scholarly journals The quasitopos hull of the construct of closure spaces

2003 ◽  
Vol 4 (1) ◽  
pp. 15 ◽  
Author(s):  
Veerle Claes ◽  
G. Sonck
Keyword(s):  
Functional Analysis ◽  
Differential Calculus ◽  
Homotopy Theory ◽  
Topological Construct ◽  
Continuous Maps ◽  
Closure Spaces ◽  
Cartesian Closed ◽  
Cartesian Closed Topological Hull ◽  
Extensional Topological Hull ◽  
Topological Constructs

<p>In the list of convenience properties for topological constructs the property of being a quasitopos is one of the most interesting ones for investigations in function spaces, differential calculus, functional analysis, homotopy theory, etc. The topological construct Cls of closure spaces and continuous maps is not a quasitopos. In this article we give an explicit description of the quasitopos topological hull of Cls using a method of F. Schwarz: we first describe the extensional topological hull of Cls and of this hull we construct the cartesian closed topological hull.</p>

scholarly journals A note on connectedness in cartesian closed categories

1997 ◽  
Vol 20 (1) ◽  
pp. 101-104
Author(s):  
Reino Vainio
Keyword(s):  
Function Space ◽  
Homotopy Theory ◽  
Continuous Convergence ◽  
Continuous Maps ◽  
Space Objects ◽  
Cartesian Closed Categories ◽  
Cartesian Closed

Primaxily working in the category of limit spaces and continuous maps we suggest a new concept of connectivity with application in all categories where function space objects satisfy natural exponential laws. In a separate Appendix we motivate the development of a homotopy theory for spaces of real-valued continuous maps endowed with the structure of continuous convergence.

scholarly journals The Cartesian Closed Topological Hull of the Category of Approach Uniform Spaces

2001 ◽  
Vol 31 (4) ◽  
pp. 1373-1404 ◽  
Author(s):  
Mark Nauwelaerts
Keyword(s):  
Uniform Spaces ◽  
Cartesian Closed ◽  
Cartesian Closed Topological Hull

Homotopy Groups and CW Complexes

2020 ◽  
pp. 11-20
Author(s):  
Loring W. Tu
Keyword(s):  
Topological Space ◽  
Algebraic Topology ◽  
Weak Topology ◽  
Homotopy Theory ◽  
Fiber Bundles ◽  
Homotopy Groups ◽  
Cw Complex ◽  
Continuous Maps ◽  
Cw Complexes ◽  
Set Of Points

This chapter discusses some results about homotopy groups and CW complexes. Throughout this book, one needs to assume a certain amount of algebraic topology. A CW complex is a topological space built up from a discrete set of points by successively attaching cells one dimension at a time. The name CW complex refers to the two properties satisfied by a CW complex: closure-finiteness and weak topology. With continuous maps as morphisms, the CW complexes form a category. It turns out that this is the most appropriate category in which to do homotopy theory. The chapter also looks at fiber bundles.

Introduction to Functional Analysis: Banach Spaces and Differential Calculus.

1983 ◽  
Vol 90 (8) ◽  
pp. 579
Author(s):  
Joe Diestel ◽  
Leopoldo Nachbin
Keyword(s):  
Functional Analysis ◽  
Banach Spaces ◽  
Differential Calculus

scholarly journals Neighborhood spaces

2002 ◽  
Vol 32 (7) ◽  
pp. 387-399 ◽  
Author(s):  
D. C. Kent ◽  
Won Keun Min
Keyword(s):  
Topological Space ◽  
Convergence Theory ◽  
Closure Operators ◽  
Continuous Maps ◽  
Closure Spaces ◽  
Bireflective Subcategory

Neighborhood spaces, pretopological spaces, and closure spaces are topological space generalizations which can be characterized by means of their associated interior (or closure) operators. The category NBD of neighborhood spaces and continuous maps contains PRTOP as a bicoreflective subcategory and CLS as a bireflective subcategory, whereas TOP is bireflectively embedded in PRTOP and bicoreflectively embedded in CLS. Initial and final structures are described in these categories, and it is shown that the Tychonov theorem holds in all of them. In order to describe a successful convergence theory in NBD, it is necessary to replace filters by more generalp-stacks.

scholarly journals Nonlinear functional analysis

1983 ◽  
Vol 2 (3) ◽  
pp. 162-165
Author(s):  
W. L. Fouché
Keyword(s):  
Functional Analysis ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Homotopy Theory ◽  
Nonlinear Functional Analysis ◽  
Nonlinear Functional ◽  
Infinite Dimensional ◽  
Contraction Theorem ◽  
New Ideas

In this article we discuss some aspects of nonlinear functional analysis. It included reviews of Banach’s contraction theorem, Schauder’s fixed point theorem, globalising techniques and applications of homotopy theory to nonlinear functional analysis. The author emphasises that fundamentally new ideas are required in order to achieve a better understanding of phenomena which contain both nonlinear and definite infinite dimensional features.

Topological representation of the λ-calculus

2000 ◽  
Vol 10 (1) ◽  
pp. 81-96 ◽  
Author(s):  
STEVEN AWODEY
Keyword(s):  
Continuous Function ◽  
Recent Result ◽  
Category Theory ◽  
Topological Model ◽  
Topological Representation ◽  
Topological Semantics ◽  
Continuous Maps ◽  
Cartesian Closed Categories ◽  
Cartesian Closed

The λ-calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of λ-conversion is shown to be deductively complete with respect to such topological semantics. It is also shown to be functionally complete, in the sense that there is always a ‘minimal’ topological model in which every continuous function is λ-definable. These results subsume earlier ones using cartesian closed categories, as well as those employing so-called Henkin and Kripke λ-models.

scholarly journals Semantics of higher inductive types

2019 ◽  
Vol 169 (1) ◽  
pp. 159-208 ◽  
Author(s):  
PETER LEFANU LUMSDAINE ◽  
MICHAEL SHULMAN
Keyword(s):  
Type Theory ◽  
Homotopy Theory ◽  
Model Category ◽  
Suitable Model ◽  
Simplicial Sets ◽  
James Construction ◽  
Quillen Model Category ◽  
Wide Range ◽  
Inductive Types ◽  
Cartesian Closed

AbstractHigher inductive typesare a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the “synthetic” development of homotopy theory within type theory, as well as in formalising ordinary set-level mathematics in type theory. In this paper, we construct models of a wide range of higher inductive types in a fairly wide range of settings.We introduce the notion ofcell monad with parameters: a semantically-defined scheme for specifying homotopically well-behaved notions of structure. We then show that any suitable model category hasweakly stable typal initial algebrasfor any cell monad with parameters. When combined with the local universes construction to obtain strict stability, this specialises to give models of specific higher inductive types, including spheres, the torus, pushout types, truncations, the James construction and general localisations.Our results apply in any sufficiently nice Quillen model category, including any right proper, simplicially locally cartesian closed, simplicial Cisinski model category (such as simplicial sets) and any locally presentable locally cartesian closed category (such as sets) with its trivial model structure. In particular, any locally presentable locally cartesian closed (∞, 1)-category is presented by some model category to which our results apply.

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