Orthogonal Factorization Systems for Double Categories as Algebras for a Monad
Dorette Pronk
Dalhousie University
Abstract:
In this talk we will introduce a notion of orthogonal factorization system
(DOFS) for double categories that interacts well the notion of double
fibration. A DOFS consists of two ordinary orthogonal factorization systems:
one for the (strict) arrows of the double category and one for the double cells
as arrows between the proarrows of the double category. In other words, we may
think of a double category with a DOFS as a pseudo-category internal to a
category of categories with an orthogonal factorization system (OFS). As
categories with an OFS are algebras for a monad, there are four options for the
morphisms in this category: strict, pseudo, lax and colax morphisms of
algebras. (Lax morphisms of these algebras are the ones that preserve the right
class of arrows.) Analogous to what was needed for double fibrations, we
require that the source, target and identity morphisms are strict morphisms of
algebras. There are then two versions of DOFS: the ones for which proarrow
composition of double cells is a lax morphism of algebras and the ones for
which it is colax.
I will discuss the details of this construction and present several examples. I
will also present a 2-monad for which the double categories with a DOFS form
the algebras, and describe the induced maps between these double categories. I
will also discuss the interaction between these factorization systems and
double fibrations.