1133:
1147:
1215:
4996:
4453:
4212:
3748:
4824:
1227:
4838:
6267:
5433:
6091:
7475:
forming a monadic adjunction. For example, the free–forgetful adjunction between groups and sets is monadic, since algebras over the associated monad are groups, as was mentioned above. In general, knowing that an adjunction is monadic allows one to reconstruct objects in
3120:. Thus, every such type of algebra gives rise to a monad on the category of sets. Importantly, the algebra type can be recovered from the monad (as the category of Eilenberg–Moore algebras), so monads can also be seen as generalizing varieties of universal algebras.
7646:
2325:
2512:
5799:
7326:
491:—they relate categories of different natures. The monad theory matters as part of the effort to capture what it is that adjunctions 'preserve'. The other half of the theory, of what can be learned likewise from consideration of
5290:
5672:
4087:
6550:
7547:
to sets is monadic. However the forgetful functor from all topological spaces to sets is not conservative since there are continuous bijective maps (between non-compact or non-Hausdorff spaces) that fail to be
6625:
6821:
5932:
3471:
3350:
1022:
7240:
6684:
6998:
6096:
2229:
4913:
1873:
2576:
6262:{\displaystyle {\begin{aligned}{\text{Alt}}^{\bullet }(R^{\oplus n})&=R(x_{1},\ldots ,x_{n})\\{\text{T}}^{\bullet }(R^{\oplus n})&=R\langle x_{1},\ldots ,x_{n}\rangle \end{aligned}}}
880:
1782:
3955:
6010:
4144:
4190:
1658:
2142:
1836:
650:
5843:
736:
6927:
4970:
3609:
3510:
7570:
7088:
6883:
4360:
6734:
6474:
3043:
2942:
1344:
2027:
in 1958 under the name "standard construction". Monad has been called "dual standard construction", "triple", "monoid" and "triad". The term "monad" is used at latest 1967, by
1948:
7822:
6046:
5721:
4790:
3099:
7449:
5201:
4580:
3994:
3852:
1537:
1432:
1404:
607:
110:
7382:
6423:
6379:
3699:
2237:
2187:
4435:
3636:
2882:
913:
234:
208:
146:
5552:
2980:
2660:
789:
515:
441:
5470:
5265:
3255:
2728:
7675:
4406:
4314:
3898:
2613:
1684:
1048:
7417:
7119:
7045:
5508:
5051:
4715:
3726:
3560:
2362:
1713:
1285:
1205:
1182:
1075:
959:
936:
763:
677:
8507:
7156:
5605:
3284:
3193:
1566:
1481:
1261:
2409:
172:
7552:. Thus, this forgetful functor is not monadic. The dual version of Beck's theorem, characterizing comonadic adjunctions, is relevant in different fields such as
7788:
7473:
7018:
6754:
6703:
6494:
6443:
6401:
6347:
6327:
6307:
6287:
6086:
6066:
5972:
5952:
5712:
5692:
5600:
5580:
5285:
5221:
5164:
5144:
5124:
5104:
5084:
5020:
4990:
4933:
4810:
4755:
4735:
4683:
4663:
4643:
4621:
4600:
4380:
4288:
4260:
3872:
3820:
3796:
3659:
3533:
3213:
3161:
3141:
3003:
2902:
2850:
2830:
2806:
2764:
2700:
2680:
2382:
2090:
2066:
1988:
1968:
1913:
1610:
1590:
1505:
1452:
1364:
1305:
1115:
1095:
829:
809:
697:
575:
547:
481:
461:
415:
395:
371:
351:
7256:
6555:
49:
Monads seem to bother a lot of people. There’s even a YouTube video called The Monads Hurt My Head! ... Shortly thereafter, the woman speaking exclaims:
8216:"The concept of a monad, which arises from category theory, has been applied by Moggi to structure the denotational semantics of programming languages."
6764:
As was mentioned above, any adjunction gives rise to a monad. Conversely, every monad arises from some adjunction, namely the free–forgetful adjunction
8006:
2011:
than monoids; however, comonoids in the category of vector spaces with its usual tensor product are important and widely studied under the name of
8442:
8381:
8209:
8108:
7990:
4519:
6630:
6499:
5428:{\displaystyle {\mathcal {D}}(X)=\left\{f:X\to :{\begin{matrix}\#{\text{supp}}(f)<+\infty \\\sum _{x\in X}f(x)=1\end{matrix}}\right\}}
4002:
8460:
8403:
7913:
4440:
In functional programming and denotational semantics, the environment monad models computations with access to some read-only data.
6770:
5848:
7714:
38:
8508:
https://medium.com/@felix.kuehl/a-monad-is-just-a-monoid-in-the-category-of-endofunctors-lets-actually-unravel-this-f5d4b7dbe5d6
7833:
3390:
295:
291:
3415:
3195:, and which maps linear maps to their tensor product. We then have a natural transformation corresponding to the embedding of
7843:
7501:
gives a necessary and sufficient condition for an adjunction to be monadic. A simplified version of this theorem states that
3380:
3304:
970:
299:
7177:
46:
We had some time to talk, and during the course of it I realized I’d become less scared of certain topics involving monads.
8275:
6932:
2335:
2195:
8271:
8041:
7497:
2101:
1844:
1132:
2530:
31:
841:
1721:
8395:
7885:
8522:
7561:
7541:
7385:
4500:
3906:
488:
5977:
4095:
8527:
4196:
1615:
1308:
1240:
240:
5435:
By inspection of the definitions, it can be shown that algebras over the distribution monad are equivalent to
4852:
2109:
1809:
7698:
7641:{\displaystyle -\otimes _{A}B:\mathbf {Mod} _{A}\rightleftarrows \mathbf {Mod} _{B}:\operatorname {forget} }
1687:
616:
5804:
1146:
702:
8298:, Actualités Sci. Ind., Publ. Math. Univ. Strasbourg, vol. 1252, Paris: Hermann, pp. viii+283 pp
7729:
7725:
7710:
7682:
6888:
3568:
3477:
3386:
3109:
1884:
1158:
962:
610:
550:
287:
244:
120:
7050:
6845:
4322:
6708:
6448:
3008:
2907:
2628:
1314:
279:
255:
mapping a category to itself). According to John Baez, a monad can be considered at least in two ways:
1921:
7793:
7718:
3294:
Under mild conditions, functors not admitting a left adjoint also give rise to a monad, the so-called
3112:. The preceding example about free groups can be generalized to any type of algebra in the sense of a
8351:
8016:
7756:
7506:
6015:
4938:
4760:
4150:
3051:
2231:
of the adjunction, and the multiplication map is constructed using the counit map of the adjunction:
1122:
7693:-modules, equipped with a descent datum (i.e., an action of the comonad given by the adjunction) to
7422:
5182:
4547:
3961:
3825:
2320:{\displaystyle T^{2}=G\circ F\circ G\circ F\xrightarrow {G\circ {\text{counit}}\circ F} G\circ F=T.}
1510:
1409:
1385:
580:
77:
3113:
2778:
2397:
1569:
832:
263:
7358:
6406:
6362:
5954:-algebra on the right is considered as a module. Then, an algebra over this monad are commutative
1803:
axioms. In fact, monads are special cases of monoids, namely they are precisely the monoids among
8422:
8341:
7973:. In BĂ©nabou, J.; Davis, R.; Dold, A.; Isbell, J.; MacLane, S.; Oberst, U.; Roos, J. -E. (eds.).
7748:
7557:
5146:
subject to associativity and unitality conditions. Such a structure is equivalent to saying that
3664:
3045:
in the natural way, as strings of length 1. Further, the multiplication of this monad is the map
2771:
2157:
1788:
8189:
4414:
3615:
2855:
1214:
885:
213:
181:
125:
5513:
2950:
2633:
259:
A monad as a generalized monoid; this is clear since a monad is a monoid in a certain category,
8456:
8438:
8399:
8377:
8249:
8205:
8146:
8104:
7986:
7909:
4409:
3117:
2767:
2731:
2525:
2069:
1916:
1367:
768:
494:
420:
5442:
5226:
4665:
in a way which is compatible with the unit and multiplication of the monad. More formally, a
3222:
2707:
1841:
Composition of monads is not, in general, a monad. For example, the double power set functor
8488:
8430:
8409:
8367:
8312:
8239:
8197:
8136:
8094:
7978:
7938:
7744:
7740:
7654:
7091:
5715:
4515:
to the singleton list . The multiplication concatenates a list of lists into a single list.
4385:
4293:
3877:
3732:
2782:
2735:
2588:
2507:{\displaystyle (-)^{*}:\mathbf {Vect} _{k}\rightleftarrows \mathbf {Vect} _{k}^{op}:(-)^{*}}
2045:
2008:
1663:
1264:
1027:
484:
374:
275:
271:
8166:
uses a stronger definition, where the two categories are isomorphic rather than equivalent.
8093:, IFIP Advances in Information and Communication Technology, vol. 323, pp. 1–19,
8083:
7977:. Lecture Notes in Mathematics. Vol. 47. Berlin, Heidelberg: Springer. pp. 1–77.
7677:
between commutative rings. This adjunction is comonadic, by Beck's theorem, if and only if
7395:
7097:
7023:
5475:
5029:
4688:
3704:
3538:
2340:
2192:
This endofunctor is quickly seen to be a monad, where the unit map stems from the unit map
1692:
1270:
1187:
1164:
1053:
941:
918:
741:
655:
8413:
8279:
8079:
7898:
7544:
7132:
3295:
3260:
3169:
1542:
1457:
1246:
175:
63:
2028:
151:
8355:
5794:{\displaystyle {\text{Sym}}^{\bullet }(M)=\bigoplus _{k=0}^{\infty }{\text{Sym}}^{k}(M)}
3355:
does not admit a left adjoint. Its codensity monad is the monad on sets sending any set
298:, allowing languages without mutable state to do things such as simulate for-loops; see
17:
8291:
7970:
7863:
7838:
7773:
7549:
7458:
7321:{\displaystyle D{\stackrel {\tilde {G}}{\to }}C^{T}{\stackrel {\text{forget}}{\to }}C,}
7003:
6842:). However, there are usually several distinct adjunctions giving rise to a monad: let
6739:
6688:
6479:
6428:
6386:
6332:
6312:
6292:
6272:
6071:
6051:
5957:
5937:
5697:
5677:
5585:
5565:
5270:
5206:
5149:
5129:
5109:
5089:
5069:
5005:
4995:
4975:
4918:
4795:
4740:
4720:
4668:
4648:
4628:
4606:
4585:
4365:
4273:
4245:
3857:
3805:
3781:
3644:
3518:
3216:
3198:
3164:
3146:
3126:
2988:
2887:
2835:
2815:
2791:
2749:
2702:
2685:
2665:
2367:
2151:. This very widespread construction works as follows: the endofunctor is the composite
2075:
2051:
2024:
1973:
1953:
1898:
1595:
1575:
1490:
1437:
1349:
1290:
1100:
1080:
814:
794:
682:
560:
532:
466:
446:
400:
380:
356:
336:
8244:
8227:
8141:
8124:
7735:
In categorical logic, an analogy has been drawn between the monad-comonad theory, and
5562:
Another useful example of a monad is the symmetric algebra functor on the category of
4452:
4211:
3747:
1263:
as the monoid's binary operation, and the second axiom is akin to the existence of an
8516:
8392:
Categorical
Foundations. Special Topics in Order, Topology, Algebra, and Sheaf Theory
4539:
3105:
1236:
8470:
8326:
7853:
7553:
4823:
2518:
1226:
8196:. NATO ASI Series. Vol. 118. Berlin, Heidelberg: Springer. pp. 233–264.
7020:. Then the above free–forgetful adjunction involving the Eilenberg–Moore category
4837:
3731:
In both functional programming and denotational semantics, the maybe monad models
1838:, which is equipped with the multiplication given by composition of endofunctors.
8201:
8099:
8030:
7943:
7886:
https://golem.ph.utexas.edu/category/2009/07/the_monads_hurt_my_head_but_no.html
7736:
7534:
4317:
3360:
2583:
1804:
330:
283:
248:
67:
8434:
8317:
5974:-algebras. There are also algebras over the monads for the alternating tensors
8501:
8372:
7858:
7768:
7752:
7713:
to express types of sequential computation (sometimes with side-effects). See
5436:
3143:
is the endofunctor on the category of vector spaces which maps a vector space
2809:
8394:. Encyclopedia of Mathematics and Its Applications. Vol. 97. Cambridge:
8253:
8150:
8034:
5667:{\displaystyle {\text{Sym}}^{\bullet }(-):{\text{Mod}}(R)\to {\text{Mod}}(R)}
4195:
In functional programming and denotational semantics, the state monad models
2007:. Every set is a comonoid in a unique way, so comonoids are less familiar in
1998:
that come from reversing the arrows everywhere in the definition just given.
831:). These are required to fulfill the following conditions (sometimes called
8495:
4535:
2012:
1484:
8234:. US-Brazil Joint Workshops on the Formal Foundations of Software Systems.
7000:
and whose arrows are the morphisms of adjunctions that are the identity on
1366:
and whose morphisms are the natural transformations between them, with the
6358:
4496:
2003:
5510:
subject to axioms resembling the behavior of convex linear combinations
1221:
1141:
7982:
1895:); this can be said quickly in the terms that a comonad for a category
252:
113:
7929:
Klin; Salamanca (2018), "Iterated
Covariant Powerset is not a Monad",
7848:
6545:{\displaystyle \mathbb {P} :{\mathcal {M}}_{A}\to {\mathcal {M}}_{A}}
1800:
2279:
6620:{\displaystyle \mathbb {P} (M)=\bigvee _{j\geq 0}M^{j}/\Sigma _{j}}
2812:
functor from the category of sets to the category of groups. Then
8346:
8011:
3393:, and analogous constructions are used in functional programming.
1207:, or see below the commutative diagrams not using these notions:
262:
A monad as a tool for studying algebraic gadgets; for example, a
4082:{\displaystyle f:S\to S\times (S\to S\times X),s\mapsto (s',f')}
148:
that satisfy the conditions like associativity. For example, if
7564:. A first example of a comonadic adjunction is the adjunction
4447:
4206:
3742:
3108:
or 'flattening' of 'strings of strings'. This amounts to two
2730:), then the formalism becomes much simpler: adjoint pairs are
8491:, a YouTube video of five short lectures (with one appendix).
6269:
where the first ring is the free anti-symmetric algebra over
3219:, and a natural transformation corresponding to the map from
6816:{\displaystyle T(-):C\rightleftarrows C^{T}:{\text{forget}}}
6715:
6531:
6514:
6455:
5927:{\displaystyle {\text{Sym}}^{\bullet }(R^{\oplus n})\cong R}
5296:
5188:
4994:
2332:
any monad can be found as an explicit adjunction of functors
1860:
1850:
1391:
1145:
1131:
4518:
In functional programming, the list monad is used to model
3385:
The following monads over the category of sets are used in
4526:, and is also used to model nondeterministic computation.
2615:. This monad is discussed, in much greater generality, by
6756:-algebras from the category of algebras over this monad.
6309:-generators and the second ring is the free algebra over
5066:
For example, for the free group monad discussed above, a
1568:
be the function between the power sets induced by taking
52:
What the heck?! How do you even explain what a monad is?
8390:
Pedicchio, Maria
Cristina; Tholen, Walter, eds. (2004).
7540:
For example, the forgetful functor from the category of
5267:
with finite support and such that their sum is equal to
5203:
on the category of sets. It is defined by sending a set
3466:{\displaystyle (-)_{*}:\mathbf {Set} \to \mathbf {Set} }
305:
A monad is also called, especially in old literature, a
8089:
Jacobs, Bart (2010), "Convexity, Duality and
Effects",
8070:Ĺšwirszcz, T. (1974), "Monadic functors and convexity",
7908:, vol. 278, Springer-Verlag, pp. 82 and 120,
7717:, and the more mathematically oriented Wikibook module
4464:
4223:
3759:
3345:{\displaystyle \mathbf {FinSet} \subset \mathbf {Set} }
1017:{\displaystyle \mu \circ T\eta =\mu \circ \eta T=1_{T}}
282:
to arbitrary categories. Monads are also useful in the
7235:{\displaystyle (F:C\to D,G:D\to C,\eta ,\varepsilon )}
5349:
8303:
8072:
Bull. Acad. Polon. Sci. SĂ©r. Sci. Math. Astron. Phys.
7796:
7776:
7657:
7573:
7461:
7425:
7398:
7361:
7259:
7180:
7135:
7100:
7053:
7026:
7006:
6935:
6891:
6848:
6773:
6742:
6711:
6691:
6679:{\displaystyle M^{j}=M\wedge _{A}\cdots \wedge _{A}M}
6633:
6558:
6502:
6482:
6451:
6431:
6409:
6389:
6365:
6335:
6315:
6295:
6275:
6094:
6074:
6054:
6018:
5980:
5960:
5940:
5851:
5807:
5724:
5700:
5680:
5608:
5588:
5568:
5516:
5478:
5445:
5293:
5273:
5229:
5209:
5185:
5152:
5132:
5112:
5106:
together with a map from the free group generated by
5092:
5072:
5032:
5008:
4978:
4941:
4921:
4855:
4798:
4763:
4743:
4723:
4691:
4671:
4651:
4631:
4609:
4588:
4550:
4417:
4388:
4368:
4325:
4316:. Thus, the endofunctor of this monad is exactly the
4296:
4276:
4248:
4153:
4098:
4005:
3964:
3909:
3880:
3860:
3828:
3808:
3784:
3707:
3667:
3647:
3618:
3571:
3541:
3521:
3480:
3418:
3307:
3263:
3225:
3201:
3172:
3149:
3129:
3054:
3011:
2991:
2953:
2910:
2890:
2858:
2838:
2818:
2794:
2752:
2710:
2688:
2668:
2636:
2591:
2533:
2412:
2370:
2343:
2240:
2198:
2160:
2112:
2078:
2054:
1976:
1956:
1924:
1901:
1847:
1812:
1724:
1695:
1666:
1618:
1598:
1578:
1545:
1513:
1493:
1460:
1440:
1412:
1388:
1352:
1317:
1293:
1273:
1249:
1190:
1167:
1103:
1083:
1056:
1030:
973:
944:
921:
888:
844:
817:
797:
771:
744:
705:
685:
658:
619:
583:
563:
535:
497:
483:
are inverse functors, the corresponding monad is the
469:
449:
423:
403:
383:
359:
339:
216:
184:
154:
128:
80:
8125:"André–Quillen cohomology of commutative S-algebras"
6993:{\displaystyle (GF,e,G\varepsilon F)=(T,\eta ,\mu )}
4522:. The covariant powerset monad is also known as the
4463: with: describe multiplication. You can help by
4222: with: describe multiplication. You can help by
3758: with: describe multiplication. You can help by
1121:
We can rewrite these conditions using the following
8496:
2224:{\displaystyle \operatorname {id} _{C}\to G\circ F}
7816:
7782:
7669:
7640:
7467:
7443:
7411:
7376:
7320:
7234:
7150:
7113:
7082:
7039:
7012:
6992:
6921:
6885:be the category whose objects are the adjunctions
6877:
6815:
6748:
6728:
6697:
6678:
6619:
6544:
6488:
6468:
6437:
6417:
6395:
6373:
6341:
6321:
6301:
6281:
6261:
6080:
6060:
6040:
6004:
5966:
5946:
5926:
5837:
5793:
5706:
5686:
5666:
5594:
5574:
5546:
5502:
5464:
5427:
5279:
5259:
5215:
5195:
5158:
5138:
5118:
5098:
5078:
5045:
5014:
4984:
4964:
4927:
4907:
4804:
4784:
4749:
4729:
4709:
4677:
4657:
4637:
4615:
4594:
4574:
4429:
4400:
4374:
4354:
4308:
4282:
4254:
4184:
4138:
4081:
3988:
3949:
3892:
3866:
3846:
3814:
3790:
3720:
3693:
3653:
3630:
3603:
3554:
3527:
3504:
3465:
3344:
3286:obtained by simply expanding all tensor products.
3278:
3249:
3207:
3187:
3155:
3135:
3093:
3037:
2997:
2974:
2944:. The unit map of this monad is given by the maps
2936:
2896:
2876:
2844:
2824:
2800:
2758:
2722:
2694:
2674:
2654:
2607:
2570:
2506:
2376:
2356:
2319:
2223:
2181:
2136:
2084:
2060:
1982:
1962:
1942:
1907:
1868:{\displaystyle {\mathcal {P}}\circ {\mathcal {P}}}
1867:
1830:
1799:The axioms of a monad are formally similar to the
1776:
1707:
1678:
1652:
1604:
1584:
1560:
1531:
1499:
1475:
1446:
1426:
1398:
1358:
1338:
1299:
1279:
1255:
1199:
1176:
1109:
1089:
1069:
1042:
1016:
953:
930:
907:
874:
823:
803:
783:
757:
730:
691:
671:
644:
601:
569:
541:
509:
475:
455:
435:
409:
389:
365:
345:
228:
202:
166:
140:
104:
7517:is an isomorphism if and only if its image under
7094:, which is by definition the full subcategory of
3123:Another monad arising from an adjunction is when
2904:and returns the underlying set of the free group
37:For the uses of monads in computer software, see
8232:Electronic Notes in Theoretical Computer Science
7931:Electronic Notes in Theoretical Computer Science
2571:{\displaystyle V^{*}:=\operatorname {Hom} (V,k)}
2068:is a monad. Its multiplication and unit are the
6353:Commutative algebras in E-infinity ring spectra
875:{\displaystyle \mu \circ T\mu =\mu \circ \mu T}
236:determined by the adjoint relation is a monad.
44:
8366:, Graduate Texts in Mathematics, vol. 5,
8296:Topologie Algébrique et Théorie des Faisceaux.
3661:to themselves, and the two disjoint points in
1777:{\displaystyle \mu _{A}\colon T(T(A))\to T(A)}
7906:Grundlehren der mathematischen Wissenschaften
6705:-times. Then there is an associated category
3367:. This and similar examples are discussed in
8:
6252:
6220:
3515:The unit is given by the inclusion of a set
3499:
3493:
2578:. The associated monad sends a vector space
1702:
1696:
1370:induced by the composition of endofunctors.
251:of some fixed category (an endofunctor is a
7513:reflects isomorphisms, i.e., a morphism in
3950:{\displaystyle \eta _{X}(x):S\to S\times X}
2623:Closure operators on partially ordered sets
2517:where both functors are given by sending a
6005:{\displaystyle {\text{Alt}}^{\bullet }(-)}
5287:. In set-builder notation, this is the set
4139:{\displaystyle \mu _{X}(f):S\to S\times X}
270:Monads are used in the theory of pairs of
8371:
8345:
8316:
8243:
8140:
8098:
7942:
7803:
7795:
7775:
7701:is widely applied in algebraic geometry.
7656:
7626:
7615:
7605:
7594:
7581:
7572:
7460:
7424:
7403:
7397:
7363:
7362:
7360:
7304:
7299:
7297:
7296:
7290:
7272:
7271:
7266:
7264:
7263:
7258:
7179:
7134:
7105:
7099:
7054:
7052:
7031:
7025:
7005:
6934:
6890:
6849:
6847:
6808:
6799:
6772:
6741:
6720:
6714:
6713:
6710:
6690:
6667:
6654:
6638:
6632:
6611:
6602:
6596:
6580:
6560:
6559:
6557:
6536:
6530:
6529:
6519:
6513:
6512:
6504:
6503:
6501:
6481:
6460:
6454:
6453:
6450:
6430:
6411:
6410:
6408:
6388:
6367:
6366:
6364:
6334:
6314:
6294:
6274:
6246:
6227:
6198:
6185:
6180:
6166:
6147:
6118:
6105:
6100:
6095:
6093:
6073:
6053:
6023:
6017:
5987:
5982:
5979:
5959:
5939:
5915:
5896:
5871:
5858:
5853:
5850:
5814:
5809:
5806:
5776:
5771:
5764:
5753:
5731:
5726:
5723:
5699:
5679:
5650:
5633:
5615:
5610:
5607:
5587:
5567:
5515:
5477:
5453:
5444:
5386:
5355:
5348:
5295:
5294:
5292:
5272:
5228:
5208:
5187:
5186:
5184:
5151:
5131:
5111:
5091:
5071:
5037:
5031:
5007:
4977:
4940:
4920:
4854:
4797:
4762:
4742:
4722:
4690:
4670:
4650:
4630:
4608:
4587:
4549:
4416:
4387:
4367:
4326:
4324:
4295:
4275:
4247:
4152:
4103:
4097:
4004:
3963:
3914:
3908:
3879:
3859:
3827:
3807:
3783:
3712:
3706:
3685:
3675:
3666:
3646:
3617:
3595:
3576:
3570:
3546:
3540:
3520:
3479:
3452:
3438:
3429:
3417:
3331:
3308:
3306:
3262:
3224:
3200:
3171:
3148:
3128:
3053:
3012:
3010:
2990:
2952:
2911:
2909:
2889:
2857:
2837:
2817:
2793:
2751:
2709:
2687:
2667:
2635:
2596:
2590:
2538:
2532:
2498:
2476:
2471:
2457:
2447:
2433:
2423:
2411:
2369:
2348:
2342:
2286:
2245:
2239:
2203:
2197:
2159:
2111:
2077:
2053:
2001:Monads are to monoids as comonads are to
1975:
1955:
1930:
1929:
1923:
1900:
1859:
1858:
1849:
1848:
1846:
1811:
1729:
1723:
1694:
1665:
1653:{\displaystyle \eta _{A}\colon A\to T(A)}
1623:
1617:
1597:
1577:
1544:
1512:
1492:
1459:
1439:
1413:
1411:
1390:
1389:
1387:
1351:
1330:
1319:
1316:
1292:
1272:
1248:
1189:
1166:
1102:
1082:
1077:denotes the identity transformation from
1061:
1055:
1029:
1008:
972:
943:
920:
893:
887:
843:
816:
796:
770:
749:
743:
716:
704:
684:
663:
657:
630:
618:
582:
562:
534:
496:
468:
448:
422:
402:
382:
358:
338:
215:
183:
153:
127:
79:
8364:Categories for the Working Mathematician
7790:. Monads described above are monads for
4908:{\displaystyle f\colon (x,h)\to (x',h')}
3368:
517:, is discussed under the dual theory of
8175:
8163:
7975:Reports of the Midwest Category Seminar
7956:
7875:
6357:There is an analogous construction for
3735:, that is, computations that may fail.
2137:{\displaystyle F:C\rightleftarrows D:G}
1887:definition is a formal definition of a
1831:{\displaystyle \operatorname {End} (C)}
7897:Barr, Michael; Wells, Charles (1985),
7881:
7879:
5439:, i.e., sets equipped with operations
4816:of the algebra such that the diagrams
1346:whose objects are the endofunctors of
645:{\displaystyle \eta \colon 1_{C}\to T}
8334:Theory and Applications of Categories
8327:"Codensity and the ultrafilter monad"
8281:Category Theory for Computing Science
8057:
7767:It is possible to define monads in a
5838:{\displaystyle {\text{Sym}}^{0}(M)=R}
5022:-algebras form a category called the
3999:The multiplication maps the function
3375:Monads used in denotational semantics
2852:. In this case, the associated monad
1161:for the explanation of the notations
731:{\displaystyle \mu \colon T^{2}\to T}
7:
6922:{\displaystyle (F,G,e,\varepsilon )}
5170:Algebras over the distribution monad
3641:The multiplication maps elements of
3604:{\displaystyle \eta _{X}:X\to X_{*}}
3505:{\displaystyle X\mapsto X\cup \{*\}}
2616:
2023:The notion of monad was invented by
1990:to itself, with a set of axioms for
1875:does not admit any monad structure.
266:can be described by a certain monad.
27:Operation in algebra and mathematics
8190:"Monads for functional programming"
8129:Journal of Pure and Applied Algebra
7689:-module. It thus allows to descend
7083:{\displaystyle \mathbf {Adj} (C,T)}
6878:{\displaystyle \mathbf {Adj} (C,T)}
6826:whose left adjoint sends an object
4355:{\displaystyle \mathrm {Hom} (E,-)}
487:. In general, adjunctions are not
7697:-modules. The resulting theory of
7491:
7339:) can be naturally endowed with a
6729:{\displaystyle {\mathcal {C}}_{A}}
6608:
6469:{\displaystyle {\mathcal {M}}_{A}}
5765:
5375:
5352:
5062:Algebras over the free group monad
4333:
4330:
4327:
3038:{\displaystyle \mathrm {Free} (X)}
3022:
3019:
3016:
3013:
2937:{\displaystyle \mathrm {Free} (X)}
2921:
2918:
2915:
2912:
1934:
1931:
1339:{\displaystyle \mathbf {End} _{C}}
1307:can alternatively be defined as a
119:from a category to itself and two
25:
7392:and the Eilenberg–Moore category
5558:Algebras over the symmetric monad
1943:{\displaystyle C^{\mathrm {op} }}
8423:"Chapter 5. Monads and Comonads"
8047:from the original on 5 Apr 2021.
7817:{\displaystyle C=\mathbf {Cat} }
7810:
7807:
7804:
7730:imperative programming languages
7715:monads in functional programming
7622:
7619:
7616:
7601:
7598:
7595:
7061:
7058:
7055:
6856:
6853:
6850:
5582:-modules for a commutative ring
4836:
4822:
4451:
4210:
3746:
3459:
3456:
3453:
3445:
3442:
3439:
3391:imperative programming languages
3338:
3335:
3332:
3324:
3321:
3318:
3315:
3312:
3309:
2467:
2464:
2461:
2458:
2443:
2440:
2437:
2434:
1420:
1417:
1414:
1326:
1323:
1320:
1225:
1213:
679:denotes the identity functor on
296:functional programming languages
292:imperative programming languages
39:monads in functional programming
8226:Mulry, Philip S. (1998-01-01).
7899:"Toposes, Triples and Theories"
7834:Distributive law between monads
6041:{\displaystyle T^{\bullet }(-)}
4965:{\displaystyle f\colon x\to x'}
4818:
4785:{\displaystyle h\colon Tx\to x}
4362:. The component of the unit at
4185:{\displaystyle s\mapsto f'(s')}
3854:. The component of the unit at
3094:{\displaystyle T(T(X))\to T(X)}
2096:Monads arising from adjunctions
1791:. These data describe a monad.
1267:(which we think of as given by
1235:The first axiom is akin to the
239:In concise terms, a monad is a
8498:covers monads in 2-categories.
8455:, Courier Dover Publications,
7971:"Introduction to bicategories"
7844:Monad (functional programming)
7661:
7611:
7444:{\displaystyle G\colon D\to C}
7435:
7368:
7300:
7277:
7267:
7229:
7211:
7193:
7181:
7145:
7139:
7077:
7065:
6987:
6969:
6963:
6936:
6916:
6892:
6872:
6860:
6792:
6783:
6777:
6570:
6564:
6525:
6207:
6191:
6172:
6140:
6127:
6111:
6035:
6029:
5999:
5993:
5921:
5889:
5880:
5864:
5826:
5820:
5788:
5782:
5743:
5737:
5661:
5655:
5647:
5644:
5638:
5627:
5621:
5538:
5526:
5497:
5485:
5407:
5401:
5366:
5360:
5342:
5330:
5327:
5307:
5301:
5254:
5242:
5239:
5196:{\displaystyle {\mathcal {D}}}
4951:
4902:
4880:
4877:
4874:
4862:
4776:
4704:
4692:
4575:{\displaystyle (T,\eta ,\mu )}
4569:
4551:
4421:
4349:
4337:
4300:
4179:
4168:
4157:
4124:
4115:
4109:
4076:
4054:
4051:
4042:
4030:
4024:
4015:
3989:{\displaystyle s\mapsto (s,x)}
3983:
3971:
3968:
3935:
3926:
3920:
3847:{\displaystyle S\to S\times X}
3832:
3682:
3668:
3622:
3588:
3484:
3449:
3426:
3419:
3381:Monad (functional programming)
3273:
3267:
3244:
3241:
3235:
3229:
3182:
3176:
3088:
3082:
3076:
3073:
3070:
3064:
3058:
3032:
3026:
2969:
2963:
2957:
2931:
2925:
2649:
2637:
2565:
2553:
2495:
2488:
2453:
2420:
2413:
2209:
2122:
1825:
1819:
1771:
1765:
1759:
1756:
1753:
1747:
1741:
1647:
1641:
1635:
1555:
1549:
1532:{\displaystyle f\colon A\to B}
1523:
1470:
1464:
1427:{\displaystyle \mathbf {Set} }
1399:{\displaystyle {\mathcal {P}}}
1034:
899:
722:
636:
602:{\displaystyle T\colon C\to C}
593:
300:Monad (functional programming)
105:{\displaystyle (T,\eta ,\mu )}
99:
81:
1:
8472:Category Theory Lecture Notes
8245:10.1016/S1571-0661(05)80241-5
8142:10.1016/S0022-4049(98)00051-6
7351:. The adjunction is called a
4520:nondeterministic computations
3409:monad adds a disjoint point:
3298:. For example, the inclusion
2662:(with a single morphism from
329:A monad is a certain type of
8091:Theoretical Computer Science
8035:"Category Theory in Context"
8015:. 2009-04-04. Archived from
7377:{\displaystyle {\tilde {G}}}
6418:{\displaystyle \mathbb {S} }
6403:-algebras for a commutative
6374:{\displaystyle \mathbb {S} }
4602:, it is natural to consider
2627:For categories arising from
1950:. It is therefore a functor
1024:(as natural transformations
882:(as natural transformations
8469:Turi, Daniele (1996–2001),
8202:10.1007/978-3-662-02880-3_8
8123:Basterra, M. (1999-12-15).
8100:10.1007/978-3-642-15240-5_1
7944:10.1016/j.entcs.2018.11.013
7343:-algebra structure for any
7090:. An initial object is the
6496:-modules, then the functor
4507:. The unit maps an element
3694:{\displaystyle (X_{*})_{*}}
2403:arises from the adjunction
2182:{\displaystyle T=G\circ F.}
1787:takes a set of sets to its
577:consists of an endofunctor
325:Introduction and definition
32:Monad (homological algebra)
8544:
8453:Category Theory in Context
8435:10.1142/9789811286018_0005
8396:Cambridge University Press
8362:MacLane, Saunders (1978),
8318:10.7146/math.scand.a-10995
8192:. In Broy, Manfred (ed.).
7419:. By extension, a functor
6012:and total tensor functors
4533:
4430:{\displaystyle e\mapsto x}
3631:{\displaystyle x\mapsto x}
3378:
2877:{\displaystyle T=G\circ F}
2742:Free-forgetful adjunctions
908:{\displaystyle T^{3}\to T}
229:{\displaystyle \eta ,\mu }
203:{\displaystyle T=G\circ F}
141:{\displaystyle \eta ,\mu }
36:
29:
8373:10.1007/978-1-4757-4721-8
7728:of impure functional and
7719:b:Haskell/Category theory
7498:Beck's monadicity theorem
7492:Beck's monadicity theorem
7455:if it has a left adjoint
7386:equivalence of categories
5547:{\displaystyle rx+(1-r)y}
4262:, the endofunctor of the
3798:, the endofunctor of the
2975:{\displaystyle X\to T(X)}
2655:{\displaystyle (P,\leq )}
2147:gives rise to a monad on
1660:, which assigns to every
529:Throughout this article,
8427:Starting Category Theory
8305:Mathematica Scandinavica
7747:, and their relation to
7651:for a ring homomorphism
7121:consisting only of free
7047:is a terminal object in
6383:which gives commutative
5223:to the set of functions
5024:Eilenberg–Moore category
4833:
4828:
4290:to the set of functions
3822:to the set of functions
2394:double dualization monad
2336:Eilenberg–Moore category
784:{\displaystyle T\circ T}
510:{\displaystyle F\circ G}
436:{\displaystyle G\circ F}
319:fundamental construction
30:Not to be confused with
18:Eilenberg–Moore category
8421:Perrone, Paolo (2024),
8188:Wadler, Philip (1993).
7724:Monads are used in the
7699:faithfully flat descent
5465:{\displaystyle x+_{r}y}
5260:{\displaystyle f:X\to }
5174:Another example is the
4757:together with an arrow
4444:The list and set monads
3401:The endofunctor of the
3250:{\displaystyle T(T(V))}
3110:natural transformations
2723:{\displaystyle x\leq y}
1287:). Indeed, a monad on
1159:natural transformations
611:natural transformations
417:, then the composition
121:natural transformations
8325:Leinster, Tom (2013),
8194:Program Design Calculi
7969:BĂ©nabou, Jean (1967).
7818:
7784:
7726:denotational semantics
7711:functional programming
7671:
7670:{\displaystyle A\to B}
7642:
7469:
7445:
7413:
7378:
7322:
7242:with associated monad
7236:
7152:
7129:-algebras of the form
7115:
7084:
7041:
7014:
6994:
6923:
6879:
6817:
6760:Monads and adjunctions
6750:
6730:
6699:
6680:
6621:
6546:
6490:
6470:
6439:
6419:
6397:
6375:
6343:
6323:
6303:
6283:
6263:
6082:
6062:
6048:giving anti-symmetric
6042:
6006:
5968:
5948:
5928:
5839:
5795:
5769:
5708:
5688:
5668:
5596:
5576:
5548:
5504:
5466:
5429:
5281:
5261:
5217:
5197:
5160:
5140:
5120:
5100:
5080:
5047:
5016:
4999:
4992:such that the diagram
4986:
4966:
4935:-algebras is an arrow
4929:
4909:
4806:
4786:
4751:
4731:
4711:
4679:
4659:
4639:
4617:
4596:
4576:
4431:
4402:
4401:{\displaystyle x\in X}
4376:
4356:
4310:
4309:{\displaystyle E\to X}
4284:
4256:
4186:
4140:
4083:
3990:
3951:
3894:
3893:{\displaystyle x\in X}
3868:
3848:
3816:
3792:
3722:
3695:
3655:
3632:
3605:
3556:
3529:
3506:
3467:
3387:denotational semantics
3346:
3280:
3251:
3209:
3189:
3157:
3137:
3104:made out of a natural
3095:
3039:
2999:
2976:
2938:
2898:
2878:
2846:
2826:
2802:
2760:
2724:
2696:
2676:
2656:
2629:partially ordered sets
2609:
2608:{\displaystyle V^{**}}
2572:
2508:
2378:
2358:
2321:
2225:
2183:
2138:
2086:
2062:
2019:Terminological history
1984:
1964:
1944:
1909:
1869:
1832:
1778:
1709:
1680:
1679:{\displaystyle a\in A}
1654:
1606:
1586:
1562:
1533:
1501:
1477:
1448:
1428:
1400:
1360:
1340:
1301:
1281:
1257:
1201:
1178:
1150:
1136:
1111:
1091:
1071:
1044:
1043:{\displaystyle T\to T}
1018:
963:horizontal composition
955:
932:
909:
876:
825:
805:
785:
759:
732:
693:
673:
646:
603:
571:
543:
511:
477:
457:
437:
411:
391:
367:
347:
288:denotational semantics
280:partially ordered sets
274:, and they generalize
230:
204:
168:
142:
106:
54:
8451:Riehl, Emily (2017),
8228:"Monads in Semantics"
7819:
7785:
7757:intuitionistic logics
7672:
7643:
7521:is an isomorphism in
7470:
7446:
7414:
7412:{\displaystyle C^{T}}
7379:
7355:if the first functor
7323:
7237:
7174:Given any adjunction
7153:
7116:
7114:{\displaystyle C^{T}}
7085:
7042:
7040:{\displaystyle C^{T}}
7015:
6995:
6924:
6880:
6818:
6751:
6731:
6700:
6681:
6622:
6552:is the monad given by
6547:
6491:
6471:
6440:
6420:
6398:
6376:
6344:
6324:
6304:
6284:
6264:
6083:
6063:
6043:
6007:
5969:
5949:
5929:
5840:
5796:
5749:
5714:to the direct sum of
5709:
5689:
5669:
5597:
5577:
5549:
5505:
5503:{\displaystyle r\in }
5467:
5430:
5282:
5262:
5218:
5198:
5161:
5141:
5121:
5101:
5081:
5048:
5046:{\displaystyle C^{T}}
5017:
4998:
4987:
4967:
4930:
4910:
4807:
4787:
4752:
4732:
4712:
4710:{\displaystyle (x,h)}
4680:
4660:
4640:
4618:
4597:
4577:
4503:) with elements from
4495:to the set of finite
4432:
4403:
4377:
4357:
4311:
4285:
4257:
4203:The environment monad
4197:stateful computations
4187:
4141:
4084:
3991:
3952:
3895:
3869:
3849:
3817:
3793:
3723:
3721:{\displaystyle X_{*}}
3696:
3656:
3633:
3606:
3557:
3555:{\displaystyle X_{*}}
3530:
3507:
3468:
3347:
3281:
3252:
3210:
3190:
3158:
3138:
3096:
3040:
3000:
2977:
2939:
2899:
2879:
2847:
2827:
2803:
2761:
2725:
2697:
2677:
2657:
2610:
2573:
2509:
2379:
2359:
2357:{\displaystyle C^{T}}
2322:
2226:
2184:
2139:
2087:
2063:
1985:
1965:
1945:
1910:
1870:
1833:
1779:
1710:
1708:{\displaystyle \{a\}}
1681:
1655:
1607:
1587:
1563:
1534:
1502:
1478:
1449:
1429:
1401:
1361:
1341:
1302:
1282:
1280:{\displaystyle \eta }
1258:
1202:
1200:{\displaystyle \mu T}
1179:
1177:{\displaystyle T\mu }
1149:
1135:
1112:
1092:
1072:
1070:{\displaystyle 1_{T}}
1045:
1019:
956:
954:{\displaystyle \mu T}
933:
931:{\displaystyle T\mu }
910:
877:
826:
806:
786:
760:
758:{\displaystyle T^{2}}
733:
694:
674:
672:{\displaystyle 1_{C}}
647:
604:
572:
544:
512:
478:
458:
438:
412:
392:
368:
348:
315:standard construction
231:
205:
169:
143:
107:
8429:, World Scientific,
7794:
7774:
7655:
7571:
7505:is monadic if it is
7459:
7423:
7396:
7359:
7257:
7178:
7151:{\displaystyle T(x)}
7133:
7098:
7051:
7024:
7004:
6933:
6889:
6846:
6771:
6740:
6709:
6689:
6631:
6556:
6500:
6480:
6449:
6429:
6407:
6387:
6363:
6333:
6313:
6293:
6273:
6092:
6072:
6068:-algebras, and free
6052:
6016:
5978:
5958:
5938:
5849:
5805:
5722:
5698:
5678:
5606:
5586:
5566:
5554:in Euclidean space.
5514:
5476:
5443:
5291:
5271:
5227:
5207:
5183:
5150:
5130:
5110:
5090:
5070:
5030:
5006:
4976:
4939:
4919:
4853:
4796:
4761:
4741:
4721:
4689:
4669:
4649:
4629:
4607:
4586:
4548:
4530:Algebras for a monad
4489:nondeterminism monad
4415:
4386:
4366:
4323:
4294:
4274:
4246:
4151:
4096:
4003:
3962:
3907:
3878:
3858:
3826:
3806:
3782:
3733:partial computations
3705:
3665:
3645:
3616:
3569:
3539:
3519:
3478:
3416:
3305:
3279:{\displaystyle T(V)}
3261:
3223:
3199:
3188:{\displaystyle T(V)}
3170:
3147:
3127:
3052:
3009:
2989:
2951:
2908:
2888:
2856:
2836:
2816:
2792:
2750:
2708:
2686:
2666:
2634:
2589:
2531:
2410:
2368:
2341:
2238:
2196:
2158:
2110:
2076:
2052:
1974:
1954:
1922:
1899:
1845:
1810:
1722:
1693:
1664:
1616:
1596:
1576:
1561:{\displaystyle T(f)}
1543:
1511:
1491:
1476:{\displaystyle T(A)}
1458:
1438:
1410:
1386:
1350:
1315:
1291:
1271:
1256:{\displaystyle \mu }
1247:
1188:
1165:
1123:commutative diagrams
1101:
1081:
1054:
1028:
971:
942:
919:
886:
842:
833:coherence conditions
815:
795:
769:
742:
703:
683:
656:
617:
581:
561:
533:
495:
467:
447:
421:
401:
381:
357:
337:
214:
182:
178:to each other, then
152:
126:
78:
8502:Monads and comonads
8356:2012arXiv1209.3606L
7709:Monads are used in
7250:can be factored as
7170:Monadic adjunctions
6476:is the category of
5166:is a group itself.
4625:, i.e., objects of
3114:variety of algebras
2832:is left adjoint of
2484:
2297:
1915:is a monad for the
1507:and for a function
1374:The power set monad
1157:See the article on
284:theory of datatypes
167:{\displaystyle F,G}
7983:10.1007/BFb0074299
7814:
7780:
7667:
7638:
7558:algebraic geometry
7480:out of objects in
7465:
7441:
7409:
7374:
7353:monadic adjunction
7318:
7232:
7148:
7111:
7080:
7037:
7010:
6990:
6919:
6875:
6813:
6746:
6726:
6695:
6676:
6617:
6591:
6542:
6486:
6466:
6435:
6415:
6393:
6371:
6339:
6319:
6299:
6279:
6259:
6257:
6078:
6058:
6038:
6002:
5964:
5944:
5924:
5835:
5791:
5704:
5684:
5664:
5592:
5572:
5544:
5500:
5462:
5425:
5418:
5397:
5277:
5257:
5213:
5193:
5177:distribution monad
5156:
5136:
5116:
5096:
5086:-algebra is a set
5076:
5043:
5012:
5000:
4982:
4962:
4925:
4905:
4802:
4782:
4747:
4727:
4707:
4675:
4655:
4635:
4613:
4592:
4572:
4427:
4398:
4382:maps each element
4372:
4352:
4306:
4280:
4252:
4182:
4136:
4079:
3986:
3947:
3890:
3874:maps each element
3864:
3844:
3812:
3788:
3718:
3691:
3651:
3628:
3601:
3552:
3525:
3502:
3463:
3342:
3276:
3247:
3205:
3185:
3153:
3133:
3091:
3035:
2995:
2985:including any set
2972:
2934:
2894:
2874:
2842:
2822:
2798:
2788:of sets, and let
2756:
2732:Galois connections
2720:
2692:
2672:
2652:
2605:
2568:
2504:
2456:
2388:Double dualization
2374:
2354:
2317:
2221:
2179:
2134:
2082:
2072:on the objects of
2058:
1980:
1960:
1940:
1905:
1865:
1828:
1774:
1705:
1676:
1650:
1602:
1582:
1558:
1529:
1497:
1473:
1444:
1424:
1396:
1368:monoidal structure
1356:
1336:
1297:
1277:
1253:
1197:
1174:
1151:
1137:
1107:
1087:
1067:
1040:
1014:
951:
928:
905:
872:
821:
801:
781:
755:
728:
689:
669:
642:
609:together with two
599:
567:
539:
507:
473:
453:
433:
407:
387:
363:
343:
333:. For example, if
226:
200:
164:
138:
102:
8504:, video tutorial.
8444:978-981-12-8600-1
8383:978-1-4419-3123-8
8211:978-3-662-02880-3
8110:978-3-642-15239-9
7992:978-3-540-35545-8
7783:{\displaystyle C}
7745:interior algebras
7741:closure operators
7468:{\displaystyle F}
7371:
7309:
7307:
7283:
7280:
7125:-algebras, i.e.,
7013:{\displaystyle C}
6811:
6749:{\displaystyle A}
6698:{\displaystyle j}
6576:
6489:{\displaystyle A}
6438:{\displaystyle A}
6396:{\displaystyle A}
6342:{\displaystyle n}
6322:{\displaystyle R}
6302:{\displaystyle n}
6282:{\displaystyle R}
6183:
6103:
6081:{\displaystyle R}
6061:{\displaystyle R}
5985:
5967:{\displaystyle R}
5947:{\displaystyle R}
5856:
5812:
5774:
5729:
5707:{\displaystyle M}
5687:{\displaystyle R}
5653:
5636:
5613:
5595:{\displaystyle R}
5575:{\displaystyle R}
5382:
5358:
5280:{\displaystyle 1}
5216:{\displaystyle X}
5159:{\displaystyle X}
5139:{\displaystyle X}
5119:{\displaystyle X}
5099:{\displaystyle X}
5079:{\displaystyle T}
5015:{\displaystyle T}
4985:{\displaystyle C}
4928:{\displaystyle T}
4844:
4843:
4805:{\displaystyle C}
4750:{\displaystyle C}
4730:{\displaystyle x}
4678:{\displaystyle T}
4658:{\displaystyle T}
4638:{\displaystyle C}
4616:{\displaystyle T}
4595:{\displaystyle C}
4481:
4480:
4410:constant function
4375:{\displaystyle X}
4283:{\displaystyle X}
4268:environment monad
4255:{\displaystyle E}
4240:
4239:
3867:{\displaystyle X}
3815:{\displaystyle X}
3791:{\displaystyle S}
3776:
3775:
3654:{\displaystyle X}
3528:{\displaystyle X}
3208:{\displaystyle V}
3156:{\displaystyle V}
3136:{\displaystyle T}
3118:universal algebra
2998:{\displaystyle X}
2897:{\displaystyle X}
2845:{\displaystyle G}
2825:{\displaystyle F}
2801:{\displaystyle F}
2768:forgetful functor
2759:{\displaystyle G}
2746:For example, let
2736:closure operators
2695:{\displaystyle y}
2675:{\displaystyle x}
2526:dual vector space
2377:{\displaystyle T}
2364:(the category of
2298:
2289:
2085:{\displaystyle C}
2070:identity function
2061:{\displaystyle C}
1983:{\displaystyle C}
1963:{\displaystyle U}
1917:opposite category
1908:{\displaystyle C}
1605:{\displaystyle A}
1592:. For every set
1585:{\displaystyle f}
1500:{\displaystyle A}
1447:{\displaystyle A}
1359:{\displaystyle C}
1300:{\displaystyle C}
1233:
1232:
1155:
1154:
1110:{\displaystyle T}
1090:{\displaystyle T}
824:{\displaystyle C}
804:{\displaystyle C}
692:{\displaystyle C}
570:{\displaystyle C}
542:{\displaystyle C}
525:Formal definition
476:{\displaystyle G}
456:{\displaystyle F}
410:{\displaystyle G}
390:{\displaystyle F}
366:{\displaystyle G}
346:{\displaystyle F}
276:closure operators
16:(Redirected from
8535:
8523:Adjoint functors
8478:
8477:
8465:
8447:
8417:
8386:
8375:
8358:
8349:
8331:
8321:
8320:
8299:
8287:
8286:
8258:
8257:
8247:
8223:
8217:
8215:
8185:
8179:
8173:
8167:
8161:
8155:
8154:
8144:
8120:
8114:
8113:
8102:
8086:
8067:
8061:
8055:
8049:
8048:
8046:
8039:
8027:
8021:
8020:
8003:
7997:
7996:
7966:
7960:
7954:
7948:
7947:
7946:
7926:
7920:
7919:
7903:
7894:
7888:
7883:
7823:
7821:
7820:
7815:
7813:
7789:
7787:
7786:
7781:
7676:
7674:
7673:
7668:
7647:
7645:
7644:
7639:
7631:
7630:
7625:
7610:
7609:
7604:
7586:
7585:
7545:Hausdorff spaces
7474:
7472:
7471:
7466:
7450:
7448:
7447:
7442:
7418:
7416:
7415:
7410:
7408:
7407:
7383:
7381:
7380:
7375:
7373:
7372:
7364:
7327:
7325:
7324:
7319:
7311:
7310:
7308:
7305:
7303:
7298:
7295:
7294:
7285:
7284:
7282:
7281:
7273:
7270:
7265:
7241:
7239:
7238:
7233:
7158:for some object
7157:
7155:
7154:
7149:
7120:
7118:
7117:
7112:
7110:
7109:
7092:Kleisli category
7089:
7087:
7086:
7081:
7064:
7046:
7044:
7043:
7038:
7036:
7035:
7019:
7017:
7016:
7011:
6999:
6997:
6996:
6991:
6928:
6926:
6925:
6920:
6884:
6882:
6881:
6876:
6859:
6822:
6820:
6819:
6814:
6812:
6809:
6804:
6803:
6755:
6753:
6752:
6747:
6735:
6733:
6732:
6727:
6725:
6724:
6719:
6718:
6704:
6702:
6701:
6696:
6685:
6683:
6682:
6677:
6672:
6671:
6659:
6658:
6643:
6642:
6626:
6624:
6623:
6618:
6616:
6615:
6606:
6601:
6600:
6590:
6563:
6551:
6549:
6548:
6543:
6541:
6540:
6535:
6534:
6524:
6523:
6518:
6517:
6507:
6495:
6493:
6492:
6487:
6475:
6473:
6472:
6467:
6465:
6464:
6459:
6458:
6444:
6442:
6441:
6436:
6424:
6422:
6421:
6416:
6414:
6402:
6400:
6399:
6394:
6380:
6378:
6377:
6372:
6370:
6348:
6346:
6345:
6340:
6328:
6326:
6325:
6320:
6308:
6306:
6305:
6300:
6288:
6286:
6285:
6280:
6268:
6266:
6265:
6260:
6258:
6251:
6250:
6232:
6231:
6206:
6205:
6190:
6189:
6184:
6181:
6171:
6170:
6152:
6151:
6126:
6125:
6110:
6109:
6104:
6101:
6087:
6085:
6084:
6079:
6067:
6065:
6064:
6059:
6047:
6045:
6044:
6039:
6028:
6027:
6011:
6009:
6008:
6003:
5992:
5991:
5986:
5983:
5973:
5971:
5970:
5965:
5953:
5951:
5950:
5945:
5933:
5931:
5930:
5925:
5920:
5919:
5901:
5900:
5879:
5878:
5863:
5862:
5857:
5854:
5844:
5842:
5841:
5836:
5819:
5818:
5813:
5810:
5800:
5798:
5797:
5792:
5781:
5780:
5775:
5772:
5768:
5763:
5736:
5735:
5730:
5727:
5716:symmetric tensor
5713:
5711:
5710:
5705:
5693:
5691:
5690:
5685:
5673:
5671:
5670:
5665:
5654:
5651:
5637:
5634:
5620:
5619:
5614:
5611:
5601:
5599:
5598:
5593:
5581:
5579:
5578:
5573:
5553:
5551:
5550:
5545:
5509:
5507:
5506:
5501:
5471:
5469:
5468:
5463:
5458:
5457:
5434:
5432:
5431:
5426:
5424:
5420:
5419:
5396:
5359:
5356:
5300:
5299:
5286:
5284:
5283:
5278:
5266:
5264:
5263:
5258:
5222:
5220:
5219:
5214:
5202:
5200:
5199:
5194:
5192:
5191:
5165:
5163:
5162:
5157:
5145:
5143:
5142:
5137:
5125:
5123:
5122:
5117:
5105:
5103:
5102:
5097:
5085:
5083:
5082:
5077:
5052:
5050:
5049:
5044:
5042:
5041:
5021:
5019:
5018:
5013:
4991:
4989:
4988:
4983:
4971:
4969:
4968:
4963:
4961:
4934:
4932:
4931:
4926:
4914:
4912:
4911:
4906:
4901:
4890:
4840:
4826:
4819:
4811:
4809:
4808:
4803:
4791:
4789:
4788:
4783:
4756:
4754:
4753:
4748:
4736:
4734:
4733:
4728:
4716:
4714:
4713:
4708:
4684:
4682:
4681:
4676:
4664:
4662:
4661:
4656:
4644:
4642:
4641:
4636:
4622:
4620:
4619:
4614:
4601:
4599:
4598:
4593:
4581:
4579:
4578:
4573:
4476:
4473:
4455:
4448:
4436:
4434:
4433:
4428:
4407:
4405:
4404:
4399:
4381:
4379:
4378:
4373:
4361:
4359:
4358:
4353:
4336:
4315:
4313:
4312:
4307:
4289:
4287:
4286:
4281:
4261:
4259:
4258:
4253:
4235:
4232:
4214:
4207:
4191:
4189:
4188:
4183:
4178:
4167:
4145:
4143:
4142:
4137:
4108:
4107:
4089:to the function
4088:
4086:
4085:
4080:
4075:
4064:
3995:
3993:
3992:
3987:
3956:
3954:
3953:
3948:
3919:
3918:
3900:to the function
3899:
3897:
3896:
3891:
3873:
3871:
3870:
3865:
3853:
3851:
3850:
3845:
3821:
3819:
3818:
3813:
3797:
3795:
3794:
3789:
3771:
3768:
3750:
3743:
3727:
3725:
3724:
3719:
3717:
3716:
3700:
3698:
3697:
3692:
3690:
3689:
3680:
3679:
3660:
3658:
3657:
3652:
3637:
3635:
3634:
3629:
3610:
3608:
3607:
3602:
3600:
3599:
3581:
3580:
3561:
3559:
3558:
3553:
3551:
3550:
3534:
3532:
3531:
3526:
3511:
3509:
3508:
3503:
3472:
3470:
3469:
3464:
3462:
3448:
3434:
3433:
3351:
3349:
3348:
3343:
3341:
3327:
3290:Codensity monads
3285:
3283:
3282:
3277:
3256:
3254:
3253:
3248:
3214:
3212:
3211:
3206:
3194:
3192:
3191:
3186:
3162:
3160:
3159:
3154:
3142:
3140:
3139:
3134:
3100:
3098:
3097:
3092:
3044:
3042:
3041:
3036:
3025:
3004:
3002:
3001:
2996:
2981:
2979:
2978:
2973:
2943:
2941:
2940:
2935:
2924:
2903:
2901:
2900:
2895:
2883:
2881:
2880:
2875:
2851:
2849:
2848:
2843:
2831:
2829:
2828:
2823:
2807:
2805:
2804:
2799:
2765:
2763:
2762:
2757:
2729:
2727:
2726:
2721:
2701:
2699:
2698:
2693:
2681:
2679:
2678:
2673:
2661:
2659:
2658:
2653:
2614:
2612:
2611:
2606:
2604:
2603:
2577:
2575:
2574:
2569:
2543:
2542:
2513:
2511:
2510:
2505:
2503:
2502:
2483:
2475:
2470:
2452:
2451:
2446:
2428:
2427:
2383:
2381:
2380:
2375:
2363:
2361:
2360:
2355:
2353:
2352:
2326:
2324:
2323:
2318:
2290:
2287:
2275:
2250:
2249:
2230:
2228:
2227:
2222:
2208:
2207:
2188:
2186:
2185:
2180:
2143:
2141:
2140:
2135:
2091:
2089:
2088:
2083:
2067:
2065:
2064:
2059:
2046:identity functor
2009:abstract algebra
1996:comultiplication
1989:
1987:
1986:
1981:
1969:
1967:
1966:
1961:
1949:
1947:
1946:
1941:
1939:
1938:
1937:
1914:
1912:
1911:
1906:
1885:categorical dual
1874:
1872:
1871:
1866:
1864:
1863:
1854:
1853:
1837:
1835:
1834:
1829:
1783:
1781:
1780:
1775:
1734:
1733:
1714:
1712:
1711:
1706:
1685:
1683:
1682:
1677:
1659:
1657:
1656:
1651:
1628:
1627:
1612:, we have a map
1611:
1609:
1608:
1603:
1591:
1589:
1588:
1583:
1567:
1565:
1564:
1559:
1538:
1536:
1535:
1530:
1506:
1504:
1503:
1498:
1482:
1480:
1479:
1474:
1453:
1451:
1450:
1445:
1433:
1431:
1430:
1425:
1423:
1406:on the category
1405:
1403:
1402:
1397:
1395:
1394:
1365:
1363:
1362:
1357:
1345:
1343:
1342:
1337:
1335:
1334:
1329:
1311:in the category
1306:
1304:
1303:
1298:
1286:
1284:
1283:
1278:
1265:identity element
1262:
1260:
1259:
1254:
1229:
1222:
1217:
1210:
1209:
1206:
1204:
1203:
1198:
1183:
1181:
1180:
1175:
1142:
1128:
1127:
1116:
1114:
1113:
1108:
1096:
1094:
1093:
1088:
1076:
1074:
1073:
1068:
1066:
1065:
1049:
1047:
1046:
1041:
1023:
1021:
1020:
1015:
1013:
1012:
960:
958:
957:
952:
937:
935:
934:
929:
914:
912:
911:
906:
898:
897:
881:
879:
878:
873:
830:
828:
827:
822:
810:
808:
807:
802:
790:
788:
787:
782:
764:
762:
761:
756:
754:
753:
737:
735:
734:
729:
721:
720:
698:
696:
695:
690:
678:
676:
675:
670:
668:
667:
651:
649:
648:
643:
635:
634:
608:
606:
605:
600:
576:
574:
573:
568:
548:
546:
545:
540:
516:
514:
513:
508:
485:identity functor
482:
480:
479:
474:
462:
460:
459:
454:
443:is a monad. If
442:
440:
439:
434:
416:
414:
413:
408:
397:left adjoint to
396:
394:
393:
388:
375:adjoint functors
372:
370:
369:
364:
352:
350:
349:
344:
272:adjoint functors
235:
233:
232:
227:
209:
207:
206:
201:
173:
171:
170:
165:
147:
145:
144:
139:
112:consisting of a
111:
109:
108:
103:
58:
21:
8543:
8542:
8538:
8537:
8536:
8534:
8533:
8532:
8528:Category theory
8513:
8512:
8485:
8475:
8468:
8463:
8450:
8445:
8420:
8406:
8389:
8384:
8361:
8329:
8324:
8302:
8292:Godement, Roger
8290:
8284:
8270:
8267:
8265:Further reading
8262:
8261:
8225:
8224:
8220:
8212:
8187:
8186:
8182:
8178:, §§VI.3, VI.9)
8174:
8170:
8162:
8158:
8122:
8121:
8117:
8111:
8088:
8069:
8068:
8064:
8056:
8052:
8044:
8040:. p. 162.
8037:
8029:
8028:
8024:
8005:
8004:
8000:
7993:
7968:
7967:
7963:
7955:
7951:
7928:
7927:
7923:
7916:
7901:
7896:
7895:
7891:
7884:
7877:
7872:
7830:
7792:
7791:
7772:
7771:
7765:
7707:
7683:faithfully flat
7653:
7652:
7614:
7593:
7577:
7569:
7568:
7494:
7457:
7456:
7421:
7420:
7399:
7394:
7393:
7357:
7356:
7286:
7255:
7254:
7176:
7175:
7172:
7131:
7130:
7101:
7096:
7095:
7049:
7048:
7027:
7022:
7021:
7002:
7001:
6931:
6930:
6887:
6886:
6844:
6843:
6795:
6769:
6768:
6762:
6738:
6737:
6736:of commutative
6712:
6707:
6706:
6687:
6686:
6663:
6650:
6634:
6629:
6628:
6607:
6592:
6554:
6553:
6528:
6511:
6498:
6497:
6478:
6477:
6452:
6447:
6446:
6427:
6426:
6405:
6404:
6385:
6384:
6361:
6360:
6355:
6331:
6330:
6311:
6310:
6291:
6290:
6271:
6270:
6256:
6255:
6242:
6223:
6210:
6194:
6179:
6176:
6175:
6162:
6143:
6130:
6114:
6099:
6090:
6089:
6070:
6069:
6050:
6049:
6019:
6014:
6013:
5981:
5976:
5975:
5956:
5955:
5936:
5935:
5911:
5892:
5867:
5852:
5847:
5846:
5845:. For example,
5808:
5803:
5802:
5770:
5725:
5720:
5719:
5696:
5695:
5676:
5675:
5609:
5604:
5603:
5584:
5583:
5564:
5563:
5560:
5512:
5511:
5474:
5473:
5449:
5441:
5440:
5417:
5416:
5379:
5378:
5317:
5313:
5289:
5288:
5269:
5268:
5225:
5224:
5205:
5204:
5181:
5180:
5172:
5148:
5147:
5128:
5127:
5108:
5107:
5088:
5087:
5068:
5067:
5064:
5059:
5033:
5028:
5027:
5026:and denoted by
5004:
5003:
4974:
4973:
4954:
4937:
4936:
4917:
4916:
4894:
4883:
4851:
4850:
4794:
4793:
4759:
4758:
4739:
4738:
4719:
4718:
4687:
4686:
4667:
4666:
4647:
4646:
4627:
4626:
4605:
4604:
4584:
4583:
4546:
4545:
4542:
4532:
4477:
4471:
4468:
4461:needs expansion
4446:
4413:
4412:
4384:
4383:
4364:
4363:
4321:
4320:
4292:
4291:
4272:
4271:
4244:
4243:
4236:
4230:
4227:
4220:needs expansion
4205:
4171:
4160:
4149:
4148:
4099:
4094:
4093:
4068:
4057:
4001:
4000:
3960:
3959:
3910:
3905:
3904:
3876:
3875:
3856:
3855:
3824:
3823:
3804:
3803:
3780:
3779:
3772:
3766:
3763:
3756:needs expansion
3741:
3739:The state monad
3708:
3703:
3702:
3681:
3671:
3663:
3662:
3643:
3642:
3614:
3613:
3591:
3572:
3567:
3566:
3542:
3537:
3536:
3517:
3516:
3476:
3475:
3425:
3414:
3413:
3399:
3397:The maybe monad
3383:
3377:
3369:Leinster (2013)
3303:
3302:
3296:codensity monad
3292:
3259:
3258:
3221:
3220:
3197:
3196:
3168:
3167:
3145:
3144:
3125:
3124:
3050:
3049:
3007:
3006:
2987:
2986:
2949:
2948:
2906:
2905:
2886:
2885:
2854:
2853:
2834:
2833:
2814:
2813:
2790:
2789:
2748:
2747:
2744:
2734:and monads are
2706:
2705:
2684:
2683:
2664:
2663:
2632:
2631:
2625:
2592:
2587:
2586:
2534:
2529:
2528:
2494:
2432:
2419:
2408:
2407:
2390:
2366:
2365:
2344:
2339:
2338:
2241:
2236:
2235:
2199:
2194:
2193:
2156:
2155:
2108:
2107:
2098:
2074:
2073:
2050:
2049:
2042:
2037:
2021:
1972:
1971:
1952:
1951:
1925:
1920:
1919:
1897:
1896:
1881:
1843:
1842:
1808:
1807:
1797:
1725:
1720:
1719:
1715:. The function
1691:
1690:
1662:
1661:
1619:
1614:
1613:
1594:
1593:
1574:
1573:
1541:
1540:
1509:
1508:
1489:
1488:
1456:
1455:
1436:
1435:
1408:
1407:
1384:
1383:
1380:power set monad
1376:
1348:
1347:
1318:
1313:
1312:
1289:
1288:
1269:
1268:
1245:
1244:
1243:if we think of
1220:
1186:
1185:
1163:
1162:
1140:
1099:
1098:
1079:
1078:
1057:
1052:
1051:
1026:
1025:
1004:
969:
968:
961:are formed by "
940:
939:
917:
916:
889:
884:
883:
840:
839:
813:
812:
793:
792:
767:
766:
765:is the functor
745:
740:
739:
712:
701:
700:
681:
680:
659:
654:
653:
626:
615:
614:
579:
578:
559:
558:
531:
530:
527:
493:
492:
465:
464:
445:
444:
419:
418:
399:
398:
379:
378:
355:
354:
335:
334:
327:
212:
211:
180:
179:
150:
149:
124:
123:
76:
75:
64:category theory
60:
57:John Baez,
56:
42:
35:
28:
23:
22:
15:
12:
11:
5:
8541:
8539:
8531:
8530:
8525:
8515:
8514:
8511:
8510:
8505:
8499:
8492:
8484:
8483:External links
8481:
8480:
8479:
8466:
8461:
8448:
8443:
8418:
8404:
8387:
8382:
8359:
8322:
8300:
8288:
8276:Wells, Charles
8266:
8263:
8260:
8259:
8218:
8210:
8180:
8168:
8164:MacLane (1978)
8156:
8135:(2): 111–143.
8115:
8109:
8062:
8060:, p. 155.
8050:
8022:
8019:on 2015-03-26.
7998:
7991:
7961:
7959:, p. 138.
7949:
7921:
7914:
7889:
7874:
7873:
7871:
7868:
7867:
7866:
7864:Monoidal monad
7861:
7856:
7851:
7846:
7841:
7839:Lawvere theory
7836:
7829:
7826:
7812:
7809:
7806:
7802:
7799:
7779:
7764:
7763:Generalization
7761:
7706:
7703:
7666:
7663:
7660:
7649:
7648:
7637:
7634:
7629:
7624:
7621:
7618:
7613:
7608:
7603:
7600:
7597:
7592:
7589:
7584:
7580:
7576:
7556:and topics in
7550:homeomorphisms
7493:
7490:
7464:
7451:is said to be
7440:
7437:
7434:
7431:
7428:
7406:
7402:
7370:
7367:
7329:
7328:
7317:
7314:
7302:
7293:
7289:
7279:
7276:
7269:
7262:
7246:, the functor
7231:
7228:
7225:
7222:
7219:
7216:
7213:
7210:
7207:
7204:
7201:
7198:
7195:
7192:
7189:
7186:
7183:
7171:
7168:
7147:
7144:
7141:
7138:
7108:
7104:
7079:
7076:
7073:
7070:
7067:
7063:
7060:
7057:
7034:
7030:
7009:
6989:
6986:
6983:
6980:
6977:
6974:
6971:
6968:
6965:
6962:
6959:
6956:
6953:
6950:
6947:
6944:
6941:
6938:
6918:
6915:
6912:
6909:
6906:
6903:
6900:
6897:
6894:
6874:
6871:
6868:
6865:
6862:
6858:
6855:
6852:
6824:
6823:
6807:
6802:
6798:
6794:
6791:
6788:
6785:
6782:
6779:
6776:
6761:
6758:
6745:
6723:
6717:
6694:
6675:
6670:
6666:
6662:
6657:
6653:
6649:
6646:
6641:
6637:
6614:
6610:
6605:
6599:
6595:
6589:
6586:
6583:
6579:
6575:
6572:
6569:
6566:
6562:
6539:
6533:
6527:
6522:
6516:
6510:
6506:
6485:
6463:
6457:
6434:
6413:
6392:
6369:
6354:
6351:
6338:
6318:
6298:
6278:
6254:
6249:
6245:
6241:
6238:
6235:
6230:
6226:
6222:
6219:
6216:
6213:
6211:
6209:
6204:
6201:
6197:
6193:
6188:
6178:
6177:
6174:
6169:
6165:
6161:
6158:
6155:
6150:
6146:
6142:
6139:
6136:
6133:
6131:
6129:
6124:
6121:
6117:
6113:
6108:
6098:
6097:
6077:
6057:
6037:
6034:
6031:
6026:
6022:
6001:
5998:
5995:
5990:
5963:
5943:
5923:
5918:
5914:
5910:
5907:
5904:
5899:
5895:
5891:
5888:
5885:
5882:
5877:
5874:
5870:
5866:
5861:
5834:
5831:
5828:
5825:
5822:
5817:
5790:
5787:
5784:
5779:
5767:
5762:
5759:
5756:
5752:
5748:
5745:
5742:
5739:
5734:
5703:
5683:
5663:
5660:
5657:
5649:
5646:
5643:
5640:
5632:
5629:
5626:
5623:
5618:
5591:
5571:
5559:
5556:
5543:
5540:
5537:
5534:
5531:
5528:
5525:
5522:
5519:
5499:
5496:
5493:
5490:
5487:
5484:
5481:
5461:
5456:
5452:
5448:
5423:
5415:
5412:
5409:
5406:
5403:
5400:
5395:
5392:
5389:
5385:
5381:
5380:
5377:
5374:
5371:
5368:
5365:
5362:
5354:
5351:
5350:
5347:
5344:
5341:
5338:
5335:
5332:
5329:
5326:
5323:
5320:
5316:
5312:
5309:
5306:
5303:
5298:
5276:
5256:
5253:
5250:
5247:
5244:
5241:
5238:
5235:
5232:
5212:
5190:
5171:
5168:
5155:
5135:
5115:
5095:
5075:
5063:
5060:
5058:
5055:
5040:
5036:
5011:
4981:
4960:
4957:
4953:
4950:
4947:
4944:
4924:
4904:
4900:
4897:
4893:
4889:
4886:
4882:
4879:
4876:
4873:
4870:
4867:
4864:
4861:
4858:
4842:
4841:
4834:
4832:
4829:
4827:
4801:
4781:
4778:
4775:
4772:
4769:
4766:
4746:
4726:
4706:
4703:
4700:
4697:
4694:
4674:
4654:
4645:acted upon by
4634:
4612:
4591:
4582:on a category
4571:
4568:
4565:
4562:
4559:
4556:
4553:
4544:Given a monad
4531:
4528:
4479:
4478:
4458:
4456:
4445:
4442:
4426:
4423:
4420:
4397:
4394:
4391:
4371:
4351:
4348:
4345:
4342:
4339:
4335:
4332:
4329:
4305:
4302:
4299:
4279:
4270:maps each set
4251:
4238:
4237:
4217:
4215:
4204:
4201:
4193:
4192:
4181:
4177:
4174:
4170:
4166:
4163:
4159:
4156:
4146:
4135:
4132:
4129:
4126:
4123:
4120:
4117:
4114:
4111:
4106:
4102:
4078:
4074:
4071:
4067:
4063:
4060:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
4029:
4026:
4023:
4020:
4017:
4014:
4011:
4008:
3997:
3996:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3957:
3946:
3943:
3940:
3937:
3934:
3931:
3928:
3925:
3922:
3917:
3913:
3889:
3886:
3883:
3863:
3843:
3840:
3837:
3834:
3831:
3811:
3802:maps each set
3787:
3774:
3773:
3753:
3751:
3740:
3737:
3715:
3711:
3701:to the one in
3688:
3684:
3678:
3674:
3670:
3650:
3639:
3638:
3627:
3624:
3621:
3611:
3598:
3594:
3590:
3587:
3584:
3579:
3575:
3549:
3545:
3524:
3513:
3512:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3473:
3461:
3458:
3455:
3451:
3447:
3444:
3441:
3437:
3432:
3428:
3424:
3421:
3398:
3395:
3376:
3373:
3359:to the set of
3353:
3352:
3340:
3337:
3334:
3330:
3326:
3323:
3320:
3317:
3314:
3311:
3291:
3288:
3275:
3272:
3269:
3266:
3246:
3243:
3240:
3237:
3234:
3231:
3228:
3217:tensor algebra
3204:
3184:
3181:
3178:
3175:
3165:tensor algebra
3152:
3132:
3102:
3101:
3090:
3087:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3034:
3031:
3028:
3024:
3021:
3018:
3015:
2994:
2983:
2982:
2971:
2968:
2965:
2962:
2959:
2956:
2933:
2930:
2927:
2923:
2920:
2917:
2914:
2893:
2873:
2870:
2867:
2864:
2861:
2841:
2821:
2797:
2755:
2743:
2740:
2719:
2716:
2713:
2703:if and only if
2691:
2671:
2651:
2648:
2645:
2642:
2639:
2624:
2621:
2602:
2599:
2595:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2541:
2537:
2515:
2514:
2501:
2497:
2493:
2490:
2487:
2482:
2479:
2474:
2469:
2466:
2463:
2460:
2455:
2450:
2445:
2442:
2439:
2436:
2431:
2426:
2422:
2418:
2415:
2396:, for a fixed
2389:
2386:
2373:
2351:
2347:
2328:
2327:
2316:
2313:
2310:
2307:
2304:
2301:
2296:
2293:
2285:
2282:
2278:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2248:
2244:
2220:
2217:
2214:
2211:
2206:
2202:
2190:
2189:
2178:
2175:
2172:
2169:
2166:
2163:
2145:
2144:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2097:
2094:
2081:
2057:
2048:on a category
2041:
2038:
2036:
2033:
2025:Roger Godement
2020:
2017:
1979:
1959:
1936:
1933:
1928:
1904:
1880:
1877:
1862:
1857:
1852:
1827:
1824:
1821:
1818:
1815:
1796:
1793:
1785:
1784:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1732:
1728:
1704:
1701:
1698:
1675:
1672:
1669:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1626:
1622:
1601:
1581:
1557:
1554:
1551:
1548:
1528:
1525:
1522:
1519:
1516:
1496:
1472:
1469:
1466:
1463:
1443:
1422:
1419:
1416:
1393:
1375:
1372:
1355:
1333:
1328:
1325:
1322:
1296:
1276:
1252:
1231:
1230:
1223:
1218:
1196:
1193:
1173:
1170:
1153:
1152:
1143:
1138:
1119:
1118:
1106:
1086:
1064:
1060:
1039:
1036:
1033:
1011:
1007:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
966:
950:
947:
927:
924:
904:
901:
896:
892:
871:
868:
865:
862:
859:
856:
853:
850:
847:
820:
800:
780:
777:
774:
752:
748:
727:
724:
719:
715:
711:
708:
688:
666:
662:
641:
638:
633:
629:
625:
622:
598:
595:
592:
589:
586:
566:
538:
526:
523:
506:
503:
500:
472:
452:
432:
429:
426:
406:
386:
373:are a pair of
362:
342:
326:
323:
268:
267:
260:
225:
222:
219:
210:together with
199:
196:
193:
190:
187:
163:
160:
157:
137:
134:
131:
101:
98:
95:
92:
89:
86:
83:
66:, a branch of
43:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8540:
8529:
8526:
8524:
8521:
8520:
8518:
8509:
8506:
8503:
8500:
8497:
8493:
8490:
8487:
8486:
8482:
8474:
8473:
8467:
8464:
8462:9780486820804
8458:
8454:
8449:
8446:
8440:
8436:
8432:
8428:
8424:
8419:
8415:
8411:
8407:
8405:0-521-83414-7
8401:
8397:
8393:
8388:
8385:
8379:
8374:
8369:
8365:
8360:
8357:
8353:
8348:
8343:
8339:
8335:
8328:
8323:
8319:
8314:
8310:
8306:
8301:
8297:
8293:
8289:
8283:
8282:
8277:
8273:
8272:Barr, Michael
8269:
8268:
8264:
8255:
8251:
8246:
8241:
8237:
8233:
8229:
8222:
8219:
8213:
8207:
8203:
8199:
8195:
8191:
8184:
8181:
8177:
8176:MacLane (1978
8172:
8169:
8165:
8160:
8157:
8152:
8148:
8143:
8138:
8134:
8130:
8126:
8119:
8116:
8112:
8106:
8101:
8096:
8092:
8085:
8081:
8077:
8073:
8066:
8063:
8059:
8054:
8051:
8043:
8036:
8032:
8026:
8023:
8018:
8014:
8013:
8008:
8002:
7999:
7994:
7988:
7984:
7980:
7976:
7972:
7965:
7962:
7958:
7953:
7950:
7945:
7940:
7936:
7932:
7925:
7922:
7917:
7915:0-387-96115-1
7911:
7907:
7900:
7893:
7890:
7887:
7882:
7880:
7876:
7869:
7865:
7862:
7860:
7857:
7855:
7852:
7850:
7847:
7845:
7842:
7840:
7837:
7835:
7832:
7831:
7827:
7825:
7800:
7797:
7777:
7770:
7762:
7760:
7758:
7754:
7750:
7746:
7742:
7738:
7733:
7731:
7727:
7722:
7720:
7716:
7712:
7704:
7702:
7700:
7696:
7692:
7688:
7684:
7680:
7664:
7658:
7635:
7632:
7627:
7606:
7590:
7587:
7582:
7578:
7574:
7567:
7566:
7565:
7563:
7559:
7555:
7551:
7546:
7543:
7538:
7536:
7532:
7528:
7524:
7520:
7516:
7512:
7508:
7504:
7500:
7499:
7489:
7487:
7483:
7479:
7462:
7454:
7438:
7432:
7429:
7426:
7404:
7400:
7391:
7387:
7365:
7354:
7350:
7346:
7342:
7338:
7334:
7315:
7312:
7291:
7287:
7274:
7260:
7253:
7252:
7251:
7249:
7245:
7226:
7223:
7220:
7217:
7214:
7208:
7205:
7202:
7199:
7196:
7190:
7187:
7184:
7169:
7167:
7165:
7161:
7142:
7136:
7128:
7124:
7106:
7102:
7093:
7074:
7071:
7068:
7032:
7028:
7007:
6984:
6981:
6978:
6975:
6972:
6966:
6960:
6957:
6954:
6951:
6948:
6945:
6942:
6939:
6913:
6910:
6907:
6904:
6901:
6898:
6895:
6869:
6866:
6863:
6841:
6837:
6833:
6829:
6805:
6800:
6796:
6789:
6786:
6780:
6774:
6767:
6766:
6765:
6759:
6757:
6743:
6721:
6692:
6673:
6668:
6664:
6660:
6655:
6651:
6647:
6644:
6639:
6635:
6612:
6603:
6597:
6593:
6587:
6584:
6581:
6577:
6573:
6567:
6537:
6520:
6508:
6483:
6461:
6432:
6390:
6382:
6352:
6350:
6349:-generators.
6336:
6316:
6296:
6276:
6247:
6243:
6239:
6236:
6233:
6228:
6224:
6217:
6214:
6212:
6202:
6199:
6195:
6186:
6167:
6163:
6159:
6156:
6153:
6148:
6144:
6137:
6134:
6132:
6122:
6119:
6115:
6106:
6088:-algebras, so
6075:
6055:
6032:
6024:
6020:
5996:
5988:
5961:
5941:
5916:
5912:
5908:
5905:
5902:
5897:
5893:
5886:
5883:
5875:
5872:
5868:
5859:
5832:
5829:
5823:
5815:
5785:
5777:
5760:
5757:
5754:
5750:
5746:
5740:
5732:
5717:
5701:
5681:
5658:
5641:
5630:
5624:
5616:
5589:
5569:
5557:
5555:
5541:
5535:
5532:
5529:
5523:
5520:
5517:
5494:
5491:
5488:
5482:
5479:
5459:
5454:
5450:
5446:
5438:
5421:
5413:
5410:
5404:
5398:
5393:
5390:
5387:
5383:
5372:
5369:
5363:
5345:
5339:
5336:
5333:
5324:
5321:
5318:
5314:
5310:
5304:
5274:
5251:
5248:
5245:
5236:
5233:
5230:
5210:
5179:
5178:
5169:
5167:
5153:
5133:
5113:
5093:
5073:
5061:
5056:
5054:
5038:
5034:
5025:
5009:
4997:
4993:
4979:
4958:
4955:
4948:
4945:
4942:
4922:
4898:
4895:
4891:
4887:
4884:
4871:
4868:
4865:
4859:
4856:
4847:
4839:
4835:
4830:
4825:
4821:
4820:
4817:
4815:
4814:structure map
4799:
4779:
4773:
4770:
4767:
4764:
4744:
4724:
4717:is an object
4701:
4698:
4695:
4672:
4652:
4632:
4624:
4610:
4589:
4566:
4563:
4560:
4557:
4554:
4541:
4540:pseudoalgebra
4537:
4529:
4527:
4525:
4521:
4516:
4514:
4510:
4506:
4502:
4498:
4494:
4490:
4486:
4475:
4472:February 2023
4466:
4462:
4459:This section
4457:
4454:
4450:
4449:
4443:
4441:
4438:
4424:
4418:
4411:
4395:
4392:
4389:
4369:
4346:
4343:
4340:
4319:
4303:
4297:
4277:
4269:
4265:
4249:
4234:
4231:February 2023
4225:
4221:
4218:This section
4216:
4213:
4209:
4208:
4202:
4200:
4198:
4175:
4172:
4164:
4161:
4154:
4147:
4133:
4130:
4127:
4121:
4118:
4112:
4104:
4100:
4092:
4091:
4090:
4072:
4069:
4065:
4061:
4058:
4048:
4045:
4039:
4036:
4033:
4027:
4021:
4018:
4012:
4009:
4006:
3980:
3977:
3974:
3965:
3958:
3944:
3941:
3938:
3932:
3929:
3923:
3915:
3911:
3903:
3902:
3901:
3887:
3884:
3881:
3861:
3841:
3838:
3835:
3829:
3809:
3801:
3785:
3770:
3767:February 2023
3761:
3757:
3754:This section
3752:
3749:
3745:
3744:
3738:
3736:
3734:
3729:
3713:
3709:
3686:
3676:
3672:
3648:
3625:
3619:
3612:
3596:
3592:
3585:
3582:
3577:
3573:
3565:
3564:
3563:
3547:
3543:
3522:
3496:
3490:
3487:
3481:
3474:
3435:
3430:
3422:
3412:
3411:
3410:
3408:
3404:
3396:
3394:
3392:
3388:
3382:
3374:
3372:
3370:
3366:
3362:
3358:
3328:
3301:
3300:
3299:
3297:
3289:
3287:
3270:
3264:
3238:
3232:
3226:
3218:
3202:
3179:
3173:
3166:
3150:
3130:
3121:
3119:
3115:
3111:
3107:
3106:concatenation
3085:
3079:
3067:
3061:
3055:
3048:
3047:
3046:
3029:
3005:into the set
2992:
2966:
2960:
2954:
2947:
2946:
2945:
2928:
2891:
2871:
2868:
2865:
2862:
2859:
2839:
2819:
2811:
2795:
2787:
2786:
2780:
2776:
2775:
2772:the category
2769:
2753:
2741:
2739:
2737:
2733:
2717:
2714:
2711:
2704:
2689:
2669:
2646:
2643:
2640:
2630:
2622:
2620:
2618:
2600:
2597:
2593:
2585:
2581:
2562:
2559:
2556:
2550:
2547:
2544:
2539:
2535:
2527:
2523:
2520:
2499:
2491:
2485:
2480:
2477:
2472:
2448:
2429:
2424:
2416:
2406:
2405:
2404:
2402:
2399:
2395:
2387:
2385:
2371:
2349:
2345:
2337:
2333:
2314:
2311:
2308:
2305:
2302:
2299:
2294:
2291:
2283:
2280:
2276:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2251:
2246:
2242:
2234:
2233:
2232:
2218:
2215:
2212:
2204:
2200:
2176:
2173:
2170:
2167:
2164:
2161:
2154:
2153:
2152:
2150:
2131:
2128:
2125:
2119:
2116:
2113:
2106:
2105:
2104:
2103:
2095:
2093:
2079:
2071:
2055:
2047:
2039:
2034:
2032:
2030:
2026:
2018:
2016:
2014:
2010:
2006:
2005:
1999:
1997:
1993:
1977:
1957:
1926:
1918:
1902:
1894:
1890:
1886:
1878:
1876:
1855:
1839:
1822:
1816:
1813:
1806:
1802:
1794:
1792:
1790:
1768:
1762:
1750:
1744:
1738:
1735:
1730:
1726:
1718:
1717:
1716:
1699:
1689:
1673:
1670:
1667:
1644:
1638:
1632:
1629:
1624:
1620:
1599:
1579:
1571:
1570:direct images
1552:
1546:
1526:
1520:
1517:
1514:
1494:
1486:
1467:
1461:
1441:
1381:
1373:
1371:
1369:
1353:
1331:
1310:
1294:
1274:
1266:
1250:
1242:
1238:
1237:associativity
1228:
1224:
1219:
1216:
1212:
1211:
1208:
1194:
1191:
1171:
1168:
1160:
1148:
1144:
1139:
1134:
1130:
1129:
1126:
1124:
1104:
1084:
1062:
1058:
1037:
1031:
1009:
1005:
1001:
998:
995:
992:
989:
986:
983:
980:
977:
974:
967:
964:
948:
945:
925:
922:
902:
894:
890:
869:
866:
863:
860:
857:
854:
851:
848:
845:
838:
837:
836:
834:
818:
798:
778:
775:
772:
750:
746:
725:
717:
713:
709:
706:
686:
664:
660:
639:
631:
627:
623:
620:
612:
596:
590:
587:
584:
564:
556:
552:
536:
524:
522:
520:
504:
501:
498:
490:
486:
470:
450:
430:
427:
424:
404:
384:
376:
360:
340:
332:
324:
322:
320:
316:
312:
308:
303:
301:
297:
293:
289:
285:
281:
277:
273:
265:
261:
258:
257:
256:
254:
250:
246:
242:
237:
223:
220:
217:
197:
194:
191:
188:
185:
177:
174:are functors
161:
158:
155:
135:
132:
129:
122:
118:
115:
96:
93:
90:
87:
84:
73:
69:
65:
59:
53:
50:
47:
40:
33:
19:
8494:John Baez's
8471:
8452:
8426:
8391:
8363:
8337:
8333:
8308:
8304:
8295:
8280:
8235:
8231:
8221:
8193:
8183:
8171:
8159:
8132:
8128:
8118:
8090:
8075:
8071:
8065:
8053:
8031:Riehl, Emily
8025:
8017:the original
8010:
8007:"RE: Monads"
8001:
7974:
7964:
7957:MacLane 1978
7952:
7934:
7930:
7924:
7905:
7892:
7854:Strong monad
7766:
7734:
7723:
7708:
7694:
7690:
7686:
7678:
7650:
7554:topos theory
7539:
7535:coequalizers
7530:
7526:
7522:
7518:
7514:
7510:
7507:conservative
7502:
7496:
7495:
7485:
7481:
7477:
7452:
7389:
7352:
7348:
7344:
7340:
7336:
7332:
7330:
7247:
7243:
7173:
7163:
7159:
7126:
7122:
6839:
6835:
6831:
6830:to the free
6827:
6825:
6763:
6359:commutative
6356:
5561:
5176:
5175:
5173:
5065:
5023:
5001:
4848:
4845:
4813:
4603:
4543:
4523:
4517:
4512:
4508:
4504:
4492:
4488:
4484:
4482:
4469:
4465:adding to it
4460:
4439:
4267:
4263:
4242:Given a set
4241:
4228:
4224:adding to it
4219:
4194:
3998:
3799:
3778:Given a set
3777:
3764:
3760:adding to it
3755:
3730:
3640:
3514:
3406:
3402:
3400:
3384:
3364:
3361:ultrafilters
3356:
3354:
3293:
3122:
3103:
2984:
2884:takes a set
2784:
2773:
2745:
2626:
2579:
2521:
2519:vector space
2516:
2400:
2393:
2391:
2384:-algebras).
2331:
2329:
2191:
2148:
2146:
2099:
2043:
2029:Jean BĂ©nabou
2022:
2002:
2000:
1995:
1991:
1892:
1888:
1882:
1840:
1805:endofunctors
1798:
1786:
1434:: For a set
1379:
1377:
1234:
1156:
1120:
554:
528:
518:
489:equivalences
328:
318:
314:
310:
306:
304:
269:
249:endofunctors
238:
116:
74:is a triple
71:
61:
55:
51:
48:
45:
8340:: 332–370,
8238:: 275–286.
7937:: 261–276,
7737:modal logic
7560:related to
5674:sending an
5437:convex sets
4849:A morphism
4812:called the
4491:maps a set
4318:hom functor
3800:state monad
2617:Kock (1970)
2584:double dual
1382:is a monad
331:endofunctor
68:mathematics
8517:Categories
8414:1034.18001
8058:Riehl 2017
7870:References
7859:Giry monad
7769:2-category
7533:preserves
7384:yields an
6929:such that
5934:where the
5002:commutes.
4685:-algebra
4534:See also:
3407:partiality
3379:See also:
2810:free group
2334:using the
2102:adjunction
2013:coalgebras
549:denotes a
8347:1209.3606
8254:1571-0661
8151:0022-4049
8078:: 39–42,
7662:→
7612:⇄
7579:⊗
7575:−
7488:-action.
7436:→
7430::
7369:~
7301:→
7278:~
7268:→
7227:ε
7221:η
7212:→
7194:→
6985:μ
6979:η
6958:ε
6914:ε
6834:-algebra
6793:⇄
6781:−
6665:∧
6661:⋯
6652:∧
6609:Σ
6585:≥
6578:⋁
6526:→
6425:-algebra
6381:-algebras
6253:⟩
6237:…
6221:⟨
6200:⊕
6187:∙
6157:…
6120:⊕
6107:∙
6033:−
6025:∙
5997:−
5989:∙
5906:…
5884:≅
5873:⊕
5860:∙
5766:∞
5751:⨁
5733:∙
5648:→
5625:−
5617:∙
5533:−
5483:∈
5391:∈
5384:∑
5376:∞
5353:#
5328:→
5240:→
4952:→
4946::
4878:→
4860::
4846:commute.
4777:→
4768::
4623:-algebras
4567:μ
4561:η
4536:F-algebra
4524:set monad
4497:sequences
4422:↦
4393:∈
4347:−
4301:→
4158:↦
4131:×
4125:→
4101:μ
4052:↦
4037:×
4031:→
4022:×
4016:→
3969:↦
3942:×
3936:→
3912:η
3885:∈
3839:×
3833:→
3714:∗
3687:∗
3677:∗
3623:↦
3597:∗
3589:→
3574:η
3548:∗
3497:∗
3491:∪
3485:↦
3450:→
3431:∗
3423:−
3329:⊂
3215:into its
3077:→
2958:→
2869:∘
2783:category
2715:≤
2647:≤
2601:∗
2598:∗
2551:
2540:∗
2500:∗
2492:−
2454:⇄
2425:∗
2417:−
2330:In fact,
2303:∘
2292:∘
2284:∘
2270:∘
2264:∘
2258:∘
2216:∘
2210:→
2171:∘
2123:⇄
2004:comonoids
1856:∘
1817:
1760:→
1736::
1727:μ
1688:singleton
1671:∈
1636:→
1630::
1621:η
1524:→
1518::
1485:power set
1275:η
1251:μ
1192:μ
1172:μ
1035:→
996:η
993:∘
990:μ
984:η
978:∘
975:μ
946:μ
926:μ
900:→
867:μ
864:∘
861:μ
855:μ
849:∘
846:μ
776:∘
723:→
710::
707:μ
637:→
624::
621:η
594:→
588::
502:∘
428:∘
294:, and in
224:μ
218:η
195:∘
136:μ
130:η
97:μ
91:η
8294:(1958),
8278:(1999),
8042:Archived
7828:See also
7529:has and
7484:and the
7388:between
5694:-module
5126:towards
5057:Examples
4959:′
4899:′
4888:′
4176:′
4165:′
4073:′
4062:′
2277:→
2040:Identity
2035:Examples
1893:cotriple
1879:Comonads
915:); here
551:category
519:comonads
245:category
8352:Bibcode
8311:: 151,
8084:0390019
7562:descent
7542:compact
7453:monadic
4499:(i.e.,
4408:to the
3163:to its
2808:be the
2781:to the
2766:be the
2582:to its
2524:to its
1889:comonad
1795:Remarks
1483:be the
1241:monoids
1050:; here
738:(where
652:(where
377:, with
253:functor
243:in the
176:adjoint
114:functor
8489:Monads
8459:
8441:
8412:
8402:
8380:
8252:
8208:
8149:
8107:
8082:
7989:
7912:
7849:Polyad
7749:models
7685:as an
7636:forget
7525:) and
7331:i.e.,
7306:forget
6810:forget
5801:where
5718:powers
4264:reader
2779:groups
2288:counit
1992:counit
1801:monoid
1572:under
1309:monoid
699:) and
307:triple
286:, the
241:monoid
8476:(PDF)
8342:arXiv
8330:(PDF)
8285:(PDF)
8045:(PDF)
8038:(PDF)
8012:Gmane
7902:(PDF)
6627:where
6445:. If
4501:lists
3535:into
3403:maybe
2770:from
2398:field
1970:from
1789:union
791:from
555:monad
311:triad
264:group
72:monad
8457:ISBN
8439:ISBN
8400:ISBN
8378:ISBN
8250:ISSN
8206:ISBN
8147:ISSN
8105:ISBN
7987:ISBN
7910:ISBN
7755:and
7739:via
7705:Uses
7509:(or
5472:for
5370:<
5357:supp
4831:and
4538:and
4485:list
4483:The
2392:The
2100:Any
2044:The
1994:and
1891:(or
1883:The
1686:the
1539:let
1454:let
1378:The
1184:and
938:and
553:. A
463:and
353:and
317:and
70:, a
8431:doi
8410:Zbl
8368:doi
8313:doi
8240:doi
8198:doi
8137:doi
8133:144
8095:doi
7979:doi
7939:doi
7935:341
7751:of
7681:is
7347:in
7162:of
6329:in
6289:in
6102:Alt
5984:Alt
5855:Sym
5811:Sym
5773:Sym
5728:Sym
5652:Mod
5635:Mod
5612:Sym
4972:of
4915:of
4792:of
4737:of
4511:in
4487:or
4467:.
4266:or
4226:.
3762:.
3405:or
3389:of
3363:on
3257:to
3116:in
2785:Set
2777:of
2774:Grp
2682:to
2548:Hom
1814:End
1487:of
1239:in
1097:to
835:):
811:to
557:on
290:of
278:on
247:of
62:In
8519::
8437:,
8425:,
8408:.
8398:.
8376:,
8350:,
8338:28
8336:,
8332:,
8309:27
8307:,
8274:;
8248:.
8236:14
8230:.
8204:.
8145:.
8131:.
8127:.
8103:,
8087:,
8080:MR
8076:22
8074:,
8033:.
8009:.
7985:.
7933:,
7904:,
7878:^
7824:.
7759:.
7753:S4
7743:,
7732:.
7721:.
7537:.
7166:.
5053:.
4437:.
4199:.
3728:.
3562::
3371:.
2738:.
2619:.
2545::=
2201:id
2092:.
2031:.
2015:.
1125::
1117:).
965:."
613::
521:.
321:.
313:,
309:,
302:.
8433::
8416:.
8370::
8354::
8344::
8315::
8256:.
8242::
8214:.
8200::
8153:.
8139::
8097::
7995:.
7981::
7941::
7918:.
7811:t
7808:a
7805:C
7801:=
7798:C
7778:C
7695:A
7691:B
7687:A
7679:B
7665:B
7659:A
7633::
7628:B
7623:d
7620:o
7617:M
7607:A
7602:d
7599:o
7596:M
7591::
7588:B
7583:A
7531:G
7527:C
7523:C
7519:G
7515:D
7511:G
7503:G
7486:T
7482:C
7478:D
7463:F
7439:C
7433:D
7427:G
7405:T
7401:C
7390:D
7366:G
7349:D
7345:Y
7341:T
7337:Y
7335:(
7333:G
7316:,
7313:C
7292:T
7288:C
7275:G
7261:D
7248:G
7244:T
7230:)
7224:,
7218:,
7215:C
7209:D
7206::
7203:G
7200:,
7197:D
7191:C
7188::
7185:F
7182:(
7164:C
7160:x
7146:)
7143:x
7140:(
7137:T
7127:T
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7107:T
7103:C
7078:)
7075:T
7072:,
7069:C
7066:(
7062:j
7059:d
7056:A
7033:T
7029:C
7008:C
6988:)
6982:,
6976:,
6973:T
6970:(
6967:=
6964:)
6961:F
6955:G
6952:,
6949:e
6946:,
6943:F
6940:G
6937:(
6917:)
6911:,
6908:e
6905:,
6902:G
6899:,
6896:F
6893:(
6873:)
6870:T
6867:,
6864:C
6861:(
6857:j
6854:d
6851:A
6840:X
6838:(
6836:T
6832:T
6828:X
6806::
6801:T
6797:C
6790:C
6787::
6784:)
6778:(
6775:T
6744:A
6722:A
6716:C
6693:j
6674:M
6669:A
6656:A
6648:M
6645:=
6640:j
6636:M
6613:j
6604:/
6598:j
6594:M
6588:0
6582:j
6574:=
6571:)
6568:M
6565:(
6561:P
6538:A
6532:M
6521:A
6515:M
6509::
6505:P
6484:A
6462:A
6456:M
6433:A
6412:S
6391:A
6368:S
6337:n
6317:R
6297:n
6277:R
6248:n
6244:x
6240:,
6234:,
6229:1
6225:x
6218:R
6215:=
6208:)
6203:n
6196:R
6192:(
6182:T
6173:)
6168:n
6164:x
6160:,
6154:,
6149:1
6145:x
6141:(
6138:R
6135:=
6128:)
6123:n
6116:R
6112:(
6076:R
6056:R
6036:)
6030:(
6021:T
6000:)
5994:(
5962:R
5942:R
5922:]
5917:n
5913:x
5909:,
5903:,
5898:1
5894:x
5890:[
5887:R
5881:)
5876:n
5869:R
5865:(
5833:R
5830:=
5827:)
5824:M
5821:(
5816:0
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5778:k
5761:0
5758:=
5755:k
5747:=
5744:)
5741:M
5738:(
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5682:R
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5659:R
5656:(
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5642:R
5639:(
5631::
5628:)
5622:(
5602:.
5590:R
5570:R
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5498:]
5495:1
5492:,
5489:0
5486:[
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5460:y
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5447:x
5422:}
5414:1
5411:=
5408:)
5405:x
5402:(
5399:f
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5388:x
5373:+
5367:)
5364:f
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5346::
5343:]
5340:1
5337:,
5334:0
5331:[
5325:X
5322::
5319:f
5315:{
5311:=
5308:)
5305:X
5302:(
5297:D
5275:1
5255:]
5252:1
5249:,
5246:0
5243:[
5237:X
5234::
5231:f
5211:X
5189:D
5154:X
5134:X
5114:X
5094:X
5074:T
5039:T
5035:C
5010:T
4980:C
4956:x
4949:x
4943:f
4923:T
4903:)
4896:h
4892:,
4885:x
4881:(
4875:)
4872:h
4869:,
4866:x
4863:(
4857:f
4800:C
4780:x
4774:x
4771:T
4765:h
4745:C
4725:x
4705:)
4702:h
4699:,
4696:x
4693:(
4673:T
4653:T
4633:C
4611:T
4590:C
4570:)
4564:,
4558:,
4555:T
4552:(
4513:X
4509:x
4505:X
4493:X
4474:)
4470:(
4425:x
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4334:m
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4229:(
4180:)
4173:s
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4119::
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4049:s
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4019:S
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3984:)
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3966:s
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3930::
3927:)
3924:x
3921:(
3916:X
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3649:X
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3583::
3578:X
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3500:}
3494:{
3488:X
3482:X
3460:t
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3436::
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3420:(
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3333:S
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3271:V
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3180:V
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3089:)
3086:X
3083:(
3080:T
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3071:)
3068:X
3065:(
3062:T
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3033:)
3030:X
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3023:e
3020:e
3017:r
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2993:X
2970:)
2967:X
2964:(
2961:T
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2932:)
2929:X
2926:(
2922:e
2919:e
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2872:F
2866:G
2863:=
2860:T
2840:G
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2796:F
2754:G
2718:y
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2690:y
2670:x
2650:)
2644:,
2641:P
2638:(
2594:V
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2430::
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2315:.
2312:T
2309:=
2306:F
2300:G
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2252:=
2247:2
2243:T
2219:F
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2177:.
2174:F
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2165:=
2162:T
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2129::
2126:D
2120:C
2117::
2114:F
2080:C
2056:C
1978:C
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1038:T
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1010:T
1006:1
1002:=
999:T
987:=
981:T
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858:=
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189:=
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