...ry|category]] with respect to a certain specified class of objects in this category. The categories of modules over a ring form the principal range of applicat
be a fixed class of objects in an [[Abelian category|Abelian category]] $ \mathfrak A $,
11 KB (1,715 words) - 22:10, 5 June 2020
''of a category''
of a [[Category|category]] $ C $
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A [[category]] $\mathcal{C}$ is monoidal if it consists of the following data:
1) a category $\mathcal{C}$;
4 KB (612 words) - 14:59, 6 April 2023
An equaliser of two morphisms $f,g$ between the objects $X, Y$ of a category $\mathfrak{K}$ is a morphism $e : W \rightarrow X$ such that $ef = eh$ and
An equaliser in the category of sets exists: it is the inclusion map on $\{ x \in X : f(x) = g(x) \}$. S
1 KB (198 words) - 18:08, 14 November 2023
$#C+1 = 27 : ~/encyclopedia/old_files/data/V096/V.0906280 Variety in a category
is a [[well-powered category]], that is, the admissible subobjects of any object form a set, then every
3 KB (532 words) - 08:28, 6 June 2020
[[Algebraic group|algebraic group]]. Let ${\rm Sch}/S$ be the category of
[[Group object|group object]] of this category is known as a group
5 KB (831 words) - 21:59, 5 March 2012
of an [[Abelian category|Abelian category]] $ \mathfrak A $
In this context, local smallness of a category is the condition: A collection of representatives of the isomorphism classe
3 KB (469 words) - 16:39, 17 March 2023
...= 25 : ~/encyclopedia/old_files/data/P075/P.0705300 Projective object of a category
of a category $ \mathfrak K $
4 KB (597 words) - 08:08, 6 June 2020
...chemes. Let $X$ be a scheme. The étale topology on $X$ is the name for the category $X_{\text{et}}$ of étale $X$-schemes the objects of which are étale morph
..._{\text{et}}$ is defined as a contravariant functor $\mathcal{F}$ from the category $X_{\text{et}}$ into that of sets (groups, etc.). A pre-sheaf $\mathcal{F}$
3 KB (538 words) - 20:07, 3 June 2017
$#C+1 = 221 : ~/encyclopedia/old_files/data/D031/D.0301280 Derived category
The notion of a derived category has been introduced by J.-L. Verdier in his 1963 notes [[#References|[a7]]]
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A
category-theoretical construction; special cases are the concept of an induced fibra
...ncyclopediaofmath.org/legacyimages/b/b015/b015310/b0153108.png" />) to the
category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or
11 KB (1,513 words) - 17:08, 7 February 2011
...d in the same way. The differentiable (or analytic) super-manifolds form a category whose morphisms are the morphisms of ringed spaces that are even on the str
...me there are more morphisms in the category of super-manifolds than in the category of vector bundles.
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are one-place covariant functors from a category $ \mathfrak K $
into a category $ \mathfrak C $.
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...theory, etc. (see [[Weil cohomology|Weil cohomology]]) are functors on the category of motives.
be the category of smooth projective varieties over a field $ k $
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A category $\mathfrak C$ in which for any two objects $X$ and $Y$ an Abelian group str
...null object (zero object, cf. [[Null object of a category|Null object of a category]]) as well as the product $X\times Y$ of any two objects $X$ and $Y$.
3 KB (490 words) - 23:53, 10 December 2018
is an object in some [[Additive category|additive category]]. The multiplication in $ \mathop{\rm End} A $
...ddition is the addition of morphisms defined by the axioms of the additive category. The identity morphism $ 1 _ {A} $
4 KB (595 words) - 19:37, 5 June 2020
defined in [[#References|[a1]]], on the [[Category|category]] $ G $-
For every contravariant [[Functor|functor]] $ M $
5 KB (732 words) - 08:45, 26 March 2023
...roup|Picard group]]; the [[Chow ring|Chow ring]]; the [[K-functor| $ K $ -functor]], and the cohomology group are a tool used in the study of the varieties t
...the case of proper flat schemes of morphisms. The scheme representing this functor is not necessarily reduced even if $ X $ is a smooth projective surface;
6 KB (923 words) - 18:31, 12 December 2019
...xiomatic theory of factorization structures $( E , M )$ for morphisms of a category $\frak A$. Here, $E$ and $M$ are classes of $\frak A$-morphisms (the requir
a) factorization structures for sources in a category;
11 KB (1,634 words) - 01:54, 13 February 2024
is a functor from the category of vector spaces over $ \mathbf R $
into the category of vector space over $ \mathbf C $.
1 KB (197 words) - 17:46, 4 June 2020