an analogy breaks, but why?

an analogy breaks, but why?

[Thomas Streicher]

Locally presentable categories are considered as the "glorified",
i.e. categorical, version of algebraic lattices. They can be
characterized as full reflective subcategories of presheaf toposes
where the embedding preserves cofiltered colimits (i.e. are Scott
continuous) which is in accordance with the fact that every algebraic
lattice arises as the image of a complete prime algebraic lattice
(down closed subsets of a poset order by subset inclusion) under a
Scott continuous closure operator.

So far so good. But in Makkai and Par'e's book it is shown that every
Grothendieck topos is in particular locally presentable. However, one
easily sees that the posetal analogue doesn't hold. Grothendieck
toposes arise as localizations of presheaf toposes. However,
localizations of complete prime algebraic lattices are precisely the
frames, i.e. cHa's, which need not be algebraic lattices.

Has anyone give a good intuitive reason why the analogy breaks?

Thomas


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