nLab codescent object

Under construction! Please be patient.

Motivation

The descent data of classical (Grothendieck’s) descent theory are usually treated in the generality of fibered categories. They correspond to cartesian lifts to the total category of the simplicial objects in the base, which are typically produced from covers (in Čech picture; this is not fully general as it presupposes certain coproducts in the base). In 1-categorical situation these data depend essentially only of a 2-truncation of the simplicial set hence the lift in a total category presents a 2-truncated pseudo-simplicial object. Descent data organize into a category of descent data. This category may be co-represented in an appropriate 2-categorical sense by a category equipped with certain data. The same situation can happen without any direct reference to descent theory. We start with a 2-truncation of a pseudosimplicial object in a bicategory and consider additional object in that bicategory and 1-cells satisfying a universal property. It can be viewed as certain weighted colimit called a codescent object. The notion is dual to the notion of descent object as defined in

• Ross Street, Correction to “Fibrations in bicategories”,

  Cahiers de Topologie et Géométrie Différentielle Catégoriques 28, no.1 (1987) 53–56 numdam

A bicolimit weakening of the notion of codescent object is a bicodescent object.

Applications

The main example outside of descent theory is related to a comparison of the category of $T$-algebras for a 2-monad $T$ in strict 2-category to lax or pseudo-$T$-algebra. We ask if the inclusion of strict into lax or pseudoalgebras has a left 2-adjoint. 2-monad $T$ helps to define coherence data for a codescent object whose defining property is equivalent to the universal property of the 2-adjunction. Bicodescent objects were used in A. Corner’s thesis to define an analogue of Day’s convolution for monoidal bicategories.

Literature

• Ross Street, Categorical and combinatorial aspects of descent theory, arXiv:math.CT/0303175
• Stephen Lack, Codescent objects and coherence, Stephen Lack, J. Pure and Appl. Algebra 175 (2002) 223-241 doi
• Alex Corner, Codescent Objects and Coherence, at Category Cafe 2014/06
• A.S. Corner, Day convolution for monoidal bicategories, PhD thesis, (2016)
• A.S. Corner, A universal characterisation of codescent objects, Theory and Applications of Categories 34 (24), 684-713 (2019)