nLab split coequalizer

Split coequalizers

Split coequalizers


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 as below

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.

As 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. This is meant in the sense in which any functor can “preserve a coequalizer”, namely that FeF e is a coequalizer of FfF f and FgF g for any functor FF. Of course this happens because FF also preserves the splitting morphisms ss and tt, but those are not part of the coequalizer diagram that is “preserved” in this statement.

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.


Beck coequalizer for algebras over a monad

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 2ATaμ ATAaA, T^2 A \;\underoverset{T a}{\mu_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, whence also called the Beck coequalizer.

See also at Eilenberg-Moore category – As a colimit completion of the Kleisli category.

See also split equalizer.


Last revised on February 23, 2024 at 22:17:31. See the history of this page for a list of all contributions to it.