Abstract
AbstractWe show that the Lascar group
$\operatorname {Gal}_L(T)$
of a first-order theory T is naturally isomorphic to the fundamental group
$\pi _1(|\mathrm {Mod}(T)|)$
of the classifying space of the category of models of T and elementary embeddings. We use this identification to compute the Lascar groups of several example theories via homotopy-theoretic methods, and in fact completely characterize the homotopy type of
$|\mathrm {Mod}(T)|$
for these theories T. It turns out that in each of these cases,
$|\operatorname {Mod}(T)|$
is aspherical, i.e., its higher homotopy groups vanish. This raises the question of which homotopy types are of the form
$|\mathrm {Mod}(T)|$
in general. As a preliminary step towards answering this question, we show that every homotopy type is of the form
$|\mathcal {C}|$
where
$\mathcal {C}$
is an Abstract Elementary Class with amalgamation for
$\kappa $
-small objects, where
$\kappa $
may be taken arbitrarily large. This result is improved in another paper.
Publisher
Cambridge University Press (CUP)
Cited by
1 articles.
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