Given an adjunction F ⊣ G connecting reasonable categories with weak equivalences, we define a new derived bar and cobar construction associated to the adjunction. This yields homotopical models of the completion and cocompletion associated to the monad and comonad of the adjunction. We discuss applications of these resolutions to spectral sequences for derived completions and Goodwillie calcul...

In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterise its n-arrows using a graphical calculus that we develop here. The hom-spaces are appropriately fibrant, indeed are nerves of categories, which indicates that all of the expected coherence equations in each dimension are present. To justify our terminology, we prove tha...

Abstract We characterize the category of Sambin’s positive topologies as result Grothendieck construction applied to a doctrine over Loc locales. then construct an adjunction between and that topological spaces Top , show well-known factors through constructed adjunction.

I provide a unified account of a constraint on adjunction observed in three complex predicate constructions in Japanese: (i) restructuring motion verb constructions, (ii) light verb constructions, (iii) infinitives with wasure‘forget’. It is shown that adjunction (i.e. adverbial modification, adjectival modification, and quantifier raising) in the lower projections are impossible in these const...

It is a well-known fact that a nondeterministic automaton can be transformed into an equivalent deterministic automaton via the powerset construction. From a categorical perspective this construction is the right adjoint to the inclusion functor from the category of deterministic automata to the category of nondeterministic automata. This is in fact an adjunction between two categories of coalg...

Adjunction is a powerful operation that makes Tree Adjoining Grammar (TAG) useful for describing the syntactic structure of natural languages. In practice, a large part of wide coverage grammars written following the TAG formalism is formed by trees that can be combined by means of the simpler kind of adjunction defined for Tree Insertion Grammar. In this article, we describe a parsing algorith...

The idea of adjunction can be traced back to the beginnings of generative syntax. Already Zellig Harris used adjunction and in [Harris, 1979] even exposes the idea that sentence meanings can be derived from kernel sentences via adjunction. Today, TAGs and the somewhat lesser known contextual grammars ([Marcus, 1967]) still use adjunction as a major syntactic operation. Recently, [Kallmeyer and ...

The TAG adjunction operation operates by splitting a tree at one node, which we will call the adjunction site. In the resulting structure, the sub-trees above and below the adjunction site are separated by, and connected with, the auxiliary tree used in the composition. As the adjunction site is thus split into two nodes, with a copy in each subtree, a natural way of formalizing the adjunction ...

This paper presents a logical formalization of Tree Adjoining Grammar (TAG). TAG deals with lexicalized trees and two operations are available: substitution and adjunction. Adjunction is generally presented as an insertion of one tree inside another, surrounding the subtree at the adjunction node. This seems to contradict standard logical ability. We prove that some logical formalisms, namely a...