On the notion of flat 2-functors

Collectanea Mathematica, 2014

Consider an extension of groups 1 → K → G → Q → 1 which enjoys the property that the quotient by the lower central series Γ c+1 produces another extension 1 → K/Γ c+1 K → G/Γ c+1 G → Q/Γ c+1 Q → 1, of nilpotent groups of class c. We say that the extension is Γ c+1-flat. Let us pull back the original extension along any homomorphism X → G. Does the pullback extension enjoy the same Γ c+1-flatness property? To answer this question we consider not only quotients by the lower central series, but any localization functor in the category of groups. In fact we start by studying the analogous question for spaces, where we replace extensions by fibration sequences. We prove that the only homotopical localization functors which behave well under pull-backs are nullifications. In the category of groups, nullifications also enjoy this property, and so do all epireflections arising from a variety of groups. In particular the answer to the question about the nilpotent quotients is positive.

Journal of Pure and Applied Algebra, 1982

The purpose of this work is to analyse the appropriate notion of sheaf when we pass from categories to 2-categories. One reason for such an analysis is, as suggested by Andre Joyal, to clarify our understanding of the second cohomology construction HZ. From the outset I must apologise to Andre in that this whole paper deals purely with the ordinary limits for 2-categories (as considered in [6] and called indexed limits). However, he may rest assured that all the results carry over with minor alterations when: 2-categories are replaced by bicategories; indexed limits are replaced by indexed bilimits [7]; arrows are never asked to be equal, only isomorphic; and, objects are never asked to be isomorphic, only equivalent. For example, in 1.1, delete 'injective on objects' so that 'chronic' is replaced by 'fully faithful'; and, in 1.4, the square which, when X is a bicategory, is only known to commute up to isomorphism, should be a bipullback. Also, equivalences and adjunctions between 2-categories should be replaced by biequivalences and biadjunctions [7]. On the other hand, I make no apology for considering categories with horns enriched in Cat (that is, 2-categories) rather than with horns enriched in the category Gpd of groupoids. Andre maintains that 'bicategories in which all 2-cells are invertible' is an important level of generality. Again the results carry over and in fact simplify somewhat due to the fact that the distinction between 'pullback' and 'comma object' disappears (see especially the notion of congruence (1.8)). However, I believe that, in the final analysis, the bicategorical version will be needed for the understanding of HZ. The plan of the paper is to establish 2-dimensional versions of the notions of regular and exact categories [2], sites, sheaves, and so on. The starting point is the premise that the right notion of cover in Cat is a set of functors with the same target which are jointly surjective on objects (in the bicategorical version this becomes 'jointly surjective up to isomorphism on objects'). The main result on regular 2-categories is that the arrows which play the role of surjectives on objects (here called acute) are in fact regular in the sense that they are quotient maps in an appropriate sense. This leads to the appropriate exactness condition which is shown to hold in Cat.

Communications in Algebra, 2001

In (V.Laan, On a generalization of strong flatness, Acta Comment. Univ. Tartuensis 2 (1998), 55-60) and (V.Laan, Pullbacks and flatness properties of acts I, manuscript) studies were initiated of flatness properties of right acts A S over a monoid S that can be described in terms when the functor A S ⊗ − preserves pullbacks. In those articles, familiar flatness properties emerged in a new light, and new properties were discovered. The present paper continues this investigation. Three additional flatness properties are obtained, and the "homological classification" problem-For which monoids S do all right S-acts have a certain flatness property?-is discussed for all of the new properties under consideration.

arXiv (Cornell University), 2010

." Several new facts added, including the material on the derived 2-functors and the proof of the Gabriel-Mitchel theorem and the Morita theory for 2-rings.

Lecture Notes in Physics, 2013

We define the category Func Î with functors F:D F¨S COTT (D F´C PO) as objects and pairs (f:D F¨DG ,ú:FÿGºf) as morphisms (ú is a natural transformation). We show that this category is closed under the common domain theoretical operations +,ÿ,÷ and ¨. The category Func Î is an O-category Tennent 1993].

arXiv (Cornell University), 2016

Given a 2-category A, a 2-functor A F −→ Cat and a distinguished 1-subcategory Σ ⊂ A containing all the objects, a σ-cone for F (with respect to Σ) is a lax cone such that the structural 2-cells corresponding to the arrows of Σ are invertible. The conical σ-limit is the universal (up to isomorphism) σ-cone. The notion of σ-limit generalizes the well known notions of pseudo and lax limit. We consider the fundamental notion of σ-filtered pair (A, Σ) which generalizes the notion of 2-filtered 2-category. We give an explicit construction of σ-filtered σ-colimits of categories, construction which allows computations with these colimits. We then state and prove a basic exactness property of the 2-category of categories, namely, that σ-filtered σ-colimits commute with finite weighted pseudo (or bi) limits. An important corollary of this result is that a σ-filtered σ-colimit of exact category valued 2-functors is exact. This corollary is essential in the 2-dimensional theory of flat and pro-representable 2-functors, that we develop elsewhere.

Communications in Algebra, 2005

We characterize those small categories with the property that flat (contravariant) functors on them are coherently axiomatized in the language of presheaves on them. They are exactly the categories with the property that every finite diagram into them has a finite set of (weakly) initial cocones.

arXiv: Category Theory, 2020

We review the complete definition of monoidal 2-categories and recover Kapranov and Voevodsky's definition from the algebraic definition of weak 3-category(or tricategory).

Canadian mathematical bulletin, 1991

Let 5 be a monoid. A right 5-system A is called strongly flat if the functor A <g>-(from the category of left S-sy stems into the category of sets) preserves pullbacksand equalizers. (This concept arises in B. Stenstrôm, Math. Nachr. 48(1971), 315-334 under the name weak flatness). The main result of the present paper is a proof that for A to be strongly flat it is in fact sufficient that A (g>-preserve only pullbacks. The approach taken is to develop an "interpolation" condition for pullback-preservation, and then to show its equivalence to Stenstrom's conditions for strong flatness.

Semigroup Forum, 1987