現代数学の系譜11 ガロア理論を読む25 [無断転載禁止]©2ch.net

[過去ﾛｸﾞ] 現代数学の系譜11 ガロア理論を読む25 [無断転載禁止]©2ch.net (716ﾚｽ)
上下前次1-新
通常表示 512ﾊﾞｲﾄ分割ﾚｽ栞
抽出解除ﾚｽ栞

このｽﾚｯﾄﾞは過去ﾛｸﾞ倉庫に格納されています｡
次ｽﾚ検索歴削→次ｽﾚ栞削→次ｽﾚ過去ﾛｸﾞﾒﾆｭｰ

148: 現代数学の系譜11 ガロア理論を読む [sage] 2016/11/05(土) 19:59:07.06 ID:DzICE8Th

https://en.wikipedia.org/wiki/Simplicial_set
Simplicial set
From Wikipedia, the free encyclopedia

In mathematics, a simplicial set is a construction in categorical homotopy theory that is a pure algebraic model of the notion of a "well-behaved" topological space.
Historically, this model arose from earlier work in combinatorial topology and in particular from the notion of simplicial complexes. Simplicial sets are used to define quasi-categories, a basic notion of higher category theory.

History and uses of simplicial sets

Simplicial sets were originally used to give precise and convenient descriptions of classifying spaces of groups. This idea was vastly extended by Grothendieck's idea of considering classifying spaces of categories, and in particular by Quillen's work of algebraic K-theory.
In this work, which earned him a Fields Medal, Quillen developed surprisingly efficient methods for manipulating infinite simplicial sets.
Later these methods were used in other areas on the border between algebraic geometry and topology. For instance, the Andre-Quillen homology of a ring is a "non-abelian homology", defined and studied in this way.

Both the algebraic K-theory and the Andre-Quillen homology are defined using algebraic data to write down a simplicial set, and then taking the homotopy groups of this simplicial set. Sometimes one simply defines the algebraic K {\displaystyle K} K-theory as the space.

In recent years, simplicial sets have been used in higher category theory and derived algebraic geometry. Quasi-categories can be thought of as categories in which the composition of morphisms is defined only up to homotopy, and information about the composition of higher homotopies is also retained.
Quasi-categories are defined as simplicial sets satisfying one additional condition, the weak Kan condition.

http://rio2016.5ch.net/test/read.cgi/math/1477804000/148

631: 現代数学の系譜11 ガロア理論を読む [sage] 2016/12/03(土) 14:03:09.06 ID:6Rgz8i9T

>>628 ついで

http://math.stackexchange.com/questions/86762/finding-a-basis-of-an-infinite-dimensional-vector-space
Finding a basis of an infinite-dimensional vector space?  asked Nov 29 '11 at 16:30 InterestedGuest

2 Answers  answered Jan 20 '12 at 19:25 Qiaochu Yuan

For many infinite-dimensional vector spaces of interest we don't care about describing a basis anyway; they often come with a topology and we can therefore get a lot out of studying dense subspaces, some of which, again, have easily describable bases.
In Hilbert spaces, for example, we care more about orthonormal bases (which are not Hamel bases in the infinite-dimensional case); these span dense subspaces in a particularly nice way.

4.  answered Jan 20 '12 at 19:09 David Wheeler
The "hard case" is essentially equivalent to this one:

Find a basis for the real numbers R over the field of the rational numbers Q.

The reals are obviously an extension field of the rationals, so they form a vector space over Q. It should be clear that such a basis has to be uncountable (for if it were countable, the reals would likewise also be countable).

It should also be clear that such a basis is a subset of {1}∪R?Q. The trouble is, that the power set of the reals is "so big" that it's not even clear how to name the sets we need to apply the axiom of choice TO. Linearly independent subsets however, DO satisfy the requirements for Zorn's Lemma, a form of the Axiom of Choice.

A relatively easy-to-follow proof of the existence of a basis for any vector space using Zorn's Lemma can be found here: http://planetmath.org/encyclopedia/EveryVectorSpaceHasABasis.html

http://rio2016.5ch.net/test/read.cgi/math/1477804000/631

上下前次1-新書関写板覧索設栞歴

ｽﾚ情報赤ﾚｽ抽出画像ﾚｽ抽出歴の未読ｽﾚ AAｻﾑﾈｲﾙ

ぬこの手ぬこTOP 0.061s