Projective classes as images of accessible functors

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