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A covariant [[Functor|functor]] $F$ from a category $C$ into a category $C_1$ which is injective on the class of morphisms of $C$. ...jects. Some authors use the term "imbedding" as a synonym for "faithful functor" .

...ace $X$ with values in a category $\def\cK{ {\mathcal K}}\cK$'' (e.g., the category of sets, groups, modules, rings, etc.) [[Functor|functor]] $F$ from the

$#C+1 = 12 : ~/encyclopedia/old_files/data/F038/F.0308160 Faithful functor A [[Functor|functor]] which is "injective on Hom-sets" . Explicitly, a functor $ F : \mathfrak C \rightarrow \mathfrak D $

$#C+1 = 21 : ~/encyclopedia/old_files/data/H047/H.0407780 Homology functor A functor on an [[Abelian category|Abelian category]] that defines a certain homological structure on it. A system $ H = {( H

.../c025680/c0256802.png" /> be categories with limits. A one-place covariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l ...ncyclopediaofmath.org/legacyimages/c/c025/c025680/c02568015.png" /> to the category of sets is continuous.

...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).

''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 \

...}}(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]]).

...ategory $A$ with a sufficient number of projective objects into an Abelian category $B$. Further, let $K_{\bullet}$ be a chain complex with values in $A$ and l ...projective resolutions of length $\le n$; or when it is considered on the category of complexes with positive degrees.

...(covariant or contravariant) set-valued [[Functor|functor]] defined on the category. More formally, let $ {\mathcal C} $ be a category and $ F: {\mathcal C} \rightarrow \mathop{\rm Set} $

''multi-place functor'' ...taking values in a [[Category|category]] and giving a one-place [[Functor|functor]] in each argument. More precisely, let $ n $

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...( - ) ^ { * } : \cal C ^ { \operatorname{op} } \rightarrow C$ is a duality functor if there exists an isomorphism $d ( A , B ) : B ^ { A } \overset{\cong}{\ri

...e, thanks to which the internal Hom-functor can be used as a right-adjoint functor to the abstract tensor product. ...\otimes: \mathfrak{M} \times \mathfrak{M} \rightarrow \mathfrak{M}$ (see [[Functor]]) and a distinguished object $I$ are given on it, and if it admits natural

$#C+1 = 33 : ~/encyclopedia/old_files/data/D031/D.0301290 Derived functor A functor "measuring" the deviation of a given functor from being exact. Let $ T ( A , C ) $

A concept in [[Category|category]] theory. Let $ {\mathcal G} : {\mathcal C} \rightarrow {\mathcal D} $ be a [[Functor|functor]] between categories $ {\mathcal C} $

...the inclusion functor, which is the unit of the adjunction (see [[Adjoint functor]]). The concept dual to that of a reflective subcategory is called a corefl ...$\mathfrak{C}$. Thus, a reflective subcategory of a complete (cocomplete) category is complete (cocomplete).

$#C+1 = 106 : ~/encyclopedia/old_files/data/F042/F.0402140 Functor ...le with the category structure. More precisely, a covariant functor from a category $ \mathfrak K $

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. Given a monad (or [[triple]]) $T$ in a [[Category|category]] $\mathcal{C}$, a $T$-algebra is a pair $( A , \alpha )$, $\alpha : T A \r

...ical structure. A [[category]] $\mathfrak{L}$ is called a subcategory of a category $\mathfrak{K}$ if $\mathrm{Ob}(\mathfrak{L})\subseteq \mathrm{Ob}(\mathfrak ...This result enables one to construct the completion of an arbitrary small category with respect to limits or co-limits.

$#C+1 = 43 : ~/encyclopedia/old_files/data/R081/R.0801340 Representable functor A covariant (or contravariant) functor $ F $

and a fixed functor $ \Delta : \mathfrak A \rightarrow \mathfrak M $( cf. [[Abelian category|Abelian category]]). The functor $ \Delta $

''to a category $\mathcal{C}$'' ...at of the direct product to that of the direct sum, etc. A contravariant [[functor]] on $\mathcal{C}$ becomes covariant on $\mathcal{C}^o$.

$#C+1 = 108 : ~/encyclopedia/old_files/data/A010/A.0100820 Adjoint functor be a covariant functor in one argument from a category $ \mathfrak K $

...}$ is locally small. In particular, the small categories form the [[closed category]] $\textsf{Cat}$ of small categories, one of the basic categories of mathem <TR><TD valign="top">[1]</TD> <TD valign="top"> F.W. Lawvere, "The category of categories as a foundation for mathematics" S. Eilenberg (ed.) et al.

...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

...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.

A concept which singles out objects in a [[Category|category]] that have intrinsically the properties of a mathematical structure with a be a category with coproducts. An object $ U \in \mathop{\rm Ob} \mathfrak N $

be an exact sequence of chain complexes in an Abelian category. Then there are morphisms They are called connecting (or boundary) morphisms. Their definition in the category of modules is especially simple: For $ h \in H _ {n} ( M _ {\mathbf . }

...variant functors from the category of $A$-modules into itself and into the category of $A$-algebras. For any two $A$-modules $M$ and $N$ there is a natural iso ...lying functor from the category of commutative unitary $A$-algebras to the category of $A$-modules.

A concept in category theory. Other names are [[Triple|triple]], monad and functor-algebra. Let $\mathfrak{S}$ be a [[category]]. A standard construction is a functor $T : \mathfrak{S} \to \mathfrak{S}$ equipped with natural transformations $

defines a functor from the category of topological (pointed) spaces into itself. Since the suspension operation is a functor, one can define a homomorphism $ \pi _ {n} ( X) \rightarrow \pi _ {n + 1

...= 77 : ~/encyclopedia/old_files/data/S085/S.0805380 Simplicial object in a category A contravariant functor $ X: \Delta \rightarrow {\mathcal C} $(

...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...of manifolds only. This led to an analogous concept of bundle functor on a category over manifolds, [[#References|[a1]]].

...~/encyclopedia/old_files/data/R080/R.0800530 Reflection of an object of a category, ''reflector of an object of a category''

...e limit is the notion of an inductive limit (direct limit or colimit) of a functor. An object $ A $ of a category $ \mathfrak K $

A group object in the category of [[super-manifold]]s. A super-group $ {\mathcal G} $ is defined by a functor $ {\mathcal G} $

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...ducts, and $M : \mathcal{C} \rightarrow \mathcal{A}$ a covariant [[Functor|functor]]. Define the objects $C ^ { n } ( \mathcal{C} , M )$ for $n \geq 0$ in the

...png" /> of open sets in Banach spaces and their analytic mappings into the category of sheaves of sets on <img align="absmiddle" border="0" src="https://www.en ...yclopediaofmath.org/legacyimages/b/b015/b015140/b01514020.png" /> into the category of sheaves of sets in <img align="absmiddle" border="0" src="https://www.en

...n of this concept of projective limit is that of the projective limit of a functor. Let $ F : \mathfrak D \rightarrow \mathfrak K $ be a functor from a [[Small category|small category]] $ \mathfrak D $

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...h|graph]], i.e. $O G$ and $A G$ are finite sets. A diagram in a [[Category|category]] $\mathcal{C}$ is defined as a diagram $G \rightarrow U \mathcal{C}$, wher

...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 $,

''of a category'' of a [[Category|category]] $ C $

A [[category]] $\mathcal{C}$ is monoidal if it consists of the following data: 1) a category $\mathcal{C}$;

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

$#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

[[Algebraic group|algebraic group]]. Let ${\rm Sch}/S$ be the category of [[Group object|group object]] of this category is known as a group

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

...= 25 : ~/encyclopedia/old_files/data/P075/P.0705300 Projective object of a category of a category $ \mathfrak K $

...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 $\mathcal{F}$

$#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]]]

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

...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.

are one-place covariant functors from a category $ \mathfrak K $ into a category $ \mathfrak C $.

...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 $

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 $Y$.

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} $

defined in [[#References|[a1]]], on the [[Category|category]] $ G $- For every contravariant [[Functor|functor]] $ M $

...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;

...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;

is a functor from the category of vector spaces over $ \mathbf R $ into the category of vector space over $ \mathbf C $.

obtained by applying the functor $ O ^ {+} $ to the simplicial scheme in 2), which is a contra-variant functor on the category $ \Delta $(

$#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 $

...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...py classes of mappings of spectra. With $X$ fixed, this is a contravariant functor of $W$ which satisfies the axioms of E.H. Brown (see [[#References|[a1]]])

$#C+1 = 16 : ~/encyclopedia/old_files/data/G043/G.0403930 Generator of a category, An object in a category $ \mathfrak C $

...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...ame { Ext } _ { A } ^ { 1 } ( T , - )$ (cf. also [[Tilting functor|Tilting functor]]). The particular case where $A$ is a hereditary algebra gives rise to the

...mathrm{Ext}^n_R(A,B)$, $n=1,2,\ldots$, are the [[derived functor]]s of the functor $\mathrm{Hom}_R(A,B)$, and may be computed using a [[projective resolution] ...ective hull]] or envelope of $A$. The notion can be defined in any Abelian category, cf. [[#References|[a1]]]. The dual notion is that of a [[projective coveri

''topologized category.'' ...[[Sheaf|sheaf]] on the category. The motivating example has as underlying category the lattice $ {\mathcal O} ( X) $

is an Abelian group; therefore, the functor $ \pi ^ {n} $ is often regarded as a functor defined only on the category of CW-complexes of dimension at most $ 2 n - 2 $,

''monad, on a category $ \mathfrak R $'' A [[Monoid|monoid]] in the [[Category|category]] of all endomorphism functors on $ \mathfrak R $.

one considers the relative Picard functor $ \mathop{\rm Pic}\nolimits _{X/S} $ in the category $ \mathop{\rm Sch}\nolimits /S $

$#C+1 = 35 : ~/encyclopedia/old_files/data/M064/M.0604480 Modules, category of The [[Category|category]] mod- $ R $

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

defines a left-exact functor from the category of sheaves of Abelian groups on $ X $ into the category of Abelian groups. The value of the corresponding $ i $-

...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...equent proof using his $T$-functor (cf. also [[Lannes-T-functor|Lannes $T$-functor]]).

...which belong to the category pro-$\text{Ho}$, associated with the homotopy category [[#References|[a8]]]. ...groups obtained from a polyhedral expansion of $X$ upon application of the functor $\pi_1$. Other classical theorems of homotopy theory also have their shape-

be a [[Category|category]] whose objects are the natural numbers including zero, where the object $ Then a functor $ F $

...utative. In other words, a co-algebra is the dual concept (in the sense of category theory) to the concept of an associative algebra over a ring $k$. ...extsf{Coalg}_k$ denote, respectively, the category of $k$-algebras and the category of $k$-co-algebras, [[#References|[a2]]]; cf. also [[Hopf algebra]]. But if

The functor $ R \Psi _ {f} $ is called the nearby cycle functor. There is a distinguished triangle

...category]] in a completely similar way, [[#References|[a1]]]. E.g., in the category of sheaves of Abelian groups on a topological space an injective resolution ...derived functors. In order to construct derived functors in a non-additive category, the technique of simplicial resolutions is used [[#References|[a4]]].

...categories of modules over rings (cf. [[Grothendieck category|Grothendieck category]]). Let $ \mathfrak A $ be an [[Abelian category|Abelian category]]. A full subcategory $ {\mathfrak A ^ \prime } $

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

...then this functor determines an equivalence of that category and its dual category (see

is open. The functor $ \mathop{\rm exp} X $ ...of compacta and continuous mappings into the same category is a covariant functor of exponential type. Here to a morphism $ f $

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. in ${}_{R}\operatorname{Mod}$, the category of finitely-generated left $R$-modules, with the following properties:

...tegory]] and which makes it possible to define the homotopy groups of this category, the homology and cohomology groups with values in an Abelian group, etc. defines a functor from $ C $

Quite generally, a torsion theory for an Abelian category $ {\mathcal C} $ from a torsion theory for the category $ R \textrm{ - Mod } $

...defines a duality between the category of algebraic tori over $k$ and the category of $\Bbb Z$-free $G$-modules of finite rank. An algebraic torus over $k$ th

be a commutative group. The functor $ D ( \Gamma ) : R \rightarrow {\mathcal G} {\mathcal r} ( \Gamma , R ^ is the category of groups. Group schemes isomorphic to such group schemes are called diagon

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

...ry, calls $\phi$ a left adjoint and $\phi'$ a right adjoint (see [[Adjoint functor]]). For the antitone analogues of residuated mappings see [[Galois correspo

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...f the work [[#References|[a1]]] of J. Lannes on unstable modules and the T-functor has been to expand this knowledge to include many cases in which the source

...the homotopy category of simply-connected rational spaces and the homotopy category of connected differential graded Lie algebras (cf. also ...imply-connected rational spaces with finite Betti numbers and the homotopy category of rational commutative differential graded algebras, $(A, d)$, such that $

..."+" denotes the functor from the category of topological spaces into the category of pointed spaces $ X ^ {+} = ( X \cup x _ {0} , x _ {0} ) $. The natural transformation functor $ \mathop{\rm ch} _ {h ^ {*} } $

...o homology, depend contravariantly, as a rule, on the objects of the basic category on which they are defined. In contrast to homology, connecting homomorphism ...es) it does not form a cohomology functor (see [[Homology functor|Homology functor]]). In the case when $ {\mathcal F} $

th right [[Derived functor|derived functor]] of the functor $ V \mapsto V ^ {\mathfrak G} $ from the category of $ \mathfrak G $-

th [[Derived functor|derived functor]] of the functor $ A \mapsto H ^ {0} ( G, A) $. in the category of $ G $-

...ng every piecewise-smooth curve in the base space of a bundle in the given category, which is compatible with the isomorphism of the corresponding fibres of th be the category of all bundles associated with $ X $.

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...hape mappings $\overline { f } : X \rightarrow Y$ as morphisms (cf. also [[Category]]; [[Topological space]]). There are strong shape categories for arbitrary

...h fibre is linear. The set of vector bundles and their morphisms forms the category $ \mathbf{Bund} $. of the category $ \mathbf{Bund} $.

$#C+1 = 30 : ~/encyclopedia/old_files/data/Z110/Z.1100010 Zariski category An abstract model of the [[Category|category]] of commutative algebras (cf. [[Commutative algebra|Commutative algebra]])

theory one constructs a contravariant functor from the category of schemes into the category of graded commutative rings . The $ K $-

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...lso essential for the [[Closed category|closed category]] structure on the category of crossed complexes. However, the reduced case, i.e. when $C _ { 0 }$ is a

respectively (cf. [[Grothendieck group|Grothendieck group]]). The functor $ K _ {0} X $ ...variant functor from the category of schemes and proper morphisms into the category of Abelian groups. In this case, for a [[Proper morphism|proper morphism]]

3) The functor $ M \mapsto \mathop{\rm Ext} _ {A} ^ {n} ( M, A) $, defined on the category of $ A $-

A class of special functors from the category of pairs of spaces into the category of graded Abelian groups. is a functor from the category $ P $

modules). The resulting contravariant functor from the category of diagonalizable $ \Gamma $ - morphisms into the category of finitely-generated Abelian groups without $ k $ -

constitute a [[Functor|functor]] $ \pi _{n} $ ...tegory|category]] of pointed pairs into the category of pointed sets. This functor is homotopy invariant, i.e. $ f _ \star = g _ \star $

...ry of commutative rings with unit element into the category of rings. This functor may be represented by the ring of polynomials $ \mathbf Z [X _{0} \dots X simultaneously, [[#References|[a3]]]: a functor $ W : \ \mathbf{Ring} \rightarrow \mathbf{Ring} $

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...} \rightarrow 0$ with $T _ { 1 }$ and $T_2$ in $\operatorname{add} T$, the category of finite direct sums of direct summands of $T$. Here, $\operatorname {p.di

...bjects that preserve the mappings. These correspondences became known as [[Functor|'''functors''']]. The principal advantages of this language — the amount ...ons in several complex variables (cf. [[Grothendieck category|Grothendieck category]]).

...ies may be characterized as those equipped with a forgetful functor to the category of sets which is monadic (cf. [[Triple|Triple]]) and preserves filtered col ...exactly those which are Abelian categories (cf. [[Abelian category|Abelian category]]). Note that the second of these classes of varieties is closed under Mori

...between two rings $R$ and $S$ it is necessary and sufficient that in the [[category]] of left $R$-modules there is a finitely-generated projective generator $U For generating objects of categories see also [[Generator of a category]].

(see [[Functor|Functor]] $ \mathop{\rm Ext} $), If in an Abelian category the functor $ \mathop{\rm Hom} $

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...hcal{B}$ and $B$ are "classifying space functors" , and $U$ is a forgetful functor. One seeks such schemes in which $\Pi$ preserves certain useful co-limits,

which is a covariant functor on the category of pairs $ ( X, A) $ which is a contravariant functor of $ ( X, A) $.

This defines a functor from the category of finite sets into the category of topological spaces. If $ B \subset A $

...ncrete case, is given explicitly. A functor of families is a contravariant functor $ {\mathcal M} $ from the category of the schemes (or spaces) into the category of sets defined as follows: $ {\mathcal M} (S) $

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. In this picture, the functorial action of the Weil functor $T _ { A }$ on mappings between the $\mathbf{R} ^ { m }$ (cf. [[Weil algebr

...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...denotes the projective dimension of $T$ and $\operatorname{add} T$ is the category of finite direct sums of direct summands of $T$ (see [[Tilting module|Tilti

is a covariant functor from the category of groups into the category of Abelian groups. If $ f : \Pi \rightarrow \Pi $

A simplicial object in the category of sets $ \mathop{\rm Ens} $( cf. [[Simplicial object in a category|Simplicial object in a category]]), that is, a system of sets ( $ n $-

...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...Category|category]] of all unstable right modules over $\mathcal{A}$. This category has enough projective objects; indeed, there is an object $G ( n )$, $n \ge

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...( \lambda ) ) _ { \theta _ { \lambda } }$ defines an [[Exact functor|exact functor]] in $\mathcal{O}$. If $r = \operatorname { dim } \mathfrak{n}^-$, let

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ''braided monoidal category, quasi-tensor category''

is a group object in the category of connected affine formal schemes over $ k $ ( be the functor that associates with an algebra $ B $

...(cohomology) theory. An axiomatic homology theory is defined on a certain category of pairs $ ( X, A) $ in the category under consideration;

...ons and identities: it is the semi-group of endomorphisms of the forgetful functor from the variety to sets.

...sdorff topological groups; its existence is then guaranteed by the adjoint functor theorem. This approach is also used for semi-groups and leads to several ot

functor, and Chern classes) under admissible monoidal transformations see [[#Refere ...valign="top">[5]</TD> <TD valign="top"> Yu.I. Manin, "Lectures on the $K$-functor in algebraic geometry" ''Russian Math. Surveys'' , '''24''' (1969) pp. 1–

...heory of representable functors (cf. [[Representable functor|Representable functor]]), formal geometry, [[Weil cohomology|Weil cohomology]]; [[K-theory| $ K ...had their influence in many branches of mathematics (commutative algebra, category theory, the theory of analytic spaces and topology).

...establishes an anti-equivalence of these categories. In particular, in the category of affine schemes, there are finite direct sums and fiber products, dual to ...d '''quasi-coherent'''. The category of $ A $-modules is equivalent to the category of quasi-coherent sheaves of $ \widetilde{A} $-modules on $ \operatorname{S

is defined as the category of finite copowers $ T( n) $ or a $ T $-algebra, in a category $ C $

[[K-functor|$K$-functor]]); it is a part of general linear algebra. It deals with the structure the ...K$-theory makes extensive use of the theory of rings, homological algebra, category theory and the theory of linear groups.

...f schemes $\Spec k(x)\to X$. An important property is the existence in the category of schemes of direct and fibre products (cf. [[Fibre product of objects in a category|Fibre product of objects in a category]]), which generalize the concept of the tensor product of rings. The underl

$#C+1 = 423 : ~/encyclopedia/old_files/data/C020/C.0200740 Category ...closed with respect to successive composition (or product) of mappings. A category $ \mathfrak C $

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...numbers) takes the basic structure of the natural numbers object $N$ in a category $\mathcal{C}$ to consist of the zero element $o : 1 \rightarrow N$ (where $

...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...impression is confirmed by the existence of a functorial classifying space functor $B ( \mu )$ of such a crossed module whose homotopy groups are $\text{Coker

a functor on the category of $ k $-

the category of left (respectively, coherent) $ \mathcal D _ {X} $-modules. Denote by the [[Derived category|derived category]] of bounded complexes of left $ \mathcal D _ {X} $-modules. Let $ f :

[[Category|Category]]), in particular, methods of ...index set $J$, the direct sum and direct product of $\{M_i\}$ exist in the category of $A$-modules. Here an element of the direct product $\prod_{i \in J} M_i$

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...e are variants of this theorem depending on the choice of the (co)homology functor $H_{*}$ (respectively, $H ^ { * }$) when studying homomorphisms $f _{*} : H

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...es $X$, and in [[#References|[a5]]], where it was shown that on a suitable category of $C ^ { * }$-algebras, $\operatorname { Ext } ( A )$ fits into a short [[

becomes a category with the $ G $-restrictions as morphisms. is a category with $ P $-restrictions $ \widetilde{f} $

...or|functor]] from the [[Category|category]] of associative algebras to the category of linear spaces. The inclusion of complexes

Instead of taking the category of all pairs of spaces as the domain of definition of $ H _ {r} $, ...y of pairs consisting of polyhedra and their subpolyhedra. However, such a category must contain along with $ ( X, A) $

...://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b1202302.png" />-category'' ...nces|[a14]]], [[#References|[a19]]], [[#References|[a22]]] is a [[Category|category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org

...ed cohomology theories|Generalized cohomology theories]]) generated by the category of vector bundles (cf. [[Vector bundle|Vector bundle]]). ...Grothendieck groups (cf. [[Grothendieck group|Grothendieck group]]) of the category of vector bundles with $ X $

denotes [[Suspension|suspension]]. Spectra of spaces form a category; a morphism of a spectrum $ \mathbf M $ ...nt spectra of spaces may be introduced, and one may construct the homotopy category of spectra [[#References|[2]]]. Postnikov systems (cf. [[Postnikov system|P

...e other. In fact, it may be claimed that, at a very basic level, logic and category theory are the same. ...also been used for presenting the foundations of mathematics, and here too category theory has something to say.

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...\rightarrow {\bf R} ^ { k }$ and so each Weil algebra gives rise to a Weil functor $T _ { A }$. (See [[Weil bundle|Weil bundle]] for more details.)

...the category of finite CW-complexes and taking values in a certain Abelian category $ A $. In the case of the functor $ k = H ^ {*} ( - ; \mathbf Z _ {2} ) $

is a contravariant functor $ X \rightarrow H ^ {*} ( X) $ from the category of varieties into the category of finite-dimensional graded anti-commutative $ K $-

is a functor from the category of pointed topological spaces into the category of (non-Abelian) groups. For any path $ \phi $

...gon can be considered as a [[functor]] from a one-object category into the category of sets. ...$M$ are usually called $M$-sets; the term "operand" is also in use. The category of all $M$-sets ($M$ fixed) forms a [[topos]]; but for this it is essential

By a concrete category one means a [[Category|category]] $ {\mathcal C} $ ...whose composition is the usual composition of mappings — in other words: a category $ {\mathcal C} $

...$, where $Y$ is non-singular. One obtains a contravariant functor into the category of graded rings, satisfying the projection formula (cf. {{Cite|Fu}}). ...therian scheme (or ring), let $K_n(X)$ denote the $n$-th $K$-group of (the category of) finitely-generated projective modules over $X$; cf. [[Algebraic K-theor

Then the category of sheaves of Abelian groups on $ X _ {et} $ is an Abelian category with a sufficient collection of injective objects. The functor $ \Gamma $

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...y|homology theory]] $h_* ^ { S }$ supposed to be defined on the [[Category|category]] of pairs of compact metric (i.e., metrizable) spaces $\bf K$, satisfying

is an exact functor. In other words, the $ A $- in the category of $ A $-

...tegory]] $\mathcal C$ with products and final object, with a contravariant functor $\mathcal P\colon \mathcal C\to\mathcal C$ (here $\mathcal P(X)$ for $X\in\ A topological category is better known as a [[Site|site]].

All realizations of the same a-complex are isomorphic, so that the functor $ a $ In other words, the functor $ p $

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...t is, it will correspond to a commutative diagram in $HoTop$, the homotopy category of spaces and homotopy classes of mappings, but more is true. The homotopie

...ry]]; [[Sets, category of|Sets, category of]]). More precisely, a concrete category is a pair $ ( {\mathcal C}, U) $ consisting of a category $ {\mathcal C} $

...establishes an isomorphism between the category of Jordan algebras and the category of $ \mathbf Z $- [[Category:Nonassociative rings and algebras]]

...n of (1) in which the tensor product is replaced by an arbitrary two-place functor $ T( A, C) $, on the category of $ \Lambda $-

...joint is faithful if and only if the co-unit is epic. (See also [[Category|Category]].)

...co-multiplication, is the underlying Hopf algebra of the (big) Witt vector functor $R \mapsto W(R)$ (see [[Witt vector]]) and it plays an important role in th [[Category:TeX done]]

...implex in $K_2$. Simplicial complexes and their simplicial mappings form a category. ...|$ is a covariant functor from the category of simplicial complexes to the category of cellular spaces. A topological space $X$ homeomorphic to the body $|K|$

...}) $, and the category of algebraic vector bundles on $ X $ is dual to the category of locally free sheaves of $ \mathcal{O}_{X} $-modules. Moreover, for an $ ...n abstract definition of Chern classes that involves the [[K-functor|$ K $-functor]] or one of the variants of [[Etale cohomology|étale cohomology]].

...tor from the category of open subsets of $X$ and their inclusions into the category of groups (rings, etc.) and their homomorphisms. The mappings $F_V^U$ are c ...has the same classical properties as the category of Abelian groups or the category of modules; in particular, one can define for sheaves direct sums, infinite

...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category. Beilinson has extended motivic cohomology to the category of (Chow) motives with coefficients in a number field $ E $.

...ed for the description of the tangent space to an arbitrary functor in the category of schemes [[#References|[1]]], [[#References|[3]]].

====Category of $\Sigma$-Algebras==== ...$-algebras together with the $\Sigma$-algebra-morphisms forms a [[Category|category]] {{Cite|W90}}.

...t spaces, is a reflective category (cf. [[Reflective subcategory|Reflexive category]]). Spaces with compact completions (i.e. pre-compact spaces) are character ...ons. Such a product, though identical with the product in the sense of the category of proximity spaces, is geometrically inconvenient and mainly serves in the

in the definition may be regarded as a pre-sheaf (i.e., a set-valued functor) on the partially ordered set $ ( S, R) $; is replaced by an arbitrary small category $ {\mathcal C} $)

...y spectral sequences of objects of an arbitrary [[Abelian category|Abelian category]] (e.g., bimodules, rings, algebras, co-algebras, Hopf algebras, etc.). ...ad) spectral sequence is obtained by applying the generalized (co)homology functor $ h _ {*} $ ($ h ^ {*} $)

Thus, the problem of homotopy types on the category of $ \mathop{\rm CW}\nolimits $ - in the category of simplicial sets (with respect to homotopy groups in the sense of Kan, se

...al for the following reason. There is the following general principle: The functor $ X \mapsto \{ \textrm{ the algebra of functions on } X \} $ is an anti-equivalence between the category of "spaces" and the category of commutative associative unital algebras, perhaps with some additional st

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...ed that $E ^{ \otimes r}$ is a completely reducible $G$-module, indeed the category of polynomial representations of $G$ is semi-simple. Weyl modules are defin

...ion theories. Each one of these theories may be related to a contravariant functor from the category of analytic spaces (or germs of analytic spaces) into the category of sets. For instance, in the theory of local deformations of a complex spa

becomes a ring object in the category of co-algebras over $ \mathbf Z $. As a coring object in the category of algebras $ U $,

[[Category|category]] of $A$-schemes $S$, there is the contravariant [[Functor|functor]] $\def\M{\mathcal{M}}\M^r(\fa)$ that to each $S$ associates the set of iso

defines a [[Functor|functor]] of the category of Abelian groups into itself. For torsion in the case of a left module ove

...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...<td valign="top">[a1]</td> <td valign="top"> R.K. Dennis, M.R. Stein, "The functor $K _ { 2 }$: A survey of computations and problems" , ''Algebraic $K$-Theor

...ormatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...</tr><tr><td valign="top">[a9]</td> <td valign="top"> M. Mrozek, "Leray functor and the cohomological Conley index for discrete dynamical systems" ''Trans

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...ctor spaces form a symmetric monoidal [[Abelian category]] with [[Monoidal category|monoidal structure]] defined by

applying the direct image functor to the structure morphism $ X \rightarrow \mathop{\rm Spec} k $ from some category) into correspondence with a ring $ C( X) $

...d above. The fundamental tools for comparing triples are, in the spirit of category theory, continuous homomorphisms between them. .... The [[homology group]], the [[cohomology ring]], and the [[K-functor|$K$-functor]], associated with the concept of a vector bundle over a topological space,

...ompatible with the topology, with proximally continuous mappings, into the category of $ T _ {2} $-

be an arbitrary [[Homology theory|homology theory]] over some admissible category of pairs of spaces and their mappings, i.e. a system which satisfies the [[ ...ms of cohomology theory and represents the cohomology theory over the same category with compact or, respectively, discrete groups $ H ^ {r} (X,\ A) $.

...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...a differential graded augmented algebra and $T ( . )$ is the tensor module functor. The point is that the underlying module structure for both ordinary Tor an

Most algebraic-topological invariants are a so-called [[functor]] on the category of topological spaces of the type studied. This means, roughly speaking, th