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

2017
*
arXiv
*
pre-print

This enables us to reproduce formally some of the classical applications of

arXiv:1509.07584v3
fatcat:lvviw4kyane7xdxua3nedsq5f4
*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). ...##
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Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory
[article]

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 ...

##
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Homotopy type theory: the logic of space
[article]

2017
*
arXiv
*
pre-print

This is an introduction to

arXiv:1703.03007v1
fatcat:osarowd7brb4jli4r6utyw6d6q
*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 ...##
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Dynamics of non-metric manifolds
[article]

2011
*
arXiv
*
pre-print

To handle this situation we use Lefschetz

arXiv:1102.5684v1
fatcat:unx7zcu4krfkfbh65ixjor46t4
*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 . ...##
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Fitch-Style Modal Lambda Calculi
[chapter]

2018
*
Lecture Notes in Computer Science
*

Fitch-style modal deduction,

doi:10.1007/978-3-319-89366-2_14
fatcat:fwvnexqftrbknem2xhdacwgpau
*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. ...##
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Fitch-Style Modal Lambda Calculi
[article]

2018
*
arXiv
*
pre-print

Fitch-style modal deduction,

arXiv:1710.08326v2
fatcat:o2xdujjh4baoji2pun754pba4i
*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. ...##
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Higher Inductive Types and Internal Parametricity for Cubical Type Theory

2021

We realize a system of generalized quotients for cubical

doi:10.1184/r1/14555691.v1
fatcat:couyfgkz7za6xl4zhsvlo4t7eu
*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. ...##
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Simulation of Piecewise Smooth Differential Algebraic Equations with Application to Gas Networks

2022

well as the piecewise linear Newton method

doi:10.18452/24688
fatcat:qxos2u6ycrhdnceiulfpfnbkxm
*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. ...