split coequalizer

Split coequalisers


For purposes of this page, a fork (some might say a “cofork”) in a category CC is a diagram of the form

AfgBeC A \;\underoverset{f}{g}{\rightrightarrows}\; B \overset{e}{\rightarrow} C

such that ef=ege f = e g. A split coequalizer is a fork together with morphisms s:CBs\colon C\to B and t:BAt\colon B\to A such that es=1 Ce s = 1_C, se=gts e = g t, and ft=1 Bf t = 1_B. This is equivalent to saying that the morphism (f,e):ge(f,e)\colon g \to e has a section in the arrow category of CC.

Coequalizers and absolute coequalizers

The name “split coequalizer” is appropriate, because in any split coequalizer diagram, the morphism ee is necessarily a coequalizer of ff and gg. For given any h:BDh\colon B\to D such that hf=hgh f = h g, the composite hsh s provides a factorization of hh through ee, since hse=hgt=hft=hh s e = h g t = h f t = h, and such a factorization is unique since ee is (split) epic. In fact, a split coequalizer is not just a coequalizer but an absolute coequalizer: one preserved by all functors.

Contractible pairs

On the other hand, suppose we are given only f,g:ABf,g\colon A\to B and t:BAt\colon B\to A such that ft=1 Bf t = 1_B and gtf=gtgg t f = g t g (which is certainly the case in any split coequalizer, since gtf=sef=seg=gtgg t f = s e f = s e g = g t g). Such a situation is sometimes called a contractible pair. In this case, any coequalizer of ff and gg is split, for if e:BCe\colon B\to C is a coequalizer of ff and gg, then the equation gtf=gtgg t f = g t g implies, by the universal property of ee, a unique morphism s:CBs\colon C\to B such that se=gts e = g t, whence ese=egt=eft=ee s e = e g t = e f t = e and so es=1 Ce s = 1_C since ee is epic.

Similarly, if e:BCe\colon B\to C splits the idempotent gtg t with section s:CBs\colon C\to B, so that es=1e s = 1 and se=gts e = g t, then we have

eg=eseg=egtg=egtf=esef=ef e g = e s e g = e g t g = e g t f = e s e f = e f

and the other identities are obvious; thus ee is a split coequalizer of ff and gg.

Split epimorphisms

Dually, if e:BCe\colon B\to C is a split epimorphism, with a splitting s:CBs\colon C\to B, say, then ee is a split coequalizer of B1seBB \;\underoverset{1}{s e}{\rightrightarrows}\; B, the morphism tt being the identity.

Moreover, ee is also the split coequalizer of its kernel pair, if the latter exists. For if AfgBA \;\underoverset{f}{g}{\rightrightarrows}\; B is this kernel pair, then the two maps se,1 B:BBs e, 1_B \colon B\to B satisfy ese=e1 Be \circ s e = e \circ 1_B, and hence induce a map t:BAt\colon B\to A such that ft=1 Bf t = 1_B and gt=seg t = s e.


The “ur-example” of a split coequalizer is the following. Let AA be an algebra for the monad TT on the category CC, with structure map a:TAAa\colon T A \to A. Then the diagram

T 2Aμ ATaTAaA, T^2 A \;\underoverset{\mu_A }{T a}{\rightrightarrows}\; T A \overset{a}{\rightarrow} A\, ,

called the canonical presentation of the algebra (A,a)(A,a), is a split coequalizer in CC (and a reflexive coequalizer in the category of TT-algebras). The splittings are given by s=η A:ATAs = \eta_A \colon A \to T A and t=η TA:TAT 2At = \eta_{T A} \colon T A \to T^2 A. (Here μ\mu and η\eta are the multiplication and unit of the monad TT.)

This split coequalizer figures prominently in Beck’s monadicity theorem.

Revised on January 10, 2012 18:23:15 by Finn Lawler (