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Adjoint Functors and Representation Dimensions
Acta Mathematica Sinica, English Series, 2006Let \(\widehat{\mathcal{C}}\) denote the category of coherent functors on a category \(\mathcal{C}\). Suppose that \(\mathcal{C}\) and \(\mathcal{D}\) are additive \(k\)-categories and that \(F,G\) is pair of adjoint functors between them. The author obtains comparisons of \(\text{ gl.dim}(\widehat{\mathcal{C}}) \) with \(\text{ gl.dim}(\widehat ...
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"ADJOINT PREENVELOPES AND ADJOINT PRECOVERS IN THE FUNCTOR CATEGORY"
Mathematical Reports, 2022"Adjoint preenvelopes and adjoint precovers are defined in the category of functors by replacing the functor Hom with ⊗. We investigate the existence and basic properties of adjoint preenvelopes and adjoint precovers. The F-projective (F-injective, F-flat) functors introduced by Mao are characterized in terms of adjoint preenvelopes and adjoint ...
SHOUTAO GUO, XIAOYAN YANG
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On Fundamental Constructions and Adjoint Functors
Canadian Mathematical Bulletin, 1966A fundamental construction of a category ζ((2), Appendice) is a triple (S, p, k), where S is a functor from ζ to itself and 2 where p:S2→S and k:1ζ→S are natural transformations such ...
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Adjointness Aspects of the Down-Set Functor
Applied Categorical Structures, 2001The authors investigate various concepts \(X\) akin to openess for frame homomorphisms. They consider the down-set construction as a functor from the category of Boolean frames to the category of frames and frame homomorphisms satisfying condition \(X\).
Banaschewski, B., Pultr, A.
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Adjunctions and Adjoint Functor Theorems
2021In this section, we will discuss adjunctions between ∞-categories. We will define them in the language of fibrations and show that they may equivalently be described by choosing a binatural transformation of bivariant mapping-space functors. We will give several sufficient criteria for a fixed functor \(f \colon \mathscr {C} \to \mathscr {D}\) to admit
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Equivalences Induced by Adjoint Functors
Communications in Algebra, 2003Abstract Let 𝒜 and ℬ be two Grothendieck categories, R : 𝒜 → ℬ, L : ℬ → 𝒜 a pair of adjoint functors, S ∈ ℬ a generator, and U = L(S). U defines a hereditary torsion class in 𝒜, which is carried by L, under suitable hypotheses, into a hereditary torsion class in ℬ. We investigate necessary and sufficient conditions which
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ORDER EXTENSIONS AS ADJOINT FUNCTORS
Quaestiones Mathematicae, 1986Abstract A standard extension (resp. standard completion) is a function Z assigning to each poset P a (closure) system ZP of subsets such that x ⋚ y iff x belongs to every Z e ZP with y e Z. A poset P is Z -complete if each Z e 2P has a join in P. A map f: P → P′ is Z—continuous if f−1 [Z′] e ZP for all Z′ e ZP′, and a Z—morphism if, in addition, for ...
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RESTRICTED ADJOINTS AND TOPOLOGICAL FUNCTORS
Quaestiones Mathematicae, 1983Abstract The idea of adjoint functors or adjoint situation is one of the most important concepts in category theory. However, many examples are known which deviate from adjoint situations in one respect or another. These gave rise to various generalizations of adjoint situations (e.g. see R. Borger and w. Tholen [l], Y. Diers [2], and F.
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“Topologically Indexed Function Spaces and Adjoint Functors”
Canadian Mathematical Bulletin, 1982AbstractLet Top denote the category of topological spaces and continuous maps. In this paper we discuss families of function spaces indexed by the elements of a topological space T, and their relationship to the characterization of right adjoints Top/S → Top/T, where S is also a topological space. After reducing the problem to the case where S is a one-
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