ALC Seminar | Speaker: Rui Prezado (CMUC, Universidade de Coimbra)
Title: Descent-theoretical aspects of categorical Galois theory
Abstract: Janelidze's categorical Galois theory [2,1] provides a unifying
perspective on various Galois-type theorems, which include, for instance,
Magid's Galois theory for commutative rings, and Grothendieck's Galois theory
for étale coverings of schemes. In fact, such results were generalized and
have found applications in other settings [6, 8, 7].
The description of these results relies on techniques from descent theory and
bidimensional category theory. In particular, the notion of \textit{effective
descent morphism} is an important tool for the study of categorical Galois
theories; thus, it is an important to obtain a good supply of such morphisms
in the categories of interest.
In [6], we find evidence that commuting properties of bilimits are
useful to prove results which yield general sufficient conditions for
morphisms in suitable categories to be of effective descent. However, in the
particular settings of [5, 9], there are results which provide
conditions sharper than those currently available via abstract techniques.
Via a careful analysis of the results in [6], plus a couple of new
insights, we have shown it is possible to bring the perspective of
[5, 9] to an abstract 2-categorical setting, we were able to
sharpen the conditions for a morphism to be of effective descent, which
sometimes even allow for characterisations of these morphisms, as was done in
[4, 3]. More importantly, we are also able to sharpen the
''non--Galoisian'' form of Janelidze's categorical Galois theorem.
This talk is based on joint work with Fernando Lucatelli Nunes.
References
[1] F. Borceux and G. Janelidze. Galois Theories. Number 72 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2001.
[2] G. Janelidze. Pure Galois theory in categories. J. Algebra, 132(2):270-286, 1990.
[3] M.M. Clementino, D. Hofmann, and R. Prezado. Topological lax comma categories.
Order, 43(3), 2026.
[4] M.M. Clementino and R. Prezado. Effective descent morphisms of ordered families. Quaest. Math., 48(8):1197-1212, 2025.
[5] I. Le Creurer. Descent of Internal Categories. PhD thesis, Université Catholique de Louvain, 1999.
[6] F. Lucatelli Nunes. Pseudo-Kan extensions and descent theory. Theory Appl. Categ., 33:15, 390-444, 2018.
[7] S. Marques. Galois descent theorem in general categories. Res. Math. Sci., 3:49, 2026.
[8] B. Noohi, I. Tomašić. Galois theory of differential schemes.
arXiv:2407.21147, 2024.
[9] R. Prezado and F. Lucatelli Nunes. Descent for internal multicategory functors. Appl. Categ. Structures, 31(11), 2023.