An object G G in a category is epicomplete provided that the only morphisms out of G G which are simultaneously epi and mono are the isomorphisms. We characterize the epicomplete objects in the category A r c h {\mathbf {Arch}} , whose objects are the archimedean lattice-ordered groups (archimedean ℓ \ell -groups) and whose morphisms are the maps preserving both group and lattice structure ( ℓ \ell -homomorphisms). Recall that a space is basically disconnected if the closure of each cozero subset is open. Theorem. The following are equivalent for G ∈ A r c h G \in {\mathbf {Arch}} . (a) G G is

A r c h {\mathbf {Arch}}

epicomplete. (b) G G is an

A r c h {\mathbf {Arch}}

extremal suboject of

D ( Y ) D(Y)

for some basically disconnected compact Hausdorff space Y Y . Here

D ( Y ) D(Y)

denotes the continuous extended real-valued functions on

Y Y

which are finite on a dense subset. (c) G G is conditionally and laterally σ \sigma -complete (meaning each countable subset of positive elements of

G G

which is either bounded or pairwise disjoint has a supremum), and

G G

is divisible. The analysis of A r c h {\mathbf {Arch}} rests on an analysis of the closely related category W {\mathbf {W}} , whose objects are of the form ( G , u ) (G,u) , where G ∈

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