Book contents
- Frontmatter
- Contents
- Diagram of interdependence
- Acknowledgements
- Introduction
- Motivation for topologists
- Part I Background
- Part II Operads
- Part III n-categories
- Appendices
- A Symmetric structures
- B Coherence for monoidal categories
- C Special cartesian monads
- D Free multicategories
- E Definitions of tree
- F Free strict n-categories
- G Initial operad-with-contraction
- References
- Index of notation
- Index
C - Special cartesian monads
Published online by Cambridge University Press: 08 January 2010
- Frontmatter
- Contents
- Diagram of interdependence
- Acknowledgements
- Introduction
- Motivation for topologists
- Part I Background
- Part II Operads
- Part III n-categories
- Appendices
- A Symmetric structures
- B Coherence for monoidal categories
- C Special cartesian monads
- D Free multicategories
- E Definitions of tree
- F Free strict n-categories
- G Initial operad-with-contraction
- References
- Index of notation
- Index
Summary
Pictures can't say ‘ain't’.
Worth (1975)We have met many monads, most of them cartesian. Some had special properties beyond being cartesian – for instance, some were the monads arising from operads, and, as will be explained, some admitted a certain explicit representation. Here we look at these special kinds of cartesian monad and prove some results supporting the theory in the main text.
First (Section C.1) we look at the monads arising from plain operads. Monads are algebraic theories, so we can ask which algebraic theories come from operads. The answer turns out to be the strongly regular theories (2.2.5).
In Section 6.2 we saw that the monad arising from an operad is cartesian. It now follows that the monad corresponding to a strongly regular theory is cartesian, a fact we used in many of the examples in Chapter 4. More precisely, we saw in 6.2.4 that a monad on Set arises from a plain operad if and only if it is cartesian and ‘augmented over the free monoid monad’, meaning that there exists a cartesian natural transformation from it into the free monoid functor, commuting with the monad structures. On the other hand, not every cartesian monad on Set possesses such an augmentation, as we shall see.
One possible drawback of the generalized operad approach to higherdimensional category theory is that it can involve monads that are rather hard to describe explicitly. For instance, the sequence Tn of ‘opetopic’ monads (Chapter 7) was generated recursively using nothing more than the existence of free operads, and to describe Tn explicitly beyond low values of n is difficult. The second and third sections of this appendix ease this difficulty. The basic result is that if a Set- valued functor preserves infinitary or ‘wide’
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- Higher Operads, Higher Categories , pp. 355 - 388Publisher: Cambridge University PressPrint publication year: 2004