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Abstract
In 1966 [7], John Isbell introduced a construction on categories which he termed the "couple category" but which has since come to be known as the Isbell envelope. The Isbell envelope, which combines the ideas of contravariant and covariant presheaves, has found applications in category theory, logic, and differential geometry. We clarify its meaning by exhibiting the assignation sending a locally small category to its Isbell envelope as the action on objects of a pseudomonad on the 2category of locally small categories; this is the Isbell monad of the title. We characterise the pseudoalgebras of the Isbell monad as categories equipped with a cylinder factorisation system; this notion, which appears to be new, is an extension of Freyd and Kelly's notion of factorisation system [5] from orthogonal classes of arrows to orthogonal classes of cocones and cones.
Original language  English 

Pages (fromto)  516537 
Number of pages  22 
Journal  Advances in Mathematics 
Volume  274 
DOIs  
Publication status  Published  9 Apr 2015 
Keywords
 Isbell envelope
 Free cocompletion
 Factorisation systems
 CATEGORIES
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Dive into the research topics of 'The Isbell monad'. Together they form a unique fingerprint.Projects
 2 Finished

Structural homotopy theory: a categorytheoretic study
Street, R., Lack, S., Verity, D., Garner, R., MQRES, M., MQRES 3 (International), M. 3., MQRES 4 (International), M. & MQRES (International), M.
1/01/13 → 31/12/16
Project: Research
