Zoran Skoda MR4011808

MR4011808

Corner, Alexander S. A universal characterisation of codescent objects. Theory Appl. Categ. 34 (2019) 684–713.

18N10 (2010:18D05)

18M05 (2010:18D10), 18A30

Categories of descent data in Grothendieck’s theory of descent are categories of cartesian sections over the Čech simplicial object of a cover in the base category of a fibered category. If the fibered category is replaced by a pseudofunctor from the base to the bicategory of categories, the simplicial object is mapped into a pseudocosimplicial category whose pseudolimit (or equivalently of its 2-truncation) recovers the category of descent data. The simplicial object is mapped into a pseudocosimplicial category and the pseudolimit of its 2-truncation recovers the category of descent data. This pseudolimit, known to Grothendieck, has been generalized to the codescent object, a bicategorical limit (including a lax limit and a bilimit variant) of a 2-truncated pseudocosimplicial object in a general bicategory by R.~Street ([Correction to “Fibrations in bicategories”, Cahiers Topologie Géom. Différentielle 28 (1987) no. 1, pp. 53-56, MR903151] and [Categorical and Combinatorial aspects of descent theory, Applied Categorical Structures 12, 537–576 (2004)]), along with the dual construction, the codescent object associated to a 2-truncated pseudosimplicial objects (coherence data in the terminology of the article). In particular, lax codescent objects for pseudosimplicial object constructed from a 2-monad $T$ has a universal property ensuring existence of a left adjoint to the inclusion of the 2-category of strict $T$-algebras into the 2-category of lax $T$-algebras, what is a coherence question [S. Lack, Codescent objects and coherence, J. Pure and Appl. Algebra 175 (2002), pp. 223-241].

The article under review introduces a characterization of codescent objects. Its motivation is to generalize Day’s convolution structure to the setting of monoidal bicategories, exhibited in the author’s thesis. The auxiliary tool used for this purpose is the notion of pseudo-extranatural transformation, generalizing the 1-categorical notion from [S. Eilenberg, G. M. Kelly, A generalization of the functorial calculus. J. Algebra, 3:366–375 (1966)]. As in the classical case, the notion is useful in large part due to the construction of composition of such transformations. A two dimensional version of the composition lemma is proven. Universal properties are studied mostly in the generality of bicodescent objects. While codescent objects are usually introduced in terms of weighted colimits this weaker notion is a bicolimit and has no description using weights. The article considers 2-truncated pseudosimplicial objects (coherence data) associated to a pseudofunctor of the form $P:\mathcal{B}^{op}\times\mathcal{B}\to\mathcal{C}$ where $\mathcal{C}$ is a bicategory with biproducts. Bicodescent object for the coherence data for $P$ is equivalent to the bicoend of $P$. Bicoends are convenient as they require less data than bicolimits. They are universal among pseudo-extranatural transformations and a version of Fubini theorem is proven for them. This then specializes to codescent objects which, in comparison to bicodescent objects, satisfy an additional uniqueness property.