Descent on 2-fibrations and 2-regular 2-categories

Journal of Pure and Applied Algebra, 2002

Dedicated to Max Kelly on the occasion of his 70th birthday: a token of affection and respect.

Journal of Pure and Applied Algebra, 2014

We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2category into 2Cat. Fibred bicategories correspond to trihomomorphisms from a bicategory into Bicat. We describe the Grothendieck construction for each kind of fibration and present a few examples of each. Fibrations in our sense, between bicategories, are closed under composition and are stable under equiv-comma. The free such fibration on a homomorphism is obtained by taking an oplax comma along an identity.

Tbilisi Mathematical Journal, 2015

A theorem of A.M. states that essential surjections of toposes bounded over a base topos S are of effective lax descent. The symmetric monad M on the 2-category of toposes bounded over S is a KZ-monad (Bunge-Carboni 1995) and the M -maps are precisely the S -essential geometric morphisms . These last two results led me to conjecture 1 and then prove 2 the general lax descent theorem that is the subject matter of this paper. By a 'Pitts KZ-monad' on a 2-category K it is meant here a locally fully faithful equivariant KZ-monad M on K that is required to satisfy an analogue of Pitts' theorem on bicomma squares along essential geometric morphisms. The main result of this paper states that, for a Pitts KZ-monad M on a 2-category K ('of spaces'), every surjective M -map is of effective lax descent. There is a dual version of this theorem for a Pitts co-KZ-monad N . These theorems have (known and new) consequences regarding (lax) descent for morphisms of toposes and locales. 2 M. Bunge in the 2-category Top S of toposes bounded over a base topos S with a natural numbers object

arXiv (Cornell University), 2023

We prove that cloven Grothendieck fibrations over a fixed base B are the pseudo-coalgebras for a lax idempotent 2-comonad on Cat /B. We show this via an original observation that the known colax idempotent 2-monad for fibrations over a fixed base has a right 2-adjoint. As an important consequence, we obtain an original cofree construction of a fibration on a functor. We also give a new, conceptual proof of the fact that the forgetful 2-functor from split fibrations to cloven fibrations over a fixed base has both a left 2-adjoint and a right 2-adjoint, in terms of coherence phenomena of strictification of pseudo-(co)algebras. The 2-monad for fibrations yields the left splitting and the 2-comonad yields the right splitting. Moreover, we show that the constructions induced by these coherence theorems recover Giraud's explicit constructions of the left and the right splittings.

2010

We construct explicitly the weights on the simplicial category so that the colimits and limits of 2-functors with those weights provide the Kleisli objects and the Eilenberg-Moore objects, respectively, in any 2-category.

Applied Categorical Structures, 1994

An elementary topological approach to Grothendieck's idea of descent is given. While being motivated by the idea of localization which is central in Sheaf Theory, we show how the theory of monads (= triples) provides a direct categorical approach to Descent Theory. Thanks to an important observation by B6nabou and Roubaud and by Beck, the monadic description covers descent also in the abstract context of a bifibred category satisfying the Beck-Chevalley condition. We present the fundamentals of fibrational descent theory without requiring any prior knowledge of fibred categories. The paper contains a number of new topological descent results as well as some new examples in the context of regular categories which demonstrate the subtlety of the descent problem in concrete situations.

1997

Methods of internal-category theory are applied to show that the split epimorphisms in a category C are exactly the morphisms which are effective for descent with respect to any fibration over C (or to any C-indexed category). In the same context, composition-cancellation rules for effective descent morphisms are established and being applied to (suitably defined) locally-split epimorphisms.

Electronic Notes in Theoretical Computer Science, 2001

We introduce the notion of pseudo-commutative monad together with that of pseudoclosed 2-category, the leading example being given by the 2-monad on Cat whose 2-category of algebras is the 2-category of small symmetric monoidal categories. We prove that for any pseudo-commutative 2-monad on Cat, its 2-category of algebras is pseudo-closed. We also introduce supplementary definitions and results, and we illustrate this analysis with further examples such as those of small categories with finite products, and examples arising from wiring, interaction, contexts, and the logic of Bunched Implication.

Journal of Pure and Applied Algebra, 2017

For a small quantaloid Q, we consider 2-monads on the 2-category Q-Cat and their lax extensions to the 2-category Q-Dist of small Q-categories and their distributors, in particular those lax extensions that are flat, in the sense that they map identity distributors to identity distributors. In fact, unlike in the discrete case, a 2-monad on Q-Cat may admit only one flat lax extension. Every ordinary monad on the comma category Set/obQ with a lax extension to Q-Rel gives rise to such a 2-monad on Q-Cat, and we describe this process globally as a coreflective embedding. The Q-presheaf and the double Q-presheaf monads are important examples of 2-monads on Q-Cat allowing flat lax extensions to Q-Dist, and so are their submonads, obtained by the restriction to conical (co)presheaves and known as the Q-Hausdorff and double Q-Hausdorff monads, which we define here in full generality, thus generalizing some previous work in the case when Q is a quantale, or just the "metric" quantale [0, ∞]. Their discretization leads naturally to various lax extensions of the relevant Set-monads used in monoidal topology.

2011

For functors $L:\A\to \B$ and $R:\B\to \A$ between any categories $\A$ and $\B$, a {\em pairing} is defined by maps, natural in $A\in \A$ and $B\in \B$, $$\xymatrix{\Mor_\B (L(A),B) \ar@<0.5ex>[r]^{\alpha} & \Mor_\A (A,R(B))\ar@<0.5ex>[l]^{\beta}}.$$ $(L,R)$ is an {\em adjoint pair} provided $\alpha$ (or $\beta$) is a bijection. In this case the composition $RL$ defines a monad on the category $\A$, $LR$ defines a comonad on the category $\B$, and there is a well-known correspondence between monads (or comonads) and adjoint pairs of functors. For various applications it was observed that the conditions for a unit of a monad was too restrictive and weakening it still allowed for a useful generalised notion of a monad. This led to the introduction of {\em weak monads} and {\em weak comonads} and the definitions needed were made without referring to this kind of adjunction. The motivation for the present paper is to show that these notions can be naturally derived from pairings of functors $(L,R,\alpha,\beta)$ with $\alpha = \alpha\dcirc \beta\dcirc \alpha$ and $\beta = \beta \dcirc\alpha\dcirc\beta$. Following closely the constructions known for monads (and unital modules) and comonads (and counital comodules), we show that any weak (co)monad on $\A$ gives rise to a regular pairing between $\A$ and the category of {\em compatible (co)modules}.