We establish adjunction and inversion of adjunction for log canonical centers of arbitrary codimension in full generality.

Doctrines are categorical structures very apt to study logics of different nature within a unified environment: the 2-category Dtn of doctrines. Modal interior operators are characterised as particular adjoints in the 2-category Dtn. We show that they can be constructed from comonads in Dtn as well as from adjunctions in it, and we compare the two constructions. Finally we show the amount of information lost in the passage from a comonad, or from an adjunction, to the modal interior operator. The basis for the present work is provided by some seminal work of John Power.

Given a monoidal category $\mathcal C$ with an object J, we construct a monoidal category $\mathcal C[{J^ \vee }]$ by freely adjoining a right dual ${J^ \vee }$ to J. We show that the canonical strong monoidal functor $\Omega :\mathcal C \to \mathcal C[{J^ \vee }]$ provides the unit for a biadjunction with the forgetful 2-functor from the 2-category of monoidal categories with a distinguished dual pair to the 2-category of monoidal categories with a distinguished object. We show that $\Omega :\mathcal C \to \mathcal C[{J^ \vee }]$ is fully faithful and provide coend formulas for homs of the form $\mathcal C[{J^ \vee }](U,\,\Omega A)$ and $\mathcal C[{J^ \vee }](\Omega A,U)$ for $A \in \mathcal C$ and $U \in \mathcal C[{J^ \vee }]$. If ${\rm{N}}$ denotes the free strict monoidal category on a single generating object 1, then ${\rm{N[}}{{\rm{1}}^ \vee }{\rm{]}}$ is the free monoidal category Dpr containing a dual pair – ˧ + of objects. As we have the monoidal pseudopushout $\mathcal C[{J^ \vee }] \simeq {\rm{Dpr}}{{\rm{ + }}_{\rm{N}}}\mathcal C$, it is of interest to have an explicit model of Dpr: we provide both geometric and combinatorial models. We show that the (algebraist’s) simplicial category Δ is a monoidal full subcategory of Dpr and explain the relationship with the free 2-category Adj containing an adjunction. We describe a generalization of Dpr which includes, for example, a combinatorial model Dseq for the free monoidal category containing a duality sequence X0 ˧ X1 ˧ X2 ˧ … of objects. Actually, Dpr is a monoidal full subcategory of Dseq.

Finnish word order is relatively free, making room for all mathematically possible word orders in many constructions. Because there is no evidence in this language for radical nonconfigurationality, explanations must be sought from syntax. It is argued in this article that morphosyntax and word order represent syntactic structure at the PF-interface. Rich morphosyntax frees word order, poor morphosyntax freezes it. The hypothesis is formalized within the context of a parsing-oriented theory of the human language faculty (UG) combining left-to-right minimalism with the dynamic syntax approach. The analysis was implemented as an algorithm and successfully tested with a corpus of 119,800 unique Finnish word orders.

The current article explores the distribution of PP-adverbs, such as this month, this year etc., within English determiner phrases. Examples extracted from English newspapers show that PP-adverbs surprisingly separate head nouns from their PP-complements (i.e. of-phrases), e.g. the election this month of the first female president. At other times, PP-adverbs follow PP-complements, e.g. the election of the first female president this month. Assuming that these PP-adverbs have a null preposition (Larson 1985; McCawley 1988; Caponigro & Pearl 2008, 2009; Shun'ichiro 2013), I put forward three possible syntactic analyses to account for the examples above: (i) adjunction of both the PP-complement and the PP-adverb; (ii) leftward movement of the head noun or the noun phrase; and (iii) rightward movement of the PP-complement. Following Stowell (1981), Higginbotham (1983) and Anderson (1984), the adjunction proposal argues that both PP-adverbs and of-phrases are adjuncts, thus being freely ordered in the nominal hierarchy (Bresnan 1982; Svenonius 1994; Stroik & Putnam 2013). In contrast, the leftward movement analysis respects Kayne's (1994) Antisymmetric Theory of Linearization and argues that the of-phrase in the examples above is still a genuine complement, but the head noun, or sometimes the noun phrase, moves leftwards to a position higher than spec,FP where PP-adverbs are situated. As for the rightward movement account, it follows the leftward movement in treating the of-phrase as a complement but differs in that it extraposes the PP-complement outside PP-adverbs and right-adjoins it inside the DP. The article shows that the first two proposals are untenable, and sometimes cannot derive the wanted data. The third account is superior in that it accounts for the required data as well as other island-sensitive facts.

We present a logical and algebraic description of right adjoint functors between generalized quasi-varieties, inspired by the work of McKenzie on category equivalence. This result is achieved by developing a correspondence between the concept of adjunction and a new notion of translation between relative equational consequences.

This paper investigates the correlative construction in Isbukun Bunun, an Austronesian language spoken in Taiwan. I show that in this language the correlative clause and its associated anaphoric element do not form a constituent at any point in the derivation. Drawing on evidence from island-insensitivity, the absence of Condition C effects and non-constituency facts, I propose that the syntactic relation between the correlative clause and the nominal correlate is derived by a base-generated adjunction structure. Moreover, I argue that the correlative clause, which behaves as a generalized quantifier, binds the nominal correlate phrase in the matrix clause, which is construed as a bound variable. The proposed quantificational binding view is further shown to capture the types of correlate phrases allowed in Isbukun Bunun correlatives.

The aim of the present paper is to extend the dualizing object approach to Stone duality to the noncommutative setting of skew Boolean algebras. This continues the study of noncommutative generalizations of different forms of Stone duality initiated in recent papers by Bauer and Cvetko-Vah, Lawson, Lawson and Lenz, Resende, and also the current author. In this paper we construct a series of dual adjunctions between the categories of left-handed skew Boolean algebras and Boolean spaces, the unital versions of which are induced by dualizing objects $\{ 0, 1, \ldots , n+ 1\} $, $n\geq 0$. We describe the categories of Eilenberg-Moore algebras of the monads of the adjunctions and construct easily understood noncommutative reflections of left-handed skew Boolean algebras, where the latter can be faithfully embedded (if $n\geq 1$) in a canonical way. As an application, we answer the question that arose in a recent paper by Leech and Spinks to describe the left adjoint to their ‘twisted product’ functor $\omega $.

It is well-known that wh-pronouns may pied-pipe their containing host phrases as they move to their final scope positions. In Finnish, such pied-piping requires further that a wh-element is situated at the left edge of host phrases, a position in which it ends up either through base generation or through wh-movement. This article investigates which independent properties define such pied-piping domains. An empirical generalization will be defended according to which a phrase constitutes such pied-piping domain if and only if it is adjoinable. The hypothesis that pied-piping domains are islands is put into question. Secondary wh-movement, pied-piping and adjunction are thus intrinsically linked with each other.

We prove the following theorem characterizing Du Bois singularities. Suppose that $Y$ is smooth and that $X$ is a reduced closed subscheme. Let $\pi : \tilde{Y} \rightarrow Y$ be a log resolution of $X$ in $Y$ that is an isomorphism outside of $X$. If $E$ is the reduced pre-image of $X$ in $\tilde{Y}$, then $X$ has Du Bois singularities if and only if the natural map $\mathcal{O}_X \rightarrow R \pi_* \mathcal{O}_E$ is a quasi-isomorphism. We also deduce Kollár's conjecture that log canonical singularities are Du Bois in the special case of a local complete intersection and prove other results related to adjunction.

Let $\mathcal{E}$ be an ample vector bundle of rank $r$ on a projective variety $X$ with only log-terminal singularities. We consider the nefness of adjoint divisors ${{K}_{X}}\,+\,\left( t-r \right)\,\det \,\mathcal{E}$ when $t\,\ge \,\dim\,X$ and $t\,>\,r$. As an application, we classify pairs $\left( X,\,\mathcal{E} \right)$ with ${{c}_{r}}$-sectional genus zero.