In category theory, a comonad is a structure dual to that of the monad .
definition
A comonade is a triple consisting of
![{\ displaystyle (T, \ varepsilon, \ psi)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5db90b316dcb02648884fe87abd97e814cfd9143)
- an endofector ,
![T \ colon C \ to C](https://wikimedia.org/api/rest_v1/media/math/render/svg/efa2fc0860bf56e30fc6dc4f16fa4ec527c9d43f)
- a natural transformation and
![{\ displaystyle \ varepsilon \ colon T \ to 1_ {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b978ef4bec9ee20518b7aa27df5fbf97e5f3680)
- a natural transformation ,
![{\ displaystyle \ psi \ colon T \ to T ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5493c9efdd38604233fbb5a2467c56712b8938f5)
that meets the following conditions:
-
and
-
.
Explicitly at the level of morphisms of this means that for each object from true
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
-
and
-
.
Coalgebras
A koalgebra for a comonad on a category is a pair consisting of an object of and a morphism such that and . A homomorphism of coalgebras is a morphism in that satisfies. The koalgebras form a category .
![{\ displaystyle (T, \ varepsilon, \ psi)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5db90b316dcb02648884fe87abd97e814cfd9143)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
![{\ displaystyle (X, \ alpha)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1010b9c96c25adfa0fe35472d06fd46fe5af1065)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
![{\ displaystyle \ alpha \ colon X \ to TX}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e42d6ef9af73ef5aa7de15366b7312a69183e8ac)
![{\ displaystyle T \ alpha \ circ \ alpha = \ psi _ {X} \ circ \ alpha}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3effb020d80c52be23b669cb21d6d8fb9c5f786d)
![{\ displaystyle \ varepsilon _ {X} \ circ \ alpha = 1_ {X}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/039496d04fe93f2acfdee91c23a2ac3ab5fc6d39)
![{\ displaystyle (X, \ alpha) \ to (Y, \ beta)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0a00ebccd87fc7c6636e97b3d7a1781d11827e3)
![f \ colon X \ to Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
![{\ displaystyle Tf \ circ \ alpha = \ beta \ circ f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7eca51c0c224170c66cc4b608016f6e4dc795ec7)
![{\ displaystyle C ^ {T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e8e64cab58c6725b265c1600efec72303fba1f0)
There is a canonical functor that is on objects . He is right adjoint to the forget functor .
![{\ displaystyle A_ {T} \ colon C \ to C ^ {T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be16ae9442987b07c07befdd465c170d1a3d9ba8)
![{\ displaystyle X \ mapsto (TX, \ psi _ {X})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bed2ee7dbc8e00c77f19123f38416f0992fa451)
![{\ displaystyle U_ {T} \ colon C ^ {T} \ to C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9af1651f12ebf04faf428bf287c50bf9f99fc23)
Comonad to an adjoint functor pair
Let there be categories and , functors, so that it is right adjoint to . One or Koeins of the adjunction are or . Then a comonade is on .
![CD](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a52d2269714cb7623ef897d9ae0ff12e5d15d4c)
![F \ colon C \ to D](https://wikimedia.org/api/rest_v1/media/math/render/svg/8299d81a3d96658bf9240b799a248872f36a3772)
![{\ displaystyle G \ colon D \ to C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1442f60677550303225bce499684efec72607a4a)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![{\ displaystyle \ eta \ colon 1 \ to FG}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e4ddacf9cfcc0ec33a0ed9519856fbe6415448)
![{\ displaystyle \ varepsilon \ colon GF \ to 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/277aa90915225b009f391248acd52a50a0984bd8)
![{\ displaystyle T = (GF, \ varepsilon, G \ eta F)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9885cbd179f8d2730a928b069d41d2a8fcb38120)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
One obtains an induced functor such that and holds. The functor is called comonadic if there is an equivalence of categories . The Monadizitätssatz of Jonathan Mock Beck 's criteria for when a functor is komonadisch.
![{\ displaystyle A \ colon D \ to C ^ {T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8de2230bee1a9c522787ecff0be2379783d0e0a4)
![{\ displaystyle U_ {T} \ circ A = G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38e6f645943fd35373edc733fab7ff89a7b2b3a3)
![{\ displaystyle A \ circ F = A_ {T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7b81aa40857f0bd1b9efa6e6ace65bab7d111e)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
If a comonad is on a category , then the comonad associated with the adjoint functor pair is again .
![T](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
![{\ displaystyle (U_ {T} \ colon C ^ {T} \ to C, A_ {T} \ colon C \ to C ^ {T})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f61264a0682fe9a1b40fbbbea28d5d17785d61f5)
![T](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
example
In the set category , the endofunctor is that of the formation of -indicated sequences, i.e. H. for every set is , and for sets and as well as mappings is given by .
![T](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
![\ mathbb {N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![{\ displaystyle T (X) = X ^ {\ mathbb {N}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c835aae1d9dbc43229bc6cbe1242c5b8e9914e6)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![{\ displaystyle f \ colon A \ to B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dec1893560fabff9fa9c17b83b71f7f97996119)
![{\ displaystyle T (f) \ colon A ^ {\ mathbb {N}} \ to B ^ {\ mathbb {N}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92b64b846ec7d905fce092ab24a1c14f0384915e)
![{\ displaystyle T (f) (s): = f \ circ s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa19ec3e77b38830185c6102e958b37b7704ce3)
Let the natural transformations and be through the families of maps and ,
![\ varepsilon](https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173)
![\ psi](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a)
![{\ displaystyle \ varepsilon _ {X}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84ef7f4c36b3fc94bc0c308e9d364d85b68bfac8)
![\ psi_X](https://wikimedia.org/api/rest_v1/media/math/render/svg/146dbea7c6af0ad8088448b940fa0ed523944c4d)
![{\ displaystyle \ varepsilon _ {X} \ colon X ^ {\ mathbb {N}} \ to X, \ varepsilon _ {X} (s): = s (0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3271da9dab70dccf982446a7b0a61b10572c7f)
![{\ displaystyle \ psi _ {X} \ colon X ^ {\ mathbb {N}} \ to (X ^ {\ mathbb {N}}) ^ {\ mathbb {N}}, \ psi _ {X} (s ) (n) (m): = s (n + m)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd513e0ce9d36f31a8eadebb860be74804378a08)
given for arbitrary quantities .
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
The triple is now a comonade in set .
![{\ displaystyle (T, \ varepsilon, \ psi)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5db90b316dcb02648884fe87abd97e814cfd9143)
The Koalgebras for are figures that satisfy and . With , is , and one can identify the coalgebras with pairs with any map .
![{\ displaystyle (T, \ varepsilon, \ psi)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5db90b316dcb02648884fe87abd97e814cfd9143)
![{\ displaystyle \ alpha \ colon X \ to X ^ {\ mathbb {N}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50939549af9ae73ccafd1d9b93003b9bce8c2451)
![{\ displaystyle \ alpha (x) (0) = x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0d4a515301677dc28cab03b412c0b943357f05)
![{\ displaystyle \ alpha (x) (n + m) = \ alpha (\ alpha (x) (n)) (m)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77236098d74249794b1d99e31ca45607d76578ee)
![{\ displaystyle \ alpha _ {1} \ colon X \ to X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96bb0a7e4c89bd0fa1de2d4b2e3ab53202acbf91)
![{\ displaystyle x \ mapsto \ alpha (x) (1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b9652836152df527c0e0f163c6208bd3e8eecd)
![{\ displaystyle \ alpha (x) (n) = \ alpha _ {1} ^ {n} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c3c87352d48559b6ddc240fb03d4924413de1fb)
![{\ displaystyle (X, \ alpha _ {1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2df549b04a05bebc66e4ae652b3e10dc4a3d03b1)
![{\ displaystyle \ alpha _ {1} \ colon X \ to X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96bb0a7e4c89bd0fa1de2d4b2e3ab53202acbf91)
Is any amount then correspond Komonadenstrukturen on bijektiv the Monoidstrukturen on . The multiplication on is . For a monoid , the structural mapping of a koalgebra can be identified with other mappings under the power law :
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![{\ displaystyle T (X) = X ^ {M}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a3c81612188f0fb6008869f2b68db14c85618e2)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![{\ displaystyle \ psi _ {M} (1_ {M}) \ in (M ^ {M}) ^ {M}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/803c7c23fb72eb7827ba4250d21df79317e1163d)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![{\ displaystyle X \ to X ^ {M}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0db26fb6515701af350087df31c8ed1e776e7c45)
![{\ displaystyle (A ^ {B}) ^ {C} = A ^ {B \ times C} = (A ^ {C}) ^ {B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c604ac75748b2ce64c13dfb69eebcfeedd13f87)
- a figure which is an algebra of the Monad is
![{\ displaystyle X \ times M \ to X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/836d25c2efeb434d7d5637bb322398060f3a96b9)
![{\ displaystyle T ^ {*} (X) = X \ times M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e532ebeae40b22b4f55fbe200e04969f444e73a)
- a monoid homomorphism , d. H. an operation from on .
![{\ displaystyle M \ to X ^ {X}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b410e7ffa0d8b02053abf48748b9a38bf91bc53a)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
literature
- Saunders Mac Lane, Categories for the Working Mathematician . Springer-Verlag, Berlin 1971. ISBN 3-540-90035-7