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Brouwer's fixed-point theorem in real-cohesive homotopy type theory [article]

Michael Shulman
2017 arXiv   pre-print
This enables us to reproduce formally some of the classical applications of homotopy theory to topology. As an example, we prove Brouwer's fixed-point theorem.  ...  In a further refinement called "real-cohesion", the shape is determined by continuous maps from the real numbers, as in classical algebraic topology.  ...  work is represented in Schreiber and Shulman (2012) , which this paper draws heavily on).  ... 
arXiv:1509.07584v3 fatcat:lvviw4kyane7xdxua3nedsq5f4

Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory [article]

David Jaz Myers
2022 arXiv   pre-print
In this paper, we will put forward a definition of orbifold in synthetic differential cohesive homotopy type theory: an orbifold is a microlinear type for which the type of identifications between any  ...  In homotopy type theory, a point of a type may have internal symmetries, and we will be able to construct examples of orbifolds by defining their type of points directly.  ...  In his paper "Brouwer's fixed point theorem in real cohesive homotopy type theory" [30] , Shulman gives us the tools to do synthetic algebraic topology in homotopy type theory by adding a system of modalities  ... 
arXiv:2205.15887v1 fatcat:etop2t4cm5gllpome5jsmd4jiq

Homotopy type theory: the logic of space [article]

Michael Shulman
2017 arXiv   pre-print
This is an introduction to type theory, synthetic topology, and homotopy type theory from a category-theoretic and topological point of view, written as a chapter for the book "New Spaces for Mathematics  ...  For instance, in [101] I used ×S 1 = S 1 to prove the Brouwer fixed point theorem synthetically.  ...  In type theory, however, we have seen that types "potentially" have both topological and homotopical structure, which we can draw out by asserting axioms such as Brouwer's theorem or Voevodsky's univalence  ... 
arXiv:1703.03007v1 fatcat:osarowd7brb4jli4r6utyw6d6q

Dynamics of non-metric manifolds [article]

Alexandre Gabard, David Gauld
2011 arXiv   pre-print
To handle this situation we use Lefschetz in place of Brouwer's fixed point theorem, to obtain: Proof.  ...  An alternative (non-foliated) proof follows either from Theorem 4.5 (Poincaré-Bendixson approach) or from Corollary 3.19 (Brouwer's fixed-point theorem), which establishes the general case of L n .  ... 
arXiv:1102.5684v1 fatcat:unx7zcu4krfkfbh65ixjor46t4

Fitch-Style Modal Lambda Calculi [chapter]

Ranald Clouston
2018 Lecture Notes in Computer Science  
Fitch-style modal deduction, in which modalities are eliminated by opening a subordinate proof, and introduced by shutting one, were investigated in the 1990s as a basis for lambda calculi.  ...  Semantics are given in cartesian closed categories equipped with an adjunction of endofunctors, with the necessity modality interpreted by the right adjoint.  ...  .: Differential cohomology in a cohesive infinity-topos. arXiv:1310.7930 (2013) 37. Shulman, M.: Brouwer's fixed-point theorem in real-cohesive homotopy type theory. Math. Structures Comput.  ... 
doi:10.1007/978-3-319-89366-2_14 fatcat:fwvnexqftrbknem2xhdacwgpau

Fitch-Style Modal Lambda Calculi [article]

Ranald Clouston
2018 arXiv   pre-print
Fitch-style modal deduction, in which modalities are eliminated by opening a subordinate proof, and introduced by shutting one, were investigated in the 1990s as a basis for lambda calculi.  ...  Semantics are given in cartesian closed categories equipped with an adjunction of endofunctors, with the necessity modality interpreted by the right adjoint.  ...  .: Differential cohomology in a cohesive infinity-topos. arXiv:1310.7930 (2013) 37. Shulman, M.: Brouwer's fixed-point theorem in real-cohesive homotopy type theory. Math. Structures Comput.  ... 
arXiv:1710.08326v2 fatcat:o2xdujjh4baoji2pun754pba4i

Higher Inductive Types and Internal Parametricity for Cubical Type Theory

Evan Cavallo
2021
We realize a system of generalized quotients for cubical type theory originally conceived in homotopy type theory, called higher inductive types, that merges the concepts of inductive type and quotient  ...  Cubical type theory provides a constructive interpretation of homotopy type theory and the Univalent Foundations, formalisms that introduced the idea of isomorphism as equality but which lack intrinsic  ...  an equivalence in the homotopy type theory and cubical type theory community.  ... 
doi:10.1184/r1/14555691.v1 fatcat:couyfgkz7za6xl4zhsvlo4t7eu

Simulation of Piecewise Smooth Differential Algebraic Equations with Application to Gas Networks

Tom Streubel, Humboldt-Universität Zu Berlin
2022
well as the piecewise linear Newton method in advance.  ...  We will consider piecewise differentiable functions in so-called abs-normal form.  ...  Caren Tischendorf who helped and enabled me to engage into my doctoral studies, after I have finished and received my diploma degree in mathematics.  ... 
doi:10.18452/24688 fatcat:qxos2u6ycrhdnceiulfpfnbkxm