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Kan extensions are partial colimits

Paolo Perrone, Walter Tholen

Abstract

One way of interpreting a left Kan extension is as taking a kind of "partial colimit", whereby one replaces parts of a diagram by their colimits. We make this intuition precise by means of the "partial evaluations" sitting in the so-called bar construction of monads. The (pseudo)monads of interest for forming colimits are the monad of diagrams and the monad of small presheaves, both on the (huge) category CAT of locally small categories. Throughout, particular care is taken to handle size issues, which are notoriously delicate in the context of free cocompletion. We spell out, with all 2-dimensional details, the structure maps of these pseudomonads. Then, based on a detailed general proof of how the "restriction-of-scalars" construction of monads extends to the case of pseudoalgebras over pseudomonads, we define a morphism of monads between them, which we call "image". This morphism allows us in particular to generalize the idea of "confinal functors" i.e. of functors which leave colimits invariant in an absolute way. This generalization includes the concept of absolute colimit as a special case. The main result of this paper spells out how a pointwise left Kan extension of a diagram corresponds precisely to a partial evaluation of its colimit. This categorical result is analogous to what happens in the case of probability monads, where a conditional expectation of a random variable corresponds to a partial evaluation of its center of mass.

Kan extensions are partial colimits

Abstract

One way of interpreting a left Kan extension is as taking a kind of "partial colimit", whereby one replaces parts of a diagram by their colimits. We make this intuition precise by means of the "partial evaluations" sitting in the so-called bar construction of monads. The (pseudo)monads of interest for forming colimits are the monad of diagrams and the monad of small presheaves, both on the (huge) category CAT of locally small categories. Throughout, particular care is taken to handle size issues, which are notoriously delicate in the context of free cocompletion. We spell out, with all 2-dimensional details, the structure maps of these pseudomonads. Then, based on a detailed general proof of how the "restriction-of-scalars" construction of monads extends to the case of pseudoalgebras over pseudomonads, we define a morphism of monads between them, which we call "image". This morphism allows us in particular to generalize the idea of "confinal functors" i.e. of functors which leave colimits invariant in an absolute way. This generalization includes the concept of absolute colimit as a special case. The main result of this paper spells out how a pointwise left Kan extension of a diagram corresponds precisely to a partial evaluation of its colimit. This categorical result is analogous to what happens in the case of probability monads, where a conditional expectation of a random variable corresponds to a partial evaluation of its center of mass.

Paper Structure

This paper contains 51 sections, 37 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Outline.
  3. Acknowledgements.
  4. Categorical setting, notation, and conventions.
  5. The monad of diagrams
  6. Functoriality
  7. Unit and multiplication
  8. The unit: one-object diagrams
  9. Diagrams of diagrams are lax cocones
  10. The multiplication: the Grothendieck construction
  11. Functoriality and naturality of the multiplication
  12. Unitors and associators
  13. Left unitor
  14. Right unitor
  15. Associator
  16. ...and 36 more sections

Key Result

Proposition 2.2

Let $D_0:{\mathsf{J}}\to{\mathsf{Cat}}\subseteq {\mathsf{CAT}}$ be a small diagram of small categories. Let $D_1:D_0\Rightarrow {\mathsf{C}}$ be a lax cocone over $D_0$ in ${\mathsf{CAT}}$, with tip ${\mathsf{C}}$ locally small (but not necessarily small). There is a unique functor ${ \int \! } D_0\

Theorems & Definitions (80)

  • Definition 2.1
  • Proposition 2.2
  • proof : Proof of \ref{['oplaxcolimit']}
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Proposition 2.6
  • Theorem 2.7
  • ...and 70 more