functor category
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2018 ◽  
Vol 240 ◽  
pp. 1-41 ◽  
Author(s):  
SONDRE KVAMME

Let $k$ be a commutative ring, let ${\mathcal{C}}$ be a small, $k$-linear, Hom-finite, locally bounded category, and let ${\mathcal{B}}$ be a $k$-linear abelian category. We construct a Frobenius exact subcategory ${\mathcal{G}}{\mathcal{P}}({\mathcal{G}}{\mathcal{P}}_{P}({\mathcal{B}}^{{\mathcal{C}}}))$ of the functor category ${\mathcal{B}}^{{\mathcal{C}}}$, and we show that it is a subcategory of the Gorenstein projective objects ${\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$ in ${\mathcal{B}}^{{\mathcal{C}}}$. Furthermore, we obtain criteria for when ${\mathcal{G}}{\mathcal{P}}({\mathcal{G}}{\mathcal{P}}_{P}({\mathcal{B}}^{{\mathcal{C}}}))={\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$. We show in examples that this can be used to compute ${\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$ explicitly.



2018 ◽  
Vol 20 (06) ◽  
pp. 1750071 ◽  
Author(s):  
Abhishek Banerjee

For a small abelian category [Formula: see text], Auslander’s formula allows us to express [Formula: see text] as a quotient of the category [Formula: see text] of coherent functors on [Formula: see text]. We consider an abelian category with the added structure of a cohereditary torsion pair [Formula: see text]. We prove versions of Auslander’s formula for the torsion-free class [Formula: see text] of [Formula: see text], for the derived torsion-free class [Formula: see text] of the triangulated category [Formula: see text] as well as the induced torsion-free class in the ind-category [Formula: see text] of [Formula: see text]. Further, for a given regular cardinal [Formula: see text], we also consider the category [Formula: see text] of [Formula: see text]-presentable objects in the functor category [Formula: see text]. Then, under certain conditions, we show that the torsion-free class [Formula: see text] can be recovered as a subquotient of [Formula: see text].



2015 ◽  
Vol 23 (4) ◽  
pp. 351-369
Author(s):  
Marco Riccardi

Summary In the first part of this article we formalize the concepts of terminal and initial object, categorical product [4] and natural transformation within a free-object category [1]. In particular, we show that this definition of natural transformation is equivalent to the standard definition [13]. Then we introduce the exponential object using its universal property and we show the isomorphism between the exponential object of categories and the functor category [12].





2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Septimiu Crivei

Flat objects of a finitely accessible additive category are described in terms of some objects of the associated functor category of , called strongly flat functors. We study closure properties of the class of strongly flat functors, and we use them to deduce the known result that every object of a finitely accessible abelian category has a flat cover.



2012 ◽  
Vol 11 (05) ◽  
pp. 1250099 ◽  
Author(s):  
NGUYEN VIET DUNG ◽  
JOSÉ LUIS GARCÍA

Let R be a right pure semisimple ring, and [Formula: see text] be a family of sources of left almost split morphisms in mod-R. We study the definable subcategory [Formula: see text] in Mod-R determined by the family [Formula: see text], and show that [Formula: see text] has several nice properties similar to those of the category Mod-R. For example, its functor category [Formula: see text] is a module category, and preinjective objects of [Formula: see text] are sources of left almost split morphisms in [Formula: see text] and in mod-R. As an application, it is shown that if R is a right pure semisimple ring with no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in mod-R (in particular, if R is right pure semisimple hereditary), then any definable subcategory of Mod-R determined by a finite set of indecomposable right R-modules contains only finitely many non-isomorphic indecomposable modules.



Author(s):  
IVO HERZOG ◽  
PHILIPP ROTHMALER

AbstractA notion of good behavior is introduced for a definable subcategory of left R-modules. It is proved that every finitely presented left R-module has a pure projective left -approximation if and only if the associated torsion class of finite type in the functor category (mod-R, Ab) is coherent, i.e., the torsion subobject of every finitely presented object is finitely presented. This yields a bijective correspondence between such well-behaved definable subcategories and preenveloping subcategories of the category Add(R-mod) of pure projective left R-modules. An example is given of a preenveloping subcategory ⊆ Add(R-mod) that does not arise from a covariantly finite subcategory of finitely presented left R-modules. As a general example of this good behavior, it is shown that if R is a ring over which every left cotorsion R-module is pure injective, then the smallest definable subcategory (R-proj) containing every finitely generated projective module is well-behaved.





2003 ◽  
Vol 178 (1) ◽  
pp. 49-71 ◽  
Author(s):  
B.A. Davey ◽  
M.R. Talukder


1998 ◽  
Vol 8 (4) ◽  
pp. 321-349 ◽  
Author(s):  
HIDEKI TSUIKI

We give a denotational semantics to a calculus λ[otimes ] with overloading and subtyping. In λ[otimes ], the interaction between overloading and subtyping causes self application, and non-normalizing terms exist for each type. Moreover, the semantics of a type depends not on that type alone, but also on infinitely many others. Thus, we need to consider infinitely many domains, which are related by an infinite number of mutually recursive equations. We solve this by considering a functor category from the poset of types modulo equivalence to a category in which each type is interpreted. We introduce a categorical constructor corresponding to overloading, and formalize the equations as a single equation in the functor category. A semantics of λ[otimes ] is then expressed in terms of the minimal solution of this equation. We prove the adequacy theorem for λ[otimes ] following the construction in Pitts (1994) and use it to derive some syntactic properties.



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