canonical presentation



Given an adjunction F,G,η,ε:XA\langle F,G,\eta,\varepsilon\rangle\colon X\to A, the canonical presentation of an object aobj(A)a\in\operatorname{obj}(A) is the fork

FGFGaFGε aε FGaFGaε aa F G F Ga\underoverset{F G\varepsilon_a}{\varepsilon_{F G a}}{\rightrightarrows}F G a \overset{\varepsilon_a}{\rightarrow}a

(this is indeed a fork, by the naturality of ε\varepsilon).


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

    T 2xThμ xTxhx T^2 x\underoverset{T h}{\mu_x}{\rightrightarrows}T x\overset{h}{\rightarrow}{x}

    for the monad 𝕋:=T=GF,η,μ=GεF\mathbb{T} := \langle T = G F,\eta,\mu=G\varepsilon F \rangle corresponding to the given adjunction and for the 𝕋\mathbb{T}-algebra x,h=Ga,Gε a\langle x,h\rangle = \langle G a , G\varepsilon_a \rangle. Hence, by the monadicity theorem, GG (in particular) reflects coequalizers for our pair.

  • The two parallel arrows FGε aF G\varepsilon_a and ε FGa\varepsilon_{F G a} appearing in the canonical presentation have a common section, namely, Fη GaF \eta_{G a} (by the triangle identities). Hence, whenever GG is monadic, the resulting coequalizer is a reflexive coequalizer.


If A=GrpA = \mathbf{Grp}, X=SetX = \mathbf{Set} and GG is the forgetful functor, then for a group aa, FGaF G a is just the free group on the elements of aa, and ε\varepsilon is the projection, taking a ”formal product” of elements of aa to the actual product in aa (since by a triangle identity we have Gε a(t)=tG\varepsilon_a(\langle t\rangle)=t where tGat\in Ga and t=η Ga(t)=\langle t\rangle=\eta_{G a}(t)= the reduced word with one letter tt).

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


Revised on February 20, 2012 21:51:03 by Tim Porter (