107,199 research outputs found
Dialectica Categories for the Lambek Calculus
We revisit the old work of de Paiva on the models of the Lambek Calculus in
dialectica models making sure that the syntactic details that were sketchy on
the first version got completed and verified. We extend the Lambek Calculus
with a \kappa modality, inspired by Yetter's work, which makes the calculus
commutative. Then we add the of-course modality !, as Girard did, to
re-introduce weakening and contraction for all formulas and get back the full
power of intuitionistic and classical logic. We also present the categorical
semantics, proved sound and complete. Finally we show the traditional
properties of type systems, like subject reduction, the Church-Rosser theorem
and normalization for the calculi of extended modalities, which we did not have
before
On the 3-arrow calculus for homotopy categories
We develop a localisation theory for certain categories, yielding a 3-arrow
calculus: Every morphism in the localisation is represented by a diagram of
length 3, and two such diagrams represent the same morphism if and only if they
can be embedded in a 3-by-3 diagram in an appropriate way. The methods to
construct this localisation are similar to the Ore localisation for a 2-arrow
calculus; in particular, we do not have to use zigzags of arbitrary length.
Applications include the localisation of an arbitrary model category with
respect to its weak equivalences as well as the localisation of its full
subcategories of cofibrant, fibrant and bifibrant objects, giving the homotopy
category in all four cases. In contrast to the approach of Dwyer, Hirschhorn,
Kan and Smith, the model category under consideration does not need to admit
functorial factorisations. Moreover, our method shows that the derived category
of any abelian (or idempotent splitting exact) category admits a 3-arrow
calculus if we localise the category of complexes instead of its homotopy
category.Comment: Applications added. Minor changes. This article is an extension of
the published versio
A calculus for flow categories
We describe a calculus of moves for modifying a framed flow category without changing the associated stable homotopy type. We use this calculus to show that if two framed flow categories give rise to the same stable homotopy type of homological width at most three, then the flow categories are move equivalent. The process we describe is essentially algorithmic and can often be performed by hand, without the aid of a computer program
Kan extensions and the calculus of modules for -categories
Various models of -categories, including quasi-categories,
complete Segal spaces, Segal categories, and naturally marked simplicial sets
can be considered as the objects of an -cosmos. In a generic
-cosmos, whose objects we call -categories, we introduce
modules (also called profunctors or correspondences) between
-categories, incarnated as as spans of suitably-defined fibrations with
groupoidal fibers. As the name suggests, a module from to is an
-category equipped with a left action of and a right action of ,
in a suitable sense. Applying the fibrational form of the Yoneda lemma, we
develop a general calculus of modules, proving that they naturally assemble
into a multicategory-like structure called a virtual equipment, which is known
to be a robust setting in which to develop formal category theory. Using the
calculus of modules, it is straightforward to define and study pointwise Kan
extensions, which we relate, in the case of cartesian closed -cosmoi,
to limits and colimits of diagrams valued in an -category, as
introduced in previous work.Comment: 84 pages; a sequel to arXiv:1506.05500; v2. new results added, axiom
circularity removed; v3. final journal version to appear in Alg. Geom. To
Model Categories for Orthogonal Calculus
We restate the notion of orthogonal calculus in terms of model categories.
This provides a cleaner set of results and makes the role of O(n)-equivariance
clearer. Thus we develop model structures for the category of n-polynomial and
n-homogeneous functors, along with Quillen pairs relating them. We then
classify n-homogeneous functors, via a zig-zag of Quillen equivalences, in
terms of spectra with an O(n)-action. This improves upon the classification of
Weiss. As an application, we develop a variant of orthogonal calculus by
replacing topological spaces with orthogonal spectra.Comment: 36 pages, added a new section introducing spaces with a group action,
minor corrections from previous versio
Comparing the orthogonal and homotopy functor calculi
Goodwillie's homotopy functor calculus constructs a Taylor tower of
approximations to F, often a functor from spaces to spaces. Weiss's orthogonal
calculus provides a Taylor tower for functors from vector spaces to spaces. In
particular, there is a Weiss tower associated to the functor which sends a
vector space V to F evaluated at the one-point compactification of V.
In this paper, we give a comparison of these two towers and show that when F
is analytic the towers agree up to weak equivalence. We include two main
applications, one of which gives as a corollary the convergence of the Weiss
Taylor tower of BO. We also lift the homotopy level tower comparison to a
commutative diagram of Quillen functors, relating model categories for
Goodwillie calculus and model categories for the orthogonal calculus.Comment: 28 pages, sequel to Capturing Goodwillie's Derivative,
arXiv:1406.042
A language for multiplicative-additive linear logic
A term calculus for the proofs in multiplicative-additive linear logic is
introduced and motivated as a programming language for channel based
concurrency. The term calculus is proved complete for a semantics in linearly
distributive categories with additives. It is also shown that proof equivalence
is decidable by showing that the cut elimination rewrites supply a confluent
rewriting system modulo equations.Comment: 16 pages without appendices, 30 with appendice
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