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A two-categorical Snake Lemma
Elena Caviglia, Luca Mesiti, Tim Van der Linden
We prove a Snake Lemma for 2-categories. Working in a 2-category with a strong bizero object, we develop 2-kernels and 2-cokernels, 2-monomorphisms and 2-epimorphisms as fully fait…
Exactness of the 2-categories of abelian and triangulated categories
Elena Caviglia, Zurab Janelidze, Luca Mesiti +1
We introduce a notion of -homological category modelled on pointed homological categories in the sense of Grandis, using bizero objects whose null -cells are zero objects in…
Clifford semigroups and the monoidal Grothendieck construction
Elena Caviglia, Peter F. Faul, Graham Manuell +1
We show that the monoidal Grothendieck construction can be applied to recover the known structure theorem for Clifford monoids, which states that they correspond to functors from a…
Generalized action monads and descent
Elena Caviglia, Sophie Marques, Luca Mesiti
We develop a monadic approach to actions of internal categories. Given an internal category, we construct a monad on a slice whose algebras are the actions of that internal categor…
Comonadic approach to pretorsion theories
Elena Caviglia, Zurab Janelidze, Luca Mesiti
We present a comonadic approach to pretorsion theories on semiexact categories, i.e. categories equipped with a closed ideal of null morphisms that admits all kernels and all coker…
Double Orthogonal Factorization Systems
C. B. Aberlé, Elena Caviglia, Matthew Kukla +4
We define strict and lax orthogonal factorization systems on double categories. These consist of an orthogonal factorization system on arrows and one on double cells that are compa…
Rectangular torsion theories
Elena Caviglia, Zurab Janelidze, Luca Mesiti
In this paper we introduce and study \emph{rectangular torsion theories}, i.e.\ those torsion theories $(\C,\T,\F)$ with $\C$ a pointed category, where the canonical functor $\C\to…
Fibrational approach to Grandis exactness for 2-categories
Elena Caviglia, Zurab Janelidze, Luca Mesiti
In an abelian category, the (bi)fibration of subobjects is isomorphic to the (bi)fibration of quotients. This property captures substantial information about the exactness structur…
Principal 2-bundles and quotient 2-stacks
Elena Caviglia
We generalize principal bundles and quotient stacks to the two-categorical context of bisites. We introduce a notion of principal 2-bundle that makes sense for a 2-category with fi…
2-stacks over bisites
Elena Caviglia
We generalize the concept of stack one dimension higher, introducing a notion of 2-stack suitable for a trihomomorphism from a 2-category equipped with a bitopology into the tricat…
Indexed Grothendieck construction
Elena Caviglia, Luca Mesiti
We produce an indexed version of the Grothendieck construction. This gives an equivalence of categories between opfibrations over a fixed base in the 2-category of 2-copresheaves a…
Generalized principal bundles and quotient stacks
Elena Caviglia
We consider the internalization of the usual notion of principal bundle in a site that has all pullbacks and a terminal object. We use this notion to consider the explicit construc…