1 The Cotangent Complex: General Theory 4 1.1 Stable Envelopes and Tangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Relative Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 The Tangent Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 The Relative Cotangent Complex...

We propose a modular high-level approach to the specification of transactions in rewriting logic, where the operational and the abstract views are related by suitable adjunctions between categories of tile theories and of rewrite theories.

We give a short proof of classical Waldhausen Additivity, and then prove Waldhausen Additivity for an ∞-version of Waldhausen K-theory. Namely, we prove that Waldhausen K-theory sends a split-exact sequence of Waldausen quasicategories A → E → B to a stable equivalence of spectra K(E) → K(A) ∨ K(B) under a few mild hypotheses. For example, each cofiber sequence in A of the form A0 → A1 → ∗ is r...

This chapter uses categorical techniques to describe relations between various sets of operators on a Hilbert space, such as self-adjoint, positive, density, effect and projection operators. These relations, including various Hilbert-Schmidt isomorphisms of the form tr(A−), are expressed in terms of dual adjunctions, and maps between them. Of particular interest is the connection with quantum s...

It is shown that the multiplicative monoids of Brauer’s centralizer algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself, and where, moreover, a kind of symmetry involving the self-adjoint functors is satisfied. As in a previous paper, of which this is a companion, it is shown that such a symmetric self-adjunction...

Tree Adjoining Grammar (TAG) is a useful formalism for describing the syntactic structure of natural languages. In practice, a large part of wide coverage TAGs is formed by trees that satisfy the restrictions imposed by Tree Insertion Grammar (TIG), a simpler formalism. This characteristic can be used to reduce the practical complexity of TAG parsing, applying the standard adjunction operation ...

In this work, we focus on adjunctions (also named isotone Galois connections) between fuzzy preordered sets; specifically, we study necessary and sufficient conditions that have to be fulfilled in order such an adjunction to exist.

We propose a modular high-level approach to the specification of transactions in rewriting logic, where the operational and the abstract views are related by suitable adjunctions between categories of tile theories and of rewrite theories.

The order-enriched category of monotonic predicate transformers over posets is a model of the reenement calculus of higher order imperative programs and pre-post speciications. This category is shown to be equivalent to the category of spans over ideal relations, and ideal relations are shown to be spans over monotonic functions between posets, using a skew span construction because the standar...