and a fixed functor $ \Delta : \mathfrak A \rightarrow \mathfrak M $(
cf. [[Abelian category|Abelian category]]). The functor $ \Delta $
3 KB (465 words) - 08:10, 6 June 2020
''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 $\mathcal{C}^o$.
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$#C+1 = 108 : ~/encyclopedia/old_files/data/A010/A.0100820 Adjoint functor
be a covariant functor in one argument from a category $ \mathfrak K $
8 KB (1,301 words) - 20:14, 4 April 2020
...}$ 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.
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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
4 KB (570 words) - 21:02, 21 December 2017
...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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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 $
2 KB (335 words) - 08:14, 6 June 2020
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 . }
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...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.
3 KB (487 words) - 18:21, 11 April 2017
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 $
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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
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...= 77 : ~/encyclopedia/old_files/data/S085/S.0805380 Simplicial object in a category
A contravariant functor $ X: \Delta \rightarrow {\mathcal C} $(
7 KB (966 words) - 21:39, 10 June 2020
...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]]].
4 KB (674 words) - 17:02, 1 July 2020
...~/encyclopedia/old_files/data/R080/R.0800530 Reflection of an object of a category,
''reflector of an object of a category''
2 KB (379 words) - 08:10, 6 June 2020
...e limit is the notion of an inductive limit (direct limit or colimit) of a functor. An object $ A $
of a category $ \mathfrak K $
7 KB (1,089 words) - 22:12, 5 June 2020
A group object in the category of [[super-manifold]]s. A super-group $ {\mathcal G} $
is defined by a functor $ {\mathcal G} $
3 KB (369 words) - 19:04, 18 July 2020
...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
9 KB (1,283 words) - 20:55, 8 February 2024
...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
6 KB (790 words) - 16:56, 7 February 2011
...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 $
5 KB (863 words) - 08:08, 6 June 2020
...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
6 KB (937 words) - 19:26, 11 November 2023