Research ArticleNo Access

Derived categories of functors and quiver sheaves

P. O. Gneri

DAMAT-UTFPR, Departamento Acadêmico de Matemática, Av. Sete de Setembro, 3165-Bloco F, 80230-901 Curitiba-PR, Brazil

E-mail Address: gneri@utfpr.edu.br

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M. Jardim

IMECC-UNICAMP, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, 13083-970 Campinas-SP, Brazil

E-mail Address: jardim@ime.unicamp.br

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D. D. Silva

https://orcid.org/0000-0002-7423-7858

DMA-UFS, Avenida Marechal Rondon S/N, São Cristovão-SE, Brazil

E-mail Address: ddsilva@ufs.br

Corresponding author.

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https://doi.org/10.1142/S0219498822501274Cited by:0 (Source: Crossref)

Abstract

Let $𝒞$ be small category and $𝒜$ an arbitrary category. Consider the category $𝒞 (𝒜)$ whose objects are functors from $𝒞$ to $𝒜$ and whose morphisms are natural transformations. Let $ℬ$ be another category, and again, consider the category $𝒞 (ℬ)$ . Now, given a functor $F : 𝒜 \to ℬ$ we construct the induced functor $F_{𝒞} : 𝒞 (𝒜) \to 𝒞 (ℬ)$ . Assuming $𝒜$ and $ℬ$ to be abelian categories, it follows that the categories $𝒞 (𝒜)$ and $𝒞 (ℬ)$ are also abelian. We have two main goals: first, to find a relationship between the derived category $D (𝒞 (𝒜))$ and the category $𝒞 (D (𝒜))$ ; second relate the functors $R (F_{𝒞})$ and ${(R F)}_{𝒞} : 𝒞 (D (𝒜)) \to 𝒞 (D (ℬ))$ . We apply the general results obtained to the special case of quiver sheaves.

Communicated by V. Futorny

Keywords:

AMSC: 18G80, 16G20, 14A30

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