Topic
Morphism
About: Morphism is a research topic. Over the lifetime, 5803 publications have been published within this topic receiving 79121 citations. The topic is also known as: arrow & 1-cell.
Papers published on a yearly basis
Papers
More filters
Posted Content•
TL;DR: The information deviation between any two finite measures cannot be increased by any statistical operations (Markov morphisms) and is invarient if and only if the morphism is sufficient for these two measures as mentioned in this paper.
Abstract: The information deviation between any two finite measures cannot be increased by any statistical operations (Markov morphisms). It is invarient if and only if the morphism is sufficient for these two measures
5,228 citations
Book•
16 May 2018
TL;DR: In this paper, the existence of solutions for the obstacle problem is investigated and the John-Nirenberg lemma is shown to be true for nonlinear potential theory with respect to a super-harmonic function.
Abstract: Introduction. 1: Weighted Sobolev spaces. 2: Capacity. 3: Supersolutions and the obstacle problem. 4: Refined Sobolev spaces. 5: Variational integrals. 6: A-harmonic functions. 7: A superharmonic functions. 8: Balayage. 9: Perron's method, barriers, and resolutivity. 10: Polar sets. 11: A-harmonic measure. 12: Fine topology. 13: Harmonic morphisms. 14: Quasiregular mappings. 15: Ap-weights and Jacobians of quasiconformal mappings. 16: Axiomatic nonlinear potential theory. Appendix I: The existence of solutions. Appendix II: The John-Nirenberg lemma. Bibliography. List of symbols. Index
2,017 citations
Book•
09 Dec 1980
TL;DR: In this article, the authors present a primer of complete lattices and complete topology of continuous lattices, including the Scott topology and meet-continuous lattices.
Abstract: O. A Primer of Complete Lattices.- 1. Generalities and notation.- 2. Complete lattices.- 3. Galois connections.- 4. Meet-continuous lattices.- I. Lattice Theory of Continuous Lattices.- 1. The "way-below" relation.- 2. The equational characterization.- 3. Irreducible elements.- 4. Algebraic lattices.- II. Topology of Continuous Lattices: The Scott Topology.- 1. The Scott topology.- 2. Scott-continuous functions.- 3. Injective spaces.- 4. Function spaces.- III. Topology of Continuous Lattices: The Lawson Topology.- 1. The Lawson topology.- 2. Meet-continuous lattices revisited.- 3. Lim-inf convergence.- 4. Bases and weights.- IV. Morphisms and Functors.- 1. Duality theory.- 2. Morphisms into chains.- 3. Projective limits and functors which preserve them.- 4. Fixed point construction for functors.- V. Spectral Theory of Continuous Lattices.- 1. The Lemma.- 2. Order generation and topological generation.- 3. Weak irreducibles and weakly prime elements.- 4. Sober spaces and complete lattices.- 5. Duality for continuous Heyting algebras.- VI. Compact Posets and Semilattices.- 1. Pospaces and topological semilattices.- 2. Compact topological semilattices.- 3. The fundamental theorem of compact semilattices.- 4. Some important examples.- 5. Chains in compact pospaces and semilattices.- VII. Topological Algebra and Lattice Theory: Applications.- 1. One-sided topological semilattices.- 2. Topological lattices.- 3. Compact pospaces and continuous Heyting algebras.- 4. Lattices with continuous Scott topology.- Listof Symbols.- List of Categories.
1,428 citations
Book•
09 Jan 2018
TL;DR: Topos theory has been studied at the graduate student level for a long time, see as discussed by the authors for an overview of the main applications of topos in algebraic geometry and logic.
Abstract: This text presents topos theory as it has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various kinds of manifolds (algebraic, analytic, etc.). Sheaves also appear in logic as carriers for models of set theory as well as for the semantics of other types of logic. Grothendieck introduced a topos as a category of sheaves for algebraic geometry. Subsequently, Lawvere and Tierney obtained elementary axioms for such (more general) categories. This introduction to topos theory begins with a number of illustrative examples that explain the origin of these ideas and then describes the sheafification process and the properties of an elementary topos. The applications to axiomatic set theory and the use in forcing (the Independence of the Continuum Hypothesis and of the Axiom of Choice) are then described. Geometric morphisms- like continuous maps of spaces and the construction of classifying topoi, for example those related to local rings and simplicial sets, next appear, followed by the use of locales (pointless spaces) and the construction of topoi related to geometric languages and logic. This is the first text to address all of these varied aspects of topos theory at the graduate student level.
1,166 citations
TL;DR: In this paper, a linear representation of a given category is given by a map V associating with any morphism ϕ: a→e of K a linear vector space map V(ϕ): V(a)→V(e).
Abstract: LetK be the structure got by forgetting the composition law of morphisms in a given category. A linear representation ofK is given by a map V associating with any morphism ϕ: a→e ofK a linear vector space map V(ϕ): V(a)→V(e). We classify thoseK having only finitely many isomorphy classes of indecomposable linear representations. This classification is related to an old paper by Yoshii [3].
862 citations