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" .
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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.)
[[Functor|functor]] $F$ from the
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$#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 $
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$#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
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.../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.
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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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''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 $i,j \in \
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...}}(Y,X)$ 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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...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.
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...(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} $
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''multi-place functor''
...taking values in a [[Category|category]] and giving a one-place [[Functor|functor]] in each argument. More precisely, let $ n $
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...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 $d ( A , B ) : B ^ { A } \overset{\cong}{\ri
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...e, thanks to which the internal Hom-functor can be used as a right-adjoint functor to the abstract tensor product.
...\otimes: \mathfrak{M} \times \mathfrak{M} \rightarrow \mathfrak{M}$ (see [[Functor]]) and a distinguished object $I$ are given on it, and if it admits natural
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$#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 ) $
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A concept in [[Category|category]] theory. Let $ {\mathcal G} : {\mathcal C} \rightarrow {\mathcal D} $
be a [[Functor|functor]] between categories $ {\mathcal C} $
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...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).
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$#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 $
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...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
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...ical structure. A [[category]] $\mathfrak{L}$ is called a subcategory of a category $\mathfrak{K}$ if $\mathrm{Ob}(\mathfrak{L})\subseteq \mathrm{Ob}(\mathfrak
...This result enables one to construct the completion of an arbitrary small category with respect to limits or co-limits.
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$#C+1 = 43 : ~/encyclopedia/old_files/data/R081/R.0801340 Representable functor
A covariant (or contravariant) functor $ F $
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