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Kan Extensions are Partial Colimits.

Paolo Perrone1, Walter Tholen2

  • 1Department of Computer Science, University of Oxford, Oxford, England, UK.

Applied Categorical Structures
|July 25, 2022
PubMed
Summary

Left Kan extensions are interpreted as partial colimits using partial evaluations within monads. This research clarifies how pointwise left Kan extensions precisely match partial colimit evaluations, analogous to conditional expectations in probability.

Keywords:
ColimitsKan extensionsPartial evaluationsPresheaves

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Area of Science:

  • Category Theory
  • Abstract Mathematics
  • Algebraic Theory

Background:

  • Interpreting left Kan extensions as partial colimits offers new insights.
  • Monads and their bar constructions are key tools in category theory.
  • Handling size issues in free cocompletion requires careful attention.

Purpose of the Study:

  • To precisely define the intuition of left Kan extensions as partial colimits.
  • To explore the structure maps of pseudomonads for forming colimits.
  • To generalize the concept of confinal functors and absolute colimits.

Main Methods:

  • Utilizing partial evaluations within the bar construction of monads.
  • Analyzing the monad of diagrams and the monad of small presheaves.
  • Developing a general proof for the restriction-of-scalars construction of monads.

Main Results:

  • A morphism of monads, termed 'image', is introduced, generalizing confinal functors.
  • Pointwise left Kan extensions are shown to correspond precisely to partial evaluations of colimits.
  • The categorical result is analogous to conditional expectations in probability monads.

Conclusions:

  • The study provides a precise categorical framework for understanding left Kan extensions as partial colimits.
  • The generalization of confinal functors and absolute colimits offers broader applications.
  • The findings establish a significant link between abstract category theory and probability theory.