14 citations · 24 across the 11 of their papers we have counts for
10 papers · 1 filter
Presheaves and cocompletions in formal category theory
Nathanael Arkor, Dylan McDermott
We study the relationship between presheaf constructions and free cocompletions in the context of formal category theory, elucidating the coincidence between the two concepts in fa…
Idempotence for relative monads
Nathanael Arkor, Andrew Slattery
We study the concept of idempotence for relative monads, which exhibits several subtleties not present for non-relative monads. In particular, there is a bifurcation of notions of…
Magmal characterisations of cocartesian categories
Nathanael Arkor
We present a survey of characterisations of cocartesian categories in terms of monoidal categories - and, more generally, magmal categories - satisfying additional properties. In p…
Exponentiable virtual double categories and presheaves for double categories
Nathanael Arkor
Given a pair of pseudo double categories and , the lax functors from to , along with their transformations, modules, and multimodulati…
Bicategories of algebras for relative pseudomonads
Nathanael Arkor, Philip Saville, Andrew Slattery
We introduce pseudoalgebras for relative pseudomonads and develop their theory. For each relative pseudomonad , we construct a free--forgetful relative pseudoadjunction that exh…
Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation
Nathanael Arkor, John Bourke, Joanna Ko
Many structures of interest in two-dimensional category theory have aspects that are inherently strict. This strictness is not a limitation, but rather plays a fundamental role in…
The nerve theorem for relative monads
Nathanael Arkor, Dylan McDermott
A fundamental result in the theory of monads is the characterisation of the category of algebras for a monad in terms of a pullback of the category of presheaves on the category of…
Adjoint functor theorems for lax-idempotent pseudomonads
Nathanael Arkor, Ivan Di Liberti, Fosco Loregian
For each pair of lax-idempotent pseudomonads and , for which is locally fully faithful and distributes over , we establish an adjoint functor theorem, relating $R…
Relative monadicity
Nathanael Arkor, Dylan McDermott
We establish a relative monadicity theorem for relative monads with dense roots in a virtual equipment, specialising to a relative monadicity theorem for enriched relative monads.…
The formal theory of relative monads
Nathanael Arkor, Dylan McDermott
We develop the theory of relative monads and relative adjunctions in a virtual equipment, extending the theory of monads and adjunctions in a 2-category. The theory of relative com…