Functional homology operations and the weak homotopy type

1989 ◽  
Vol 45 (5) ◽  
pp. 400-406
Author(s):  
V. A. Smirnov
Keyword(s):  
Homotopy Type ◽  
Functional Homology ◽  
Homology Operations ◽  
Weak Homotopy Type ◽  
Weak Homotopy

On simple maps

2013 ◽  
Author(s):  
Friedhelm Waldhausen ◽  
Bjørn Jahren ◽  
John Rognes
Keyword(s):  
Homotopy Type ◽  
Geometric Realization ◽  
Homotopy Equivalence ◽  
Mapping Cylinder ◽  
Simplicial Sets ◽  
Weak Homotopy ◽  
Formal Properties ◽  
Weak Homotopy Equivalence

This chapter deals with simple maps of finite simplicial sets, along with some of their formal properties. It begins with a discussion of simple maps of simplicial sets, presenting a proposition for the conditions that qualify a map of finite simplicial sets as a simple map. In particular, it considers a simple map as a weak homotopy equivalence. Weak homotopy equivalences have the 2-out-of-3 property, which combines the composition, right cancellation and left cancellation properties. The chapter proceeds by defining some relevant terms, such as Euclidean neighborhood retract, absolute neighborhood retract, Čech homotopy type, and degeneracy operator. It also describes normal subdivision of simplicial sets, geometric realization and subdivision, the reduced mapping cylinder, how to make simplicial sets non-singular, and the approximate lifting property.

The smooth case

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Homotopy Type ◽  
Unit Vector ◽  
Smooth Case ◽  
Open Set ◽  
Birational Invariant

This chapter examines the simplifications occurring in the proof of the main theorem in the smooth case. It begins by stating the theorem about the existence of an F-definable homotopy h : I × unit vector X → unit vector X and the properties for h. It then presents the proof, which depends on two lemmas. The first recaps the proof of Theorem 11.1.1, but on a Zariski dense open set V₀ only. The second uses smoothness to enable a stronger form of inflation, serving to move into V₀. The chapter also considers the birational character of the definable homotopy type in Remark 12.2.4 concerning a birational invariant.

scholarly journals Homotopy type-theoretic interpretations of constructive set theories

2019 ◽  
pp. 1-32
Author(s):  
Cesare Gallozzi
Keyword(s):  
Comparative Analysis ◽  
Set Theory ◽  
Homotopy Type ◽  
Type Theory ◽  
Axiom Of Choice ◽  
Homotopy Type Theory ◽  
Constructive Set Theory

Abstract We introduce a family of (k, h)-interpretations for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of constructive set theory into type theory, in which sets and formulas are interpreted as types of homotopy level k and h, respectively. Depending on the values of the parameters k and h, we are able to interpret different theories, like Aczel’s CZF and Myhill’s CST. We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The rest of the paper is devoted to characterising and analysing the interpretations considered. The formulas valid in the prop-as-hprop interpretation are characterised in terms of the axiom of unique choice. We also analyse the interpretations of CST into homotopy type theory, providing a comparative analysis with Aczel’s interpretation. This is done by formulating in a logic-enriched type theory the key principles used in the proofs of the two interpretations. Finally, we characterise a class of sentences valid in the (k, ∞)-interpretations in terms of the ΠΣ axiom of choice.

scholarly journals Functional homology between the yeast regulatory proteins GAL4 and LAC9: LAC9-mediated transcriptional activation in Kluyveromyces lactis involves protein binding to a regulatory sequence homologous to the GAL4 protein-binding site.

1987 ◽  
Vol 7 (12) ◽  
pp. 4400-4406 ◽  
Author(s):  
K D Breunig ◽  
P Kuger
Keyword(s):  
Protein Binding ◽  
Binding Site ◽  
Transcriptional Activation ◽  
Kluyveromyces Lactis ◽  
Regulatory Protein ◽  
Binding Activity ◽  
Protein Binding Site ◽  
Regulatory Sequence ◽  
Functional Homology ◽  
Beta Galactosidase

As shown previously, the beta-galactosidase gene of Kluyveromyces lactis is transcriptionally regulated via an upstream activation site (UASL) which contains a sequence homologous to the GAL4 protein-binding site in Saccharomyces cerevisiae (M. Ruzzi, K.D. Breunig, A.G. Ficca, and C.P. Hollenberg, Mol. Cell. Biol. 7:991-997, 1987). Here we demonstrate that the region of homology specifically binds a K. lactis regulatory protein. The binding activity was detectable in protein extracts from wild-type cells enriched for DNA-binding proteins by heparin affinity chromatography. These extracts could be used directly for DNase I and exonuclease III protection experiments. A lac9 deletion strain, which fails to induce the beta-galactosidase gene, did not contain the binding factor. The homology of LAC9 protein with GAL4 (J.M. Salmeron and S. A. Johnston, Nucleic Acids Res. 14:7767-7781, 1986) strongly suggests that LAC9 protein binds directly to UASL and plays a role similar to that of GAL4 in regulating transcription.

scholarly journals Homotopy type of posets and lattice complementation

1981 ◽  
Vol 30 (1) ◽  
pp. 90-100 ◽  
Author(s):  
Anders Björner
Keyword(s):  
Homotopy Type

scholarly journals 3-Phosphoinositide-dependent protein kinase-1 (PDK1): structural and functional homology with the Drosophila DSTPK61 kinase

1997 ◽  
Vol 7 (10) ◽  
pp. 776-789 ◽  
Author(s):  
Dario R. Alessi ◽  
Maria Deak ◽  
Antonio Casamayor ◽  
F. Barry Caudwell ◽  
Nick Morrice ◽  
...  
Keyword(s):  
Protein Kinase ◽  
Dependent Protein Kinase ◽  
Functional Homology ◽  
Dependent Protein
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