Projective classes as images of accessible functors

Abstract

Abstract We are dealing with projective classes (in short $\textrm {PC}$) over first-order vocabularies with no restrictions on the (possibly infinite) arities of relation or operation symbols. We verify that $\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })=\textrm {RPC}(\mathbin {\mathscr {L}}_{\infty \lambda })$ for any infinite cardinal $\lambda $, and that if $\lambda $ is singular, then $\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })=\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda ^+})$. If $\lambda $ is regular, then a class of structures over a $\lambda $-ary vocabulary is $\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })$-definable iff it is the image of a $\lambda $-continuous functor on a $\lambda $-accessible category; we also provide separating counterexamples for the non $\lambda $-ary case. We prove that many $\textrm {PC}$ classes of structures, previously known not to be closed under elementary equivalence over any $\mathbin {\mathscr {L}}_{\infty \lambda }$, are not even $\textrm {co}\textrm {-}\textrm {PC}$ over $\mathbin {\mathscr {L}}_{\infty \infty }$. Those classes arise from diverse contexts including convex $\ell $-subgroup lattices of lattice-ordered groups, ideal lattices of rings, nonstable K$_0$-theory of rings, coordinatization of sectionally complemented modular lattices and real spectra of commutative unital rings. For example, the class of posets of finitely generated two-sided ideals of all unital rings is $\textrm {PC}$ but not $\textrm {co}\textrm {-}\textrm {PC}$ over $\mathbin {\mathscr {L}}_{\infty \infty }$. We also provide a negative solution to a problem, raised in 2011 by Gillibert and the author, asking whether essential surjectivity of a ‘well-behaved’ functor on objects entails its essential surjectivity on diagrams indexed by arbitrary finite posets.