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comonad
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(Definition)
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Let
 be a category. A comonad over
 consists of an endofunctor
 along with natural transformations
 and
 . The natural transformation  is coassociative, in the sense that the diagram
is commutative. Moreover,  is counitary, in the sense that the diagram
also commutes. Observe that the defining diagrams for a comonad are precisely dual to those for a monad. Thus one could more briefly define a comonad over a category
 as a monad over the opposite category
 .
Just as one can define algebras over a monad it is possible to define coalgebras over a comonad.
- 1
- S. A. Joni and G.-C. Rota, Coalgebras and bialgebras in combinatorics, Stud. Appl. Math., 61 (1979), pp. 93-139.
- 2
- S. Mac Lane. Categories for the Working Mathematician, 2nd ed. Springer-Verlag, 1997
- 3
- S. Mac Lane and I. Moerdijk. Sheaves and Geometry in Logic: A First Introduction to Topos Theory, Springer-Verlag, 1992.
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"comonad" is owned by mps. [ full author list (2) | owner history (1) ]
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Cross-references: coalgebras, algebras, opposite category, monad, commutative, diagram, coassociative, natural transformations, endofunctor, category
There is 1 reference to this entry.
This is version 7 of comonad, born on 2006-12-13, modified 2007-01-23.
Object id is 8623, canonical name is Comonad.
Accessed 838 times total.
Classification:
| AMS MSC: | 18C15 (Category theory; homological algebra :: Categories and theories :: Triples , algebras for a triple, homology and derived functors for triples) |
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Pending Errata and Addenda
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