A general limit lifting theorem for 2-dimensional monad theory

DOI10.1016/J.JPAA.2017.09.018zbMath1420.18017arXiv1702.03303OpenAlexW2743147803MaRDI QIDQ1743022

Martín Szyld

Publication date: 12 April 2018

Published in: Journal of Pure and Applied Algebra (Search for Journal in Brave)

Abstract: We give a definition of weak morphism of $T$-algebras, for a $2$-monad $T$, with respect to an arbitrary family $Omega$ of $2$-cells of the base $2$-category. By considering particular choices of $Omega$, we recover the concepts of lax, pseudo and strict morphisms of $T$-algebras. We give a general notion of weak limit, and define what it means for such a limit to be compatible with another family of $2$-cells. These concepts allow us to prove a limit lifting theorem which unifies and generalizes three different previously known results of $2$-dimensional monad theory. Explicitly, by considering the three choices of $Omega$ above our theorem has as corollaries the lifting of oplax (resp. $sigma$, which generalizes lax and pseudo, resp. strict) limits to the $2$-categories of lax (resp. pseudo, resp. strict) morphisms of $T$-algebras.

Full work available at URL: https://arxiv.org/abs/1702.03303

Mathematics Subject Classification ID

Cites Work

• Unnamed Item
• Enhanced 2-categories and limits for lax morphisms
• Limits for lax morphisms
• Two-dimensional monad theory
• Limits indexed by category-valued 2-functors
• Formal category theory: Adjointness for 2-categories
• Kan extensions in enriched category theory
• The formal theory of monads
• Elementary observations on 2-categorical limits

Related Items (4)

Bilimits are bifinal objects ⋮ Two-dimensional monad theory ⋮ Pointwise Kan extensions along 2-fibrations and the 2-category of elements ⋮ Lifting PIE limits with strict projections

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