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Kleisli categories, T-categories and internal categories

Dominique Bourn

Abstract

We investigate the properties of the Kleisli category KlT of a monad (T,λ,μ) on a category E and in particular the existence of (some kind of) pullbacks. This culminates when the monad is cartesian. In this case, we show that any T-category in E in the sense of A. Burroni coincides with a special kind of internal category in KlT . So, it is the case in particular for T -operads and T -multicategories. More unexpectedly, this, in turn, sheds new lights on internal categories and n-categories.

Kleisli categories, T-categories and internal categories

Abstract

We investigate the properties of the Kleisli category KlT of a monad (T,λ,μ) on a category E and in particular the existence of (some kind of) pullbacks. This culminates when the monad is cartesian. In this case, we show that any T-category in E in the sense of A. Burroni coincides with a special kind of internal category in KlT . So, it is the case in particular for T -operads and T -multicategories. More unexpectedly, this, in turn, sheds new lights on internal categories and n-categories.
Paper Structure (36 sections, 51 theorems, 6 equations)

This paper contains 36 sections, 51 theorems, 6 equations.

Table of Contents

  1. Monads
  2. Basics
  3. Cartesian monads
  4. Cartesian adjoint pairs
  5. Autonomous adjoint pairs
  6. Kleisli category of a monad
  7. Consequences of constraints on $\lambda$
  8. Consequences of constraints on $T$
  9. Consequences of constraints on $\mu$
  10. Internal categories
  11. Basics
  12. The cartesian monad $(T_{X_{\bullet}},\lambda_{X_{\bullet}},\mu_{X_{\bullet}})$ on $\mathbb{E} /X_0$
  13. Internal groupoids
  14. Basics
  15. The fibration of points
  16. ...and 21 more sections

Key Result

Proposition 1.1

Let $(T,\lambda,\mu)$ be a monad on $\mathbb{E}$ where $\mu$ is cartesian; the two conditions are equivalent: 1) $\lambda$ is the equalizer of $\lambda T$ and $T\lambda$, and 2) $\lambda$ is cartesian.

Theorems & Definitions (59)

  • Proposition 1.1
  • Lemma 1.1
  • Definition 1.1
  • Proposition 1.2
  • Definition 1.2
  • Definition 1.3
  • Proposition 1.3
  • Proposition 1.4
  • Corollary 1.1
  • Proposition 2.1
  • ...and 49 more