...t a formal axiomatic theory, obtained within a definite [[Meta-theory|meta-theory]].
...imbedded (in a structure-preserving way) into (a power of) the particular category under consideration.
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[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]
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...n every normal epimorphism is a cokernel. In an [[Abelian category|Abelian category]] every epimorphism is normal. The concept of a normal epimorphism is dual
[[Category:Category theory; homological algebra]]
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...ace $X$ with values in a category $\def\cK{ {\mathcal K}}\cK$'' (e.g., the category of sets, groups, modules, rings, etc.)
[[Category|category]] of open sets of $X$ and their natural inclusion mappings into $\cK$. Depe
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...nother. Two categories are equivalent if and only if their [[Skeleton of a category|skeletons]] are isomorphic.
...ts (cf. the editorial comments to [[Category]] for the notion of a Kleisli category of a triple).
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A [[subcategory]] $\mathfrak C$ of a [[category]] $\mathfrak K$ such that for any objects $A$ and $B$ from $\mathfrak C$ on
...lass of its objects. Conversely, any subclass of the class of objects of a category $\mathfrak K$ uniquely defines a full subcategory, for which it serves as t
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A category $\mathfrak K$ in which subcategories of epimorphisms $\mathfrak E$ and of m
...mathfrak E\cap\mathfrak M$ coincides with the class of isomorphisms in the category $\mathfrak R$.
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''of a family of objects in a category''
...t of a family of objects in a category|product of a family of objects in a category]].
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''in a category''
...t). An equivalent definition of a monomorphism is: For any object $X$ of a category $\mathfrak{K}$ the mapping of sets induced by $\mu$,
2 KB (279 words) - 05:35, 12 January 2017
''of a category''
...ects and the class of morphisms, respectively. The class of morphisms of a category $\mathfrak{K}$ is usually denoted by $\operatorname{Mor} \mathfrak{K}$.
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...rns out to be the kernel of its cokernel. In an [[Abelian category|Abelian category]] every monomorphism is normal. The concept of a normal monomorphism is dua
...isomorphism of $G$ onto a normal subgroup of $H$. However, in an additive category the concepts of normal monomorphism and regular monomorphism coincide.
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''category of sequences''
...relation. Then $\mathbb{Z}$ can be considered as a [[Small category|small category]] with integers as objects and all possible pairs $(i,j)$, where $i,j \in \
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...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...[[Closed monoidal category|closed monoidal category]] (cf. also [[Category|Category]]). A [[Functor|functor]] $( - ) ^ { * } : \cal C ^ { \operatorname{op} } \
3 KB (375 words) - 17:46, 1 July 2020
...f a category|Null object of a category]]). An axiomatic description of the category of groups was given by P. Leroux [[#References|[3]]].
...oup object|Group object]]) in $K$ and the homomorphisms between them; this category has some of the properties of $K$; in particular, it is complete if $K$ is
3 KB (379 words) - 05:17, 12 January 2017
[[Category:Topology]]
...set|nowhere dense sets]] in $X$, otherwise $E$ is said to be of the second category (cp. with Chapter 9 of {{Cite|Ox}}).
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$#C+1 = 28 : ~/encyclopedia/old_files/data/Q076/Q.0706870 Quotient category
be an arbitrary [[Category|category]], and suppose that an equivalence relation $ \sim $
2 KB (279 words) - 08:09, 6 June 2020
''terminal object, of a category''
...ight null object of $\mathfrak{K}$. A left null or ''initial object'' of a category is defined in the dual way.
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An [[Abelian category]] with a set of generators (cf. [[Generator of a category]]) and satisfying the following axiom: There exist [[coproduct]]s (sums) of
...y]]) are Grothendieck categories. A full subcategory $\mathfrak{S}$ of the category ${}_R \mathfrak{M}$ of left $R$-modules is known as a ''localizing subcateg
2 KB (366 words) - 19:42, 30 October 2016
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
A [[category]] $\mathfrak{C}$ such that the following axioms are satisfied:
These conditions are equivalent to the following: $\mathfrak{C}$ is a category with given products such that the functors
2 KB (374 words) - 20:31, 27 December 2017
...e subsets often do not); see [[#References|[a1]]], for example. In lattice theory, least upper bounds of directed subsets again play a distinctive part; see
[[Category:Order, lattices, ordered algebraic structures]]
2 KB (292 words) - 06:36, 14 October 2014
...ral transformations, the category of modules over $\Gamma$ is an [[Abelian category]], so one can do [[homological algebra]] with these objects.
...ilarly, equivariant local cohomology can be described using modules over a category depending on the space in question.
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[[Category:Linear and multilinear algebra; matrix theory]]
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...re all sets belonging to $U$, with morphisms and composition as above. The category of sets may be denoted by $\mathfrak S$, ENS, $\mathsf{Set}$ or Me.
...t every epimorphism is split is equivalent to the [[axiom of choice]]. The category of sets has a unique [[Bicategory(2)|bicategory]] (factorization) structure
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[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]
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A [[category]] with an additional structure, thanks to which the internal Hom-functor ca
A category $\mathfrak{M}$ is said to be closed if a [[bifunctor]] $\otimes: \mathfrak{
3 KB (412 words) - 20:13, 22 December 2017
...}}(Y,X)$ defines a contravariant functor $h_X$ from $\mathcal{C}$ into the category of sets. For any object $F$ of $\hat{\mathcal{C}}$ there exists a natural b
...ieck functor it is possible to define algebraic structures on objects of a category (cf. [[Group object]]; [[Group scheme]]).
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[[Category:Number theory]]
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$#C+1 = 35 : ~/encyclopedia/old_files/data/M064/M.0604480 Modules, category of
The [[Category|category]] mod- $ R $
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[[Category:Number theory]]
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[[Category:Classical measure theory]]
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[[Category:Classical measure theory]]
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...pological vector space which is not a set of the [[Category of a set|first category]] is ultra-barrelled. If a [[locally convex space]] is ultra-barrelled, it
...gn="top">[1]</TD> <TD valign="top"> R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)</TD></TR>
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...system with multiple inputs and multiple outputs; see [[Automatic control theory]].
[[Category:Control theory and optimization]]
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...an 4 cannot, in general, be solved by radicals (see [[Galois theory|Galois theory]]).
...Many questions of the theory of radicals have been studied within category theory. See also [[Radical of a group|Radical of a group]]; [[Radical in a class o
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...lexes or simplicial decompositions. Simplicial spaces are the objects of a category whose morphisms $X\to Y$ are mappings such that every simplex of the triang
...gical spaces (cf. [[Simplicial object in a category|Simplicial object in a category]]).
2 KB (252 words) - 16:30, 9 April 2014
[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]
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...s, the exponential law makes the [[category of sets]] a [[Cartesian-closed category]].
* Benjamin C. Pierce, ''Basic Category Theory for Computer Scientists'', MIT Press (1991) {{ISBN|0262660717}}
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[[Category:Group theory and generalizations]]
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...kernel of a homomorphism of groups, rings, etc. Let $\mathfrak{K}$ be a [[category]] with zero or [[null morphism]]s. A morphism $\mu : K \to A$ is called a k
...$ contains a null object (cf. [[Null object of a category|Null object of a category]]).
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[[Category:Classical measure theory]]
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...reflective if it contains a reflection (cf. [[Reflection of an object of a category]]) for every object of $\mathfrak{K}$. Equivalently, $\mathfrak{C}$ is refl
...$\mathfrak{C}$. Thus, a reflective subcategory of a complete (cocomplete) category is complete (cocomplete).
4 KB (670 words) - 09:05, 26 November 2023
The ''fibre product of objects in a category'' is
...t|(inverse or projective) limit]]. Let $\def\fK{ {\mathfrak K}}\fK$ be a [[category]] and let $\def\a{\alpha}\a : A\to C$ and $\def\b{\beta}\b : B\to C$ be giv
3 KB (575 words) - 10:30, 23 November 2013
...mappings of sets. A [[Morphism|morphism]] $\pi : A \to B$ in a [[Category|category]] $\mathfrak{N}$ is called an epimorphism if $\alpha \, \pi = \beta \, \pi$
...ct of two epimorphisms is an epimorphism. Therefore, all epimorphisms of a category $\mathfrak{N}$ form a subcategory of $\mathfrak{N}$ (denoted by $\operatorn
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See also [[Duality|Duality]] in the theory of [[topological vector space]]s.
[[Category:Linear and multilinear algebra; matrix theory]]
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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
...oherent sheaf is similarly defined on a [[Topologized category|topologized category]] with a sheaf of rings.
gives rise to an equivalence of the category of quasi-coherent sheaves of $ {\mathcal A} $-
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...of several arguments, defined on categories, taking values in a [[Category|category]] and giving a one-place [[Functor|functor]] in each argument. More precise
be given. Construct the Cartesian product category $ \mathfrak K = \overline{\mathfrak K}\; _ {1} \times \dots \times \overl
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...between ($\mathcal{U}$-) categories, and in order to admit other "large" category-theoretic constructions.
...olland (1977) ((especially the article of D.A. Martin on Descriptive set theory))</TD></TR>
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[[Category:Linear and multilinear algebra; matrix theory]]
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''of a category''
...concept of a [[Bicategory(2)|bicategory]]. Suppose that in the [[Category|category]] $ \mathfrak K $
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* Biggs, Norman ''Algebraic graph theory'' 2nd ed. Cambridge University Press (1994) {{ISBN|0-521-45897-8}} {{ZBL|07
[[Category:Graph theory]]
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...mathrm{id}_Y$. In a wider sense, a section of any morphism in an arbitrary category is a right-inverse morphism.
[[Category:Set theory]]
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in an [[Abelian category|Abelian category]] $ C $
...ith enough injective objects (e.g., a [[Grothendieck category|Grothendieck category]] has this property). In such categories an object is injective if and only
4 KB (643 words) - 22:12, 5 June 2020
A [[category]] in which any morphism that is both a [[monomorphism]] and an [[epimorphis
* P.T. Johnstone,"Topos theory", Academic Press (1977) {{MR|0470019}} {{ZBL|0368.18001}}
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[[Category:Group theory and generalizations]]
[[Category:Geometry]]
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''of an object in a category''
be some class of epimorphisms in a [[Category|category]] $ \mathfrak K $
3 KB (460 words) - 08:09, 6 June 2020
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
''of a category''
of a [[Category|category]] $ C $
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...le with the category structure. More precisely, a covariant functor from a category $ \mathfrak K $
into a category $ \mathfrak C $
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...= 77 : ~/encyclopedia/old_files/data/S085/S.0805380 Simplicial object in a category
from the category $ \Delta $,
7 KB (966 words) - 21:39, 10 June 2020
...clopedia/old_files/data/P075/P.0705030 Product of a family of objects in a category
be an indexed family of objects in the category $ \mathfrak K $.
3 KB (545 words) - 08:07, 6 June 2020
...well-chosen levels $p$ are used in mathematical statistics and probability theory to characterize the dispersion (scatter) of probability distributions. E.g.
...ign="top">[1]</TD> <TD valign="top"> G.U. Yale, "An introduction to the theory of statistics" , Griffin (1916)</TD></TR></table>
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...similar way. The gradations by the integers play an important role in the theory of projective algebraic varieties or schemes.
...</TD> <TD valign="top"> C. Nâstâsescu, F. van Oystaeyen, "Graded ring theory" , North-Holland (1982)</TD></TR></table>
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...= 1 \}$ except, possibly, for a [[First category (set of)|set of the first category]] on $\Gamma$, are either Plessner points or Meier points. By definition, a
...set, of the [[Plessner theorem]], which is formulated in terms of measure theory. A sharpening of Meier's theorem is given in [[#References|[3]]].
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...induced fibration in topology, and extension of the ring of scalars in the
theory of modules.
...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
...Riesz spaces and arbitrary Riesz homomorphisms is dually equivalent to the
category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...a distinguished strong unit and unit-preserving morphisms and the familiar
category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
8 KB (1,097 words) - 22:47, 29 November 2014
...tegory]], and as a rule, the morphisms (cf. [[Morphism|Morphism]]) in this
category are the mappings preserving the relations of <img align="absmiddle" border=
<TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Zykov, "The
theory of finite graphs" , '''1''' , Novosibirsk (1969) (In Russian)</TD></TR>
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[[Category:Number theory]]
356 bytes (49 words) - 18:41, 9 November 2014
...separated by a spatial distance and a time duration. In special relativity theory the square of an interval is
In general relativity theory one considers the interval between two infinitesimally-close events:
1 KB (224 words) - 14:51, 30 December 2018
$#C+1 = 90 : ~/encyclopedia/old_files/data/A010/A.0100200 Abelian category
...for an abstract construction of homological algebra [[#References|[4]]]. A category $ \mathfrak A $
10 KB (1,515 words) - 18:19, 31 March 2020
...izable in the category of schemes and require its extensions. However, the category of algebraic spaces is closed under these constructions, which renders the
...e category of schemes becomes identical with a complete subcategory of the category of algebraic spaces.
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[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]
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The ''Grothendieck group of an additive category''
...ng property. More exactly, let $C$ be a small [[Additive category|additive category]] with set of objects $\mathrm{Ob}(C)$ and let $G$ be an Abelian group. A m
4 KB (701 words) - 06:11, 26 March 2023
...n="top">[a3]</TD> <TD valign="top"> J.A. Bondy, U.S.R. Murthy, "Graph theory with applications" , Macmillan (1976)</TD></TR></table>
[[Category:Graph theory]]
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A construction that arose originally in set theory and topology, and then found numerous applications in many areas of mathema
be a functor from a [[Small category|small category]] $ \mathfrak D $
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[[Category:TeX done]]
...gn="top">[a2]</TD> <TD valign="top"> P. Odifreddi, "Classical recursion theory" , North-Holland (1989) pp. Chapt. II; esp. pp. 199ff</TD></TR><TR><TD va
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In the Zermelo–Fraenkel axiom system for [[Set theory|set theory]], the sum-set axiom expresses that the union of a set of sets is a set.
If the sets $A_\alpha$ are disjoint, then in the category $\mathbf{Set}$ the union of the objects $A_\alpha$ is the sum of these obje
2 KB (282 words) - 19:46, 8 November 2014
...ng the multiplication of which is commutative (cf. [[Commutativity]]). The theory of associative-commutative rings with a unit is called [[commutative algebr
[[Category:Algebra]]
293 bytes (40 words) - 20:57, 2 November 2014
...ry the term "pullback" is also used, cf. [[Fibre product of objects in a category]].
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[[Category:Linear and multilinear algebra; matrix theory]]
333 bytes (59 words) - 22:26, 14 November 2014
...theory of Kleinian groups (cf. [[Kleinian group|Kleinian group]]) and the theory of dynamical systems (cf. e.g. [[Limit set of a trajectory|Limit set of a t
[[Category:General topology]]
348 bytes (56 words) - 19:35, 19 October 2014
...example, the terminology "class" . And then, speaking more formally, set theory deals with objects called classes (cf. [[Class|Class]]), for which there is
...and allows one to consider, for example, such "large" collections as the category of all sets, groups, topological spaces, etc.
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A construction that first appeared in set theory, and then became widely used in algebra, topology and other areas of mathem
of a category $ \mathfrak K $
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[[Category:Topology]]
...set of a Baire space is itself a Baire space. By the [[Baire theorem|Baire category theorem]], any [[Complete metric space|complete metric space]] is a Baire s
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[[Category:Field theory and polynomials]]
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[[Category:Branching processes]]
* {{Ref|H}} Th.E. Harris, "The theory of branching processes", Springer (1963)
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The name derives from the representation theory of groups: a permutation (respectively, $ R $-
monic) and epimorphisms; hence if the domain category $ \mathfrak C $
2 KB (249 words) - 19:41, 20 January 2021
are one-place covariant functors from a category $ \mathfrak K $
into a category $ \mathfrak C $.
4 KB (489 words) - 16:56, 23 November 2023
''in the theory of functions of a complex variable''
...align="top">[a3]</TD> <TD valign="top"> J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390</TD></TR></t
2 KB (260 words) - 17:48, 1 November 2014
...e basic terms in classical statistics and [[Probability theory|probability theory]]. In the axiomatic approach it is defined as any decomposition of the spac
[[Category:Probability and statistics]]
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...oincide. In this sense, the theory of discrete spaces is equivalent to the theory of [[partially ordered set]]s. If $(P,{\sqsubseteq})$ is a [[pre-order]]ed
[[Category:General topology]]
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[[Category:Classical measure theory]]
...re-measurable function. In all classical particular cases in which measure theory is developed in topological spaces, e.g. in Euclidean spaces, the concept o
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...on–Bourbaki theorem is especially useful for generalizations of the Galois theory of finite, normal and separable field extensions.
...ubfields of $P$ containing $F$ and the subgroups of $G$ (cf. also [[Galois theory]]). The elements of $G$ are linear operators on the [[vector space]] $P$ ov
3 KB (470 words) - 19:10, 9 November 2016
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...ch nowadays (as of 2000) plays a very important role in the representation theory of algebras. One can also consider the dual notion of cotilting modules.
3 KB (459 words) - 16:55, 1 July 2020
''situation, $n$-tuple of strategies, in the theory of non-cooperative games''
...ult of a choice by all coalitions of action (see [[Games, theory of|Games, theory of]]) of their strategies with regard to the connections between the strate
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...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...all category|small category]], $\mathcal{A}$ an [[Abelian category|Abelian category]] with exact infinite products, and $M : \mathcal{C} \rightarrow \mathcal{A
9 KB (1,283 words) - 20:55, 8 February 2024
...="top">[a1]</TD> <TD valign="top"> I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)</TD></TR></table>
[[Category:Number theory]]
667 bytes (98 words) - 18:57, 18 October 2014
$#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
...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...n subset is of the second category in itself (cf. also [[Category of a set|Category of a set]]). A space $X$ is Baire if and only if the intersection of each c
4 KB (610 words) - 08:48, 18 February 2024