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<p>We arrange morphisms and comorphisms of sites as the horizontal and vertical cells of a double category of sites; using the formalism of extensions and restrictions of presheaves, we explain how one can define a sheafification double functor from this double category to the quintet double category of Grothendieck topoi. We describe properties of this double functor and recover some classical results of topos theory through a new notion of <i>locally exact square</i>, generalizing exact squares in the presence of topologies. We also describe a 2-comonad on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textbf{Cat}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">Cat</mi> </math></EquationSource> </InlineEquation> for which lax morphisms of coalgebras are morphisms of sites and colax morphisms are comorphisms of sites, explaining the arrangement as a double category.</p>

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Morphisms and Comorphisms of Sites I Double-Categories of Sites

  • 期刊论文

  • 发表日期：

    2026年4月4日

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  • Olivia Caramello,
  • Axel Osmond

摘要

We arrange morphisms and comorphisms of sites as the horizontal and vertical cells of a double category of sites; using the formalism of extensions and restrictions of presheaves, we explain how one can define a sheafification double functor from this double category to the quintet double category of Grothendieck topoi. We describe properties of this double functor and recover some classical results of topos theory through a new notion of locally exact square, generalizing exact squares in the presence of topologies. We also describe a 2-comonad on \(\textbf{Cat}\) Cat for which lax morphisms of coalgebras are morphisms of sites and colax morphisms are comorphisms of sites, explaining the arrangement as a double category.