Limits in double categories, revisited (UCLouvain-ULB-VUB Category Theory Seminar)

by Bryce Clarke (Tallinn University of Technology)

Monday Sep 29, 2025, 4:15 PM → 5:15 PM Europe/Brussels

Room G.6.52 (ULB-VUB)

 Marco Grandis and Robert Paré introduced the study of limits
in double categories, generalising weighted limits in 2-categories. They
showed that a double category admits all limits indexed by double
categories if and only if it admits products, equalisers, and
tabulators. Unfortunately, their definition fails to capture many
interesting limit-like constructions in double categories, such as
restrictions, companions, conjoints, and local limits. In this talk, I
will introduce the notion of limit indexed by a loose distributor, which
captures all of these concepts as examples. The main theorem will be to
show that a double category admits all limits indexed by loose
distributors if and only if it admits parallel limits and restrictions.
The talk will focus on exhibiting many examples of limits in the double
category Rel(C) of relations in a regular category C. In particular, we
show that biproducts in the category of relations arise naturally as
colimits in the double category of relations indexed by a loose
distributor.