Classical scope-assignment strategies for multi-quantifier sentences involve quantifier phrase (QP)-movement (e.g., [11], [12]). More recent continuation-based approaches provide a compelling alternative, for they interpret QPs in situ — without reso

We give an abstract characterization of the category of co-semi-analytic functors and describe an action of semi-analytic functors on co-semi-analytic functors.

We show that the problem of whether a finite set of regular-linear axioms defines a rigid theory is undecidable.

In this paper we characterise the categories of Lawvere theories and equational theories that correspond to the categories of analytic and polynomial monads onSet, and hence also to the categories of the symmetric and rigid operads inSet. We show that the category of analytic monads is equivalent to the category of regular-linear theories. The category of polynomial monads is equivalent to the category of rigid theories, that is, regular-linear theories satisfying an additional global condition. This solves a problem posed by A. Carboni and P. T. Johnstone. The Lawvere theories corresponding to these monads are identifiedviasome factorisation systems. We also show that the categories of analytic monads and finitary endofunctors onSetare monadic over the category of analytic functors. The corresponding monad for analytic monads distributes over the monad for finitary endofunctors and hence the category of (finitary) monads onSetis monadic over the category of analytic functors. This extends a result of M. Barr.

Abstract A. M.Pitts in [Pi] proved that is a bi-Heyting category satisfying the Lawvere condition. We show that the embedding Φ: → Sh(P0, J0) into the topos of sheaves, (P0 is the category of finite rooted posets and open maps, J0 the canonical topology on P0) given by H ↦ HA(H, (−)) : P0 → Set preserves the structure mentioned above, finite coproducts, and subobject classifier; it is also conservative. This whole structure on can be derived from that of Sh(P0, J0) via the embedding Φ. We also show that the equivalence relations in are not effective in general. On the way to these results we establish a new kind of duality between and a category of sheaves equipped with certain structure defined in terms of Ehrenfeucht games. Our methods are model-theoretic and combinatorial as opposed to proof-theoretic as in [Pi].