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Descent in Locally Presentable Categories

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Abstract

Let \(\mathbb {V}=(VV, \otimes , I)\) be a symmetric monoidal category such that \(\mathcal {V}\) is locally presentable and that all functors \(V\otimes - : \mathcal {V} \rightarrow \mathcal {V}\) for \(V \in \mathcal {V}\) preserve reflexive coequalizers and directed colimits. It is proved that any pure morphism of commutative 𝕍-monoids is an effective descent morphism with respect to the indexed category given by commutative 𝕍-monoids and modules over them. As a by-product, we prove that pure morphisms in a locally presentable category are effective for codescent.

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Author notes

  1. Bachuki Mesablishvili

    Present address: Tbilisi Centre for Mathematical Sciences, Chavchavadze Ave. 75, 3/35, Tbilisi, 0168, Republic of Georgia

Authors and Affiliations

  1. A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University, 6, Tamarashvili Str., Tbilisi, 0177, Georgia

    Bachuki Mesablishvili

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  1. Bachuki Mesablishvili
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Correspondence to Bachuki Mesablishvili.

Additional information

Dedicated to George Janelidze on his sixtieth birthday

The wok was partially supported by Volkswagen Foundation (Ref.: I/85989) and Shota Rustaveli National Science Foundation Grant DI/12/5-103/11.

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Mesablishvili, B. Descent in Locally Presentable Categories. Appl Categor Struct 22, 715–726 (2014). https://doi.org/10.1007/s10485-013-9343-6

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