available online