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3064:. Composition of functors is associative where defined. Identity of composition of functors is the identity functor. This shows that functors can be considered as morphisms in categories of categories, for example in the
2370:
3075:: the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation. Functors between one-object categories correspond to monoid
1948:
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4862:. In the category of topological spaces (without distinguished point), one considers homotopy classes of generic curves, but they cannot be composed unless they share an endpoint. Thus one has the
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maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which
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There are many constructions in mathematics that would be functors but for the fact that they "turn morphisms around" and "reverse composition". We then define a
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1843:βas "covariant". This terminology originates in physics, and its rationale has to do with the position of the indices ("upstairs" and "downstairs") in
6218:
3079:. So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.
2307:
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5240:. Functions get mapped to group homomorphisms between free groups. Free constructions exist for many categories based on structured sets. See
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1913:
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2126:
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165:
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893:
454:
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1953:
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4981:, a covariant functor from the category of pointed differentiable manifolds to the category of real vector spaces. Likewise,
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3469:
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2182:βwhereas it acts "in the opposite way" on the "vector coordinates" (but "in the same way" as on the basis covectors:
1646:
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6943:
6834:
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5282:. This is a functor which is contravariant in the first and covariant in the second argument, i.e. it is a functor
5015:
3206:
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to its underlying function of sets is a functor. Functors like these, which "forget" some structure, are termed
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3088:
1528:
in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as a
989:
550:
6824:
6780:
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31:
4037:
3476:. Similar remarks apply to the colimit functor (which assigns to every functor its colimit, and is covariant).
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assigns to each functor its limit. The existence of this functor can be proved by realizing that it is the
6870:
6587:
6353:
6333:
6256:
6168:
6096:
5915:
5666:
5610:
262:
61:
2241:). This terminology is contrary to the one used in category theory because it is the covectors that have
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5044:
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615:
5975:
5713:
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2053:{\displaystyle {\boldsymbol {\omega }}'={\boldsymbol {\omega }}{\boldsymbol {\Lambda }}^{\textsf {T}}.}
1538:
2063:
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6276:
6172:
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its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed
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353:
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1844:
1704:
65:
37:"Functoriality" redirects here. For the Langlands functoriality conjecture in number theory, see
6106:
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4484:
5460:
which is contravariant in the first argument and covariant in the second, i.e. it is a functor
3866:
3735:
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5892:
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5775:
5767:
5580:. An important goal in many settings is to determine whether a given functor is representable.
5201:
5174:
4732:
4087:
3224:
2835:
2437:
1533:
1509:{\displaystyle \mathrm {Covariant} \circ \mathrm {Contravariant} \to \mathrm {Contravariant} }
1369:{\displaystyle \mathrm {Contravariant} \circ \mathrm {Contravariant} \to \mathrm {Covariant} }
73:
69:
57:
38:
6049:
6015:
5702:, the category of Haskell types) between existing types to functions between some new types.
6953:
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6749:
6635:
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4523:
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3404:
3397:, maps an object to itself and a morphism to itself. The identity functor is an endofunctor.
2913:
1816:
1811:
6948:
6913:
6661:
6227:
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6121:
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5214:
Going in the opposite direction of forgetful functors are free functors. The free functor
5098:
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4008:
3939:
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756:
314:
81:
49:
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5002:
can be considered as a category with a single object whose morphisms are the elements of
4973:
is a covariant functor from the category of differentiable manifolds to the category of
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2639:
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1734:
1567:. Some authors prefer to write all expressions covariantly. That is, instead of saying
963:
732:
524:
290:
233:
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193:
4985:
is a contravariant functor, essentially the composition of the tangent space with the
1234:{\displaystyle \mathrm {Covariant} \circ \mathrm {Covariant} \to \mathrm {Covariant} }
6978:
6933:
6688:
6520:
6397:
6323:
6193:
which generates examples of categorical constructions in the category of finite sets.
6130:
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5793:
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5637:
5193:
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4974:
4791:
3682:
112:
100:
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6343:
6184:
5633:
4597:
3076:
2942:
is a natural example; it is contravariant in one argument, covariant in the other.
1698:
6703:
4858:. We thus obtain a functor from the category of pointed topological spaces to the
6025:
6683:
6555:
6425:
6160:
5411:
5241:
5084:
2999:
2939:
2917:
2123:) acts on the "covector coordinates" "in the same way" as on the basis vectors:
108:
45:
6187:
of recorded talks relevant to categories, logic and the foundations of physics.
6114:
6735:
6673:
6286:
5732:
5727:
5709:
5674:
5233:
4970:
4966:
4891:) of all real-valued continuous functions on that space. Every continuous map
4601:
2365:{\displaystyle F^{\mathrm {op} }\colon C^{\mathrm {op} }\to D^{\mathrm {op} }}
17:
6729:
6420:
6142:
5080:
3497:
96:
6178:
4622:, i.e. topological spaces with distinguished points. The objects are pairs
2060:
In this formalism it is observed that the coordinate transformation symbol
5961:
It's not entirely clear that
Haskell datatypes truly form a category. See
6798:
6430:
6328:
6164:
4876:
4865:
4766:
3289:
3228:
689:
348:
6067:
6768:
6758:
6407:
6152:
6087:, a wiki project dedicated to the exposition of categorical mathematics
1760:
1521:
Note that contravariant functors reverse the direction of composition.
4869:
instead of the fundamental group, and this construction is functorial.
6763:
5632:
of groups or vector spaces, construction of free groups and modules,
3072:
5976:
https://wiki.haskell.org/Category_theory/Functor#Functors_in_Haskell
30:
This article is about the mathematical concept. For other uses, see
5889:
Sheaves in geometry and logic: a first introduction to topos theory
4474:{\displaystyle \{0,1\}\mapsto f(\{0,1\})=\{f(0),f(1)\}=\{\{\},X\}.}
6645:
6196:
2969:
1943:{\displaystyle \mathbf {x} '={\boldsymbol {\Lambda }}\mathbf {x} }
185:
177:
2234:{\displaystyle \mathbf {e} ^{i}=\Lambda _{j}^{i}\mathbf {e} ^{j}}
2175:{\displaystyle \mathbf {e} _{i}=\Lambda _{i}^{j}\mathbf {e} _{j}}
6855:
6071:
6014:
Simmons, Harold (2011), "Functors and natural transformations",
5962:
2889:{\displaystyle \left(F^{\mathrm {op} }\right)^{\mathrm {op} }=F}
6859:
6585:
6238:
6200:
6136:
5097:
denotes the category of vector spaces over a fixed field, with
2953:
variables. So, for example, a bifunctor is a multifunctor with
122:
3856:{\displaystyle F(X)={\mathcal {P}}(X)=\{\{\},\{0\},\{1\},X\}}
3071:
A small category with a single object is the same thing as a
953:{\displaystyle F(\mathrm {id} _{X})=\mathrm {id} _{F(X)}\,\!}
514:{\displaystyle F(\mathrm {id} _{X})=\mathrm {id} _{F(X)}\,\!}
76:, and maps between these algebraic objects are associated to
4879:(with continuous maps as morphisms) to the category of real
3794:
3595:
3547:
3367:
A functor that maps a category to that same category; e.g.,
6084:
4271:{\displaystyle \{0\}\mapsto f(\{0\})=\{f(0)\}=\{\{\}\},\ }
3243:) under inclusion. Like every partially ordered set, Open(
182:
A category with objects X, Y, Z and morphisms f, g, g β f
2512:
maps objects and morphisms in the identical way as does
2001:{\displaystyle \omega '_{i}=\Lambda _{i}^{j}\omega _{j}}
5420:
We can generalize the previous example to any category
4356:{\displaystyle \{1\}\mapsto f(\{1\})=\{f(1)\}=\{X\},\ }
1803:{\displaystyle \Gamma {\mathord {\left(T^{*}M\right)}}}
143:
6149:" An informal introduction to higher order categories.
2751:{\displaystyle G\colon C_{1}^{\mathrm {op} }\to C_{2}}
2116:{\displaystyle {\boldsymbol {\Lambda }}^{\textsf {T}}}
1697:
There is a convention which refers to "vectors"βi.e.,
95:
were borrowed by mathematicians from the philosophers
5110:
5073:
is a group, then this action is a group homomorphism.
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Universal constructions often give rise to pairs of
1690:
Contravariant functors are also occasionally called
1380:
The composite of two functors of opposite variance:
1129:
The composite of two functors of the same variance:
6748:
6712:
6660:
6653:
6604:
6513:
6455:
6406:
6361:
6352:
6249:
6133:" β by Jean-Pierre Marquis. Extensive bibliography.
5597:be categories. The collection of all functors from
138:
may be too technical for most readers to understand
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3247:) forms a small category by adding a single arrow
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6077:and the variations discussed and linked to there.
4883:is given by assigning to every topological space
4820:. This operation is compatible with the homotopy
949:
635:
510:
6115:Abstract and Concrete Categories-The Joy of Cats
4776:, with the group operation of concatenation. If
1903:{\displaystyle {x'}^{\,i}=\Lambda _{j}^{i}x^{j}}
706:Covariance and contravariance (computer science)
6137:List of academic conferences on category theory
4977:. Doing this constructions pointwise gives the
3297:. For instance, by assigning to every open set
2949:is a generalization of the functor concept to
1680:{\displaystyle F\colon C\to D^{\mathrm {op} }}
1636:{\displaystyle F\colon C^{\mathrm {op} }\to D}
1599:is a contravariant functor, they simply write
6871:
6212:
6163:. Manipulation and visualization of objects,
4875:A contravariant functor from the category of
1759:βas "contravariant" and to "covectors"βi.e.,
888:such that the following two conditions hold:
449:such that the following two conditions hold:
8:
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3745:
5846:Popescu, Nicolae; Popescu, Liliana (1979).
4824:and the composition of loops, and we get a
4566:in this example mapped to the power set of
210:must preserve the composition of morphisms
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6864:
6856:
6840:
6830:
6657:
6601:
6582:
6358:
6246:
6235:
6219:
6205:
6197:
6181:, a YouTube channel about category theory.
3472:. This requires a suitable version of the
2968:Two important consequences of the functor
1044:{\displaystyle F(g\circ f)=F(f)\circ F(g)}
605:{\displaystyle F(g\circ f)=F(g)\circ F(f)}
6100:
5774:, New York: Springer-Verlag, p. 30,
5669:. For instance, the programming language
5109:
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3612:{\displaystyle f(U)\in {\mathcal {P}}(Y)}
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166:Learn how and when to remove this message
150:, without removing the technical details.
5949:
5833:
5809:
5800:, Routledge & Kegan, pp. 13β14.
5772:Categories for the Working Mathematician
4077:{\displaystyle \{\}\mapsto f(\{\})=\{\}}
3309:, one obtains a presheaf of algebras on
3046:then one can form the composite functor
2827:{\displaystyle G^{\mathrm {op} }\circ F}
2789:{\displaystyle G\circ F^{\mathrm {op} }}
2259:Covariance and contravariance of vectors
5821:
5759:
4586:, that need not be the case in general.
4546:. Also note that although the function
3965:is the function which sends any subset
3419:to the constant functor at that object.
3305:of real-valued continuous functions on
2834:. Note that, following the property of
2029:
2017:
877:{\displaystyle F(f)\colon F(Y)\to F(X)}
438:{\displaystyle F(f)\colon F(X)\to F(Y)}
68:, where algebraic objects (such as the
5585:Relation to other categorical concepts
4986:
3564:{\displaystyle U\in {\mathcal {P}}(X)}
3201:, the category of sets and functions,
2695:{\displaystyle F\colon C_{0}\to C_{1}}
39:Langlands program Β§ Functoriality
5999:, vol. 2 (2nd ed.), Dover,
5605:forms the objects of a category: the
5262:one can assign the abelian group Hom(
5144:which is covariant in both arguments.
3725:{\displaystyle f^{-1}(V)\subseteq X.}
3450:is complete), then the limit functor
3216:Presheaves (over a topological space)
148:make it understandable to non-experts
64:. Functors were first considered in
7:
6113:J. Adamek, H. Herrlich, G. Stecker,
5061:can be considered as an "action" of
1763:, elements of the space of sections
1701:, elements of the space of sections
6127:Stanford Encyclopedia of Philosophy
5852:. Dordrecht: Springer. p. 12.
4133:, so this could also be written as
6017:An Introduction to Category Theory
5079:Assigning to every real (complex)
3022:One can compose functors, i.e. if
2874:
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2068:
1974:
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1524:Ordinary functors are also called
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1312:
1309:
1306:
1303:
1300:
1297:
1294:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
929:
926:
908:
905:
639:{\displaystyle f\colon X\to Y\,\!}
490:
487:
469:
466:
25:
5644:generalize several of the above.
5609:. Morphisms in this category are
5576:. Functors like these are called
5178:. Another example is the functor
4182:{\displaystyle (F(f))(\{\})=\{\}}
3283:. Contravariant functors on Open(
2934:. It can be seen as a functor in
2912:) is a functor whose domain is a
2629:{\displaystyle F^{\mathrm {op} }}
2584:as a category, and similarly for
2557:{\displaystyle C^{\mathrm {op} }}
2505:{\displaystyle F^{\mathrm {op} }}
2429:{\displaystyle D^{\mathrm {op} }}
2397:{\displaystyle C^{\mathrm {op} }}
2249:, whereas vectors in general are
1560:{\displaystyle C^{\mathrm {op} }}
103:, respectively. The latter used
6839:
6829:
6820:
6819:
6572:
5712:
5661:Functor (functional programming)
2221:
2191:
2162:
2132:
2101:
2085:{\displaystyle \Lambda _{i}^{j}}
2035:
1936:
1931:
1919:
1125:Variance of functor (composite)
688:That is, functors must preserve
127:
5026:-set. Likewise, a functor from
4872:Algebra of continuous functions
4596:The map which assigns to every
3621:contravariant power set functor
3407:is defined as the functor from
3197:.In the special case when J is
3145:(Category theoretical) presheaf
5916:Gubareni, Nadezhda MikhaΔlovna
5798:The Logical Syntax of Language
5616:Functors are often defined by
5348:, then the group homomorphism
5304:with group homomorphisms). If
5208:(abelian group homomorphisms).
4509:
4491:
4441:
4435:
4426:
4420:
4408:
4390:
4384:
4329:
4323:
4311:
4299:
4293:
4241:
4235:
4223:
4211:
4205:
4167:
4158:
4155:
4152:
4146:
4140:
4120:
4114:
4091:
4062:
4053:
4047:
4021:
4015:
3952:
3946:
3917:
3911:
3879:
3873:
3805:
3799:
3786:
3780:
3710:
3704:
3648:{\displaystyle f\colon X\to Y}
3639:
3606:
3600:
3587:
3581:
3558:
3552:
3525:{\displaystyle f\colon X\to Y}
3516:
3190:{\displaystyle D\colon C\to J}
3181:
3134:{\displaystyle D\colon J\to C}
3125:
2987:into a commutative diagram in
2735:
2679:
2656:. For example, when composing
2344:
2294:{\displaystyle F\colon C\to D}
2285:
1720:
1711:
1659:
1627:
1592:{\displaystyle F\colon C\to D}
1583:
1462:
1334:
1199:
1108:{\displaystyle g\colon Y\to Z}
1099:
1076:{\displaystyle f\colon X\to Y}
1067:
1038:
1032:
1023:
1017:
1008:
996:
943:
937:
918:
900:
871:
865:
859:
856:
850:
841:
835:
814:{\displaystyle f\colon X\to Y}
805:
769:
763:
671:{\displaystyle g\colon Y\to Z}
662:
628:
599:
593:
584:
578:
569:
557:
504:
498:
479:
461:
432:
426:
420:
417:
411:
402:
396:
375:{\displaystyle f\colon X\to Y}
366:
327:
321:
1:
5963:https://wiki.haskell.org/Hask
5665:Functors sometimes appear in
5065:on an object in the category
4992:Group actions/representations
4954:Tangent and cotangent bundles
4720:. To every topological space
3470:Freyd adjoint functor theorem
3349:. Such a functor is called a
700:Covariance and contravariance
6026:10.1017/CBO9780511863226.004
5456:. This defines a functor to
5168:to its underlying set and a
5022:on a particular set, i.e. a
3674:{\displaystyle V\subseteq Y}
3619:. One can also consider the
3345:to the identity morphism on
3276:{\displaystyle U\subseteq V}
3066:category of small categories
2900:Bifunctors and multifunctors
6514:Constructions on categories
6055:Encyclopedia of Mathematics
5928:Algebras, rings and modules
5192:to its underlying additive
4641:is a topological space and
4522:consequently generates the
4034:, which in this case means
3415:which sends each object in
3329:which maps every object of
3165:is a contravariant functor
1726:{\displaystyle \Gamma (TM)}
7001:
6621:Higher-dimensional algebra
5658:
5302:category of abelian groups
5123:{\displaystyle V\otimes W}
4957:The map which sends every
4769:classes of loops based at
4620:pointed topological spaces
4515:{\displaystyle f(\{0,1\})}
4104:denotes the mapping under
703:
36:
29:
27:Mapping between categories
6894:
6815:
6594:
6581:
6570:
6245:
6234:
5032:category of vector spaces
4724:with distinguished point
4618:Consider the category of
3894:{\displaystyle f(0)=\{\}}
3763:{\displaystyle X=\{0,1\}}
2092:(representing the matrix
1687:) and call it a functor.
789:associates each morphism
6167:, categories, functors,
5655:Computer implementations
5640:limits. The concepts of
5051:. In general, a functor
4097:{\displaystyle \mapsto }
3411:to the functor category
2758:, one should use either
2245:in general and are thus
32:Functor (disambiguation)
6431:Cokernels and quotients
6354:Universal constructions
6169:natural transformations
6091:Hillman, Chris (2001).
5924:Kirichenko, Vladimir V.
5611:natural transformations
5436:one can assign the set
5101:as morphisms, then the
4959:differentiable manifold
4189:. For the other values,
3655:to the map which sends
3532:to the map which sends
3109:is a covariant functor
3014:) is an isomorphism in
2564:does not coincide with
729:associates each object
6929:Essentially surjective
6588:Higher category theory
6334:Natural transformation
6093:"A Categorical Primer"
5667:functional programming
5578:representable functors
5417:Representable functors
5204:) become morphisms in
5124:
5014:is then nothing but a
4580:
4560:
4540:
4516:
4475:
4357:
4272:
4183:
4127:
4098:
4078:
4028:
3999:
3979:
3959:
3930:
3929:{\displaystyle f(1)=X}
3895:
3857:
3764:
3726:
3675:
3649:
3613:
3565:
3526:
3482:The power set functor
3341:and every morphism in
3277:
3191:
3135:
2890:
2828:
2790:
2752:
2696:
2650:
2636:is distinguished from
2630:
2598:
2578:
2558:
2526:
2506:
2474:
2454:
2430:
2398:
2366:
2295:
2235:
2176:
2117:
2086:
2054:
2002:
1944:
1904:
1837:
1836:{\displaystyle T^{*}M}
1804:
1753:
1727:
1681:
1637:
1593:
1561:
1510:
1370:
1235:
1109:
1077:
1045:
974:
954:
878:
815:
776:
743:
672:
640:
606:
535:
515:
439:
376:
334:
301:
250:
244:
224:
204:
183:
5978:for more information.
5125:
5045:linear representation
4805:can be composed with
4794:, then every loop in
4731:, one can define the
4581:
4561:
4541:
4517:
4476:
4358:
4273:
4184:
4128:
4099:
4079:
4029:
4000:
3980:
3960:
3931:
3896:
3858:
3765:
3727:
3676:
3650:
3614:
3566:
3527:
3496:maps each set to its
3278:
3237:partially ordered set
3192:
3136:
2891:
2829:
2791:
2753:
2697:
2651:
2631:
2599:
2579:
2559:
2527:
2507:
2475:
2455:
2431:
2399:
2367:
2296:
2236:
2177:
2118:
2087:
2055:
2003:
1945:
1905:
1838:
1805:
1754:
1728:
1682:
1638:
1594:
1562:
1511:
1371:
1236:
1110:
1078:
1046:
975:
955:
879:
816:
777:
744:
712:contravariant functor
673:
641:
607:
536:
516:
440:
377:
335:
302:
245:
225:
205:
189:
181:
6457:Algebraic categories
6191:Interactive Web page
6173:universal properties
5849:Theory of categories
5618:universal properties
5270:) consisting of all
5108:
4907:algebra homomorphism
4881:associative algebras
4822:equivalence relation
4570:
4550:
4530:
4485:
4366:
4281:
4193:
4137:
4126:{\displaystyle F(f)}
4108:
4088:
4038:
4027:{\displaystyle f(U)}
4009:
3989:
3969:
3958:{\displaystyle F(f)}
3940:
3905:
3867:
3774:
3736:
3688:
3659:
3627:
3575:
3536:
3504:
3261:
3169:
3113:
3101:, a diagram of type
2842:
2800:
2762:
2706:
2660:
2640:
2608:
2588:
2568:
2536:
2516:
2484:
2464:
2444:
2408:
2376:
2308:
2273:
2186:
2127:
2096:
2064:
2012:
1954:
1914:
1851:
1817:
1767:
1740:
1705:
1647:
1603:
1571:
1539:
1386:
1246:
1135:
1087:
1055:
990:
964:
894:
829:
793:
775:{\displaystyle F(X)}
757:
733:
650:
616:
551:
525:
455:
390:
354:
333:{\displaystyle F(X)}
315:
291:
234:
214:
194:
72:) are associated to
6626:Homotopy hypothesis
6304:Commutative diagram
6020:, pp. 72β107,
5912:Hazewinkel, Michiel
5620:; examples are the
5272:group homomorphisms
5247:Homomorphism groups
5083:its real (complex)
4809:to yield a loop in
3432:, if every functor
3303:associative algebra
2981:commutative diagram
2916:. For example, the
2734:
2438:opposite categories
2218:
2159:
2081:
1987:
1969:
1889:
6339:Universal property
6141:Baez, John, 1996,"
6120:2015-04-21 at the
5881:Mac Lane, Saunders
5812:, p. 19, def. 1.2.
5768:Mac Lane, Saunders
5720:Mathematics portal
5688:polytypic function
5613:between functors.
5448:of morphisms from
5202:ring homomorphisms
5175:forgetful functors
5170:group homomorphism
5147:Forgetful functors
5130:defines a functor
5120:
5087:defines a functor.
4877:topological spaces
4860:category of groups
4826:group homomorphism
4652:. A morphism from
4576:
4556:
4536:
4512:
4471:
4353:
4268:
4179:
4123:
4094:
4074:
4024:
3995:
3975:
3955:
3926:
3891:
3853:
3760:
3722:
3671:
3645:
3609:
3561:
3522:
3500:and each function
3479:Power sets functor
3369:polynomial functor
3333:to a fixed object
3273:
3187:
3131:
3038:is a functor from
3026:is a functor from
2886:
2824:
2786:
2748:
2715:
2692:
2646:
2626:
2594:
2574:
2554:
2522:
2502:
2470:
2450:
2426:
2394:
2362:
2291:
2253:since they can be
2231:
2204:
2172:
2145:
2113:
2082:
2067:
2050:
1998:
1973:
1957:
1940:
1900:
1875:
1833:
1800:
1752:{\displaystyle TM}
1749:
1723:
1677:
1633:
1589:
1557:
1526:covariant functors
1506:
1366:
1231:
1105:
1073:
1051:for all morphisms
1041:
970:
950:
874:
811:
772:
739:
725:as a mapping that
690:identity morphisms
668:
636:
612:for all morphisms
602:
531:
511:
435:
372:
330:
297:
280:is a mapping that
251:
240:
220:
200:
184:
74:topological spaces
66:algebraic topology
6972:
6971:
6944:Full and faithful
6853:
6852:
6811:
6810:
6807:
6806:
6789:monoidal category
6744:
6743:
6616:Enriched category
6568:
6567:
6564:
6563:
6541:Quotient category
6536:Opposite category
6451:
6450:
6035:978-1-107-01087-1
6006:978-0-486-47187-7
5965:for more details.
5937:978-1-4020-2690-4
5898:978-0-387-97710-2
5836:, pp. 19β20.
5824:, Exercise 3.1.4.
5781:978-3-540-90035-1
5642:limit and colimit
5514:are morphisms in
5344:are morphisms in
5006:. A functor from
4790:is a morphism of
4733:fundamental group
4615:Fundamental group
4591:Dual vector space
4579:{\displaystyle X}
4559:{\displaystyle f}
4539:{\displaystyle X}
4352:
4267:
3998:{\displaystyle X}
3978:{\displaystyle U}
3468:and invoking the
3446:(for instance if
3225:topological space
2908:(also known as a
2836:opposite category
2649:{\displaystyle F}
2597:{\displaystyle D}
2577:{\displaystyle C}
2525:{\displaystyle F}
2480:. By definition,
2473:{\displaystyle D}
2453:{\displaystyle C}
2109:
2043:
1534:opposite category
973:{\displaystyle X}
960:for every object
742:{\displaystyle X}
534:{\displaystyle X}
521:for every object
300:{\displaystyle X}
243:{\displaystyle f}
223:{\displaystyle g}
203:{\displaystyle F}
176:
175:
168:
70:fundamental group
16:(Redirected from
6992:
6880:
6873:
6866:
6857:
6843:
6842:
6833:
6832:
6823:
6822:
6658:
6636:Simplex category
6611:Categorification
6602:
6583:
6576:
6546:Product category
6531:Kleisli category
6526:Functor category
6371:Terminal objects
6359:
6294:Adjoint functors
6247:
6236:
6221:
6214:
6207:
6198:
6110:
6105:. Archived from
6104:
6063:
6038:
6009:
5993:Jacobson, Nathan
5979:
5972:
5966:
5959:
5953:
5947:
5941:
5940:
5920:Gubareni, Nadiya
5908:
5902:
5901:
5877:
5871:
5870:
5868:
5866:
5843:
5837:
5831:
5825:
5819:
5813:
5807:
5801:
5791:
5785:
5784:
5764:
5738:Functor category
5722:
5717:
5716:
5684:
5679:
5649:adjoint functors
5607:functor category
5575:
5557:
5513:
5493:
5473:
5447:
5424:. To every pair
5409:
5391:
5359:
5343:
5323:
5295:
5228:sends every set
5227:
5187:
5163:
5143:
5129:
5127:
5126:
5121:
5060:
4942:
4923:
4904:
4857:
4842:
4813:with base point
4798:with base point
4789:
4760:
4719:
4699:
4681:
4666:
4636:
4593:
4592:
4585:
4583:
4582:
4577:
4565:
4563:
4562:
4557:
4545:
4543:
4542:
4537:
4524:trivial topology
4521:
4519:
4518:
4513:
4480:
4478:
4477:
4472:
4362:
4360:
4359:
4354:
4350:
4277:
4275:
4274:
4269:
4265:
4188:
4186:
4185:
4180:
4132:
4130:
4129:
4124:
4103:
4101:
4100:
4095:
4083:
4081:
4080:
4075:
4033:
4031:
4030:
4025:
4004:
4002:
4001:
3996:
3984:
3982:
3981:
3976:
3964:
3962:
3961:
3956:
3935:
3933:
3932:
3927:
3900:
3898:
3897:
3892:
3862:
3860:
3859:
3854:
3798:
3797:
3769:
3767:
3766:
3761:
3732:For example, if
3731:
3729:
3728:
3723:
3703:
3702:
3680:
3678:
3677:
3672:
3654:
3652:
3651:
3646:
3618:
3616:
3615:
3610:
3599:
3598:
3570:
3568:
3567:
3562:
3551:
3550:
3531:
3529:
3528:
3523:
3495:
3466:diagonal functor
3459:
3441:
3405:diagonal functor
3400:Diagonal functor
3377:Identity functor
3328:
3316:Constant functor
3282:
3280:
3279:
3274:
3256:
3196:
3194:
3193:
3188:
3140:
3138:
3137:
3132:
3055:
2979:transforms each
2959:
2933:
2914:product category
2895:
2893:
2892:
2887:
2879:
2878:
2877:
2868:
2864:
2863:
2862:
2833:
2831:
2830:
2825:
2817:
2816:
2815:
2795:
2793:
2792:
2787:
2785:
2784:
2783:
2757:
2755:
2754:
2749:
2747:
2746:
2733:
2732:
2723:
2701:
2699:
2698:
2693:
2691:
2690:
2678:
2677:
2655:
2653:
2652:
2647:
2635:
2633:
2632:
2627:
2625:
2624:
2623:
2603:
2601:
2600:
2595:
2583:
2581:
2580:
2575:
2563:
2561:
2560:
2555:
2553:
2552:
2551:
2531:
2529:
2528:
2523:
2511:
2509:
2508:
2503:
2501:
2500:
2499:
2479:
2477:
2476:
2471:
2459:
2457:
2456:
2451:
2435:
2433:
2432:
2427:
2425:
2424:
2423:
2403:
2401:
2400:
2395:
2393:
2392:
2391:
2371:
2369:
2368:
2363:
2361:
2360:
2359:
2343:
2342:
2341:
2325:
2324:
2323:
2303:opposite functor
2300:
2298:
2297:
2292:
2265:Opposite functor
2240:
2238:
2237:
2232:
2230:
2229:
2224:
2217:
2212:
2200:
2199:
2194:
2181:
2179:
2178:
2173:
2171:
2170:
2165:
2158:
2153:
2141:
2140:
2135:
2122:
2120:
2119:
2114:
2112:
2111:
2110:
2104:
2091:
2089:
2088:
2083:
2080:
2075:
2059:
2057:
2056:
2051:
2046:
2045:
2044:
2038:
2032:
2024:
2020:
2007:
2005:
2004:
1999:
1997:
1996:
1986:
1981:
1965:
1949:
1947:
1946:
1941:
1939:
1934:
1926:
1922:
1909:
1907:
1906:
1901:
1899:
1898:
1888:
1883:
1871:
1870:
1864:
1863:
1842:
1840:
1839:
1834:
1829:
1828:
1812:cotangent bundle
1809:
1807:
1806:
1801:
1799:
1798:
1797:
1793:
1789:
1788:
1758:
1756:
1755:
1750:
1732:
1730:
1729:
1724:
1686:
1684:
1683:
1678:
1676:
1675:
1674:
1642:
1640:
1639:
1634:
1626:
1625:
1624:
1598:
1596:
1595:
1590:
1566:
1564:
1563:
1558:
1556:
1555:
1554:
1515:
1513:
1512:
1507:
1505:
1461:
1417:
1375:
1373:
1372:
1367:
1365:
1333:
1289:
1240:
1238:
1237:
1232:
1230:
1198:
1166:
1114:
1112:
1111:
1106:
1082:
1080:
1079:
1074:
1050:
1048:
1047:
1042:
979:
977:
976:
971:
959:
957:
956:
951:
947:
946:
932:
917:
916:
911:
883:
881:
880:
875:
825:with a morphism
820:
818:
817:
812:
781:
779:
778:
773:
748:
746:
745:
740:
677:
675:
674:
669:
645:
643:
642:
637:
611:
609:
608:
603:
540:
538:
537:
532:
520:
518:
517:
512:
508:
507:
493:
478:
477:
472:
444:
442:
441:
436:
381:
379:
378:
373:
347:associates each
339:
337:
336:
331:
306:
304:
303:
298:
284:associates each
249:
247:
246:
241:
229:
227:
226:
221:
209:
207:
206:
201:
171:
164:
160:
157:
151:
131:
130:
123:
21:
7000:
6999:
6995:
6994:
6993:
6991:
6990:
6989:
6975:
6974:
6973:
6968:
6890:
6884:
6854:
6849:
6803:
6773:
6740:
6717:
6708:
6665:
6649:
6600:
6590:
6577:
6560:
6509:
6447:
6416:Initial objects
6402:
6348:
6241:
6230:
6228:Category theory
6225:
6157:category theory
6131:Category Theory
6122:Wayback Machine
6090:
6048:
6045:
6036:
6013:
6007:
5991:
5988:
5983:
5982:
5973:
5969:
5960:
5956:
5952:, p. 20, ex. 2.
5950:Jacobson (2009)
5948:
5944:
5938:
5910:
5909:
5905:
5899:
5879:
5878:
5874:
5864:
5862:
5860:
5845:
5844:
5840:
5834:Jacobson (2009)
5832:
5828:
5820:
5816:
5810:Jacobson (2009)
5808:
5804:
5792:
5788:
5782:
5766:
5765:
5761:
5756:
5718:
5711:
5708:
5682:
5677:
5663:
5657:
5587:
5559:
5555:
5548:
5541:
5534:
5519:
5518:, then the map
5512:
5505:
5495:
5492:
5485:
5475:
5461:
5437:
5393:
5389:
5382:
5375:
5368:
5361:
5349:
5342:
5335:
5325:
5322:
5315:
5305:
5283:
5215:
5196:. Morphisms in
5179:
5151:
5131:
5106:
5105:
5090:Tensor products
5052:
5042:
4983:cotangent space
4925:
4909:
4892:
4855:
4844:
4840:
4829:
4819:
4804:
4777:
4775:
4758:
4747:
4743:
4741:
4730:
4718:
4711:
4701:
4687:
4679:
4668:
4664:
4653:
4647:
4634:
4623:
4590:
4589:
4568:
4567:
4548:
4547:
4528:
4527:
4483:
4482:
4364:
4363:
4279:
4278:
4191:
4190:
4135:
4134:
4106:
4105:
4086:
4085:
4036:
4035:
4007:
4006:
3987:
3986:
3967:
3966:
3938:
3937:
3903:
3902:
3865:
3864:
3772:
3771:
3734:
3733:
3691:
3686:
3685:
3657:
3656:
3625:
3624:
3573:
3572:
3534:
3533:
3502:
3501:
3483:
3474:axiom of choice
3451:
3433:
3396:
3390:
3378:
3364:
3320:
3259:
3258:
3257:if and only if
3248:
3167:
3166:
3149:For categories
3111:
3110:
3093:For categories
3085:
3047:
2966:
2954:
2938:arguments. The
2921:
2920:is of the type
2902:
2850:
2846:
2845:
2840:
2839:
2803:
2798:
2797:
2771:
2760:
2759:
2738:
2704:
2703:
2682:
2669:
2658:
2657:
2638:
2637:
2611:
2606:
2605:
2586:
2585:
2566:
2565:
2539:
2534:
2533:
2514:
2513:
2487:
2482:
2481:
2462:
2461:
2442:
2441:
2411:
2406:
2405:
2379:
2374:
2373:
2347:
2329:
2311:
2306:
2305:
2271:
2270:
2267:
2219:
2189:
2184:
2183:
2160:
2130:
2125:
2124:
2099:
2094:
2093:
2062:
2061:
2033:
2015:
2010:
2009:
1988:
1952:
1951:
1917:
1912:
1911:
1890:
1856:
1854:
1849:
1848:
1820:
1815:
1814:
1780:
1779:
1775:
1765:
1764:
1738:
1737:
1703:
1702:
1662:
1645:
1644:
1612:
1601:
1600:
1569:
1568:
1542:
1537:
1536:
1532:functor on the
1384:
1383:
1244:
1243:
1133:
1132:
1085:
1084:
1053:
1052:
988:
987:
962:
961:
924:
903:
892:
891:
827:
826:
791:
790:
755:
754:
753:with an object
731:
730:
708:
702:
648:
647:
614:
613:
549:
548:
523:
522:
485:
464:
453:
452:
388:
387:
352:
351:
313:
312:
289:
288:
232:
231:
212:
211:
192:
191:
172:
161:
155:
152:
144:help improve it
141:
132:
128:
121:
82:category theory
50:category theory
48:, specifically
42:
35:
28:
23:
22:
15:
12:
11:
5:
6998:
6996:
6988:
6987:
6977:
6976:
6970:
6969:
6967:
6966:
6961:
6956:
6951:
6946:
6941:
6936:
6931:
6926:
6921:
6916:
6911:
6906:
6901:
6895:
6892:
6891:
6885:
6883:
6882:
6875:
6868:
6860:
6851:
6850:
6848:
6847:
6837:
6827:
6816:
6813:
6812:
6809:
6808:
6805:
6804:
6802:
6801:
6796:
6791:
6777:
6771:
6766:
6761:
6755:
6753:
6746:
6745:
6742:
6741:
6739:
6738:
6733:
6722:
6720:
6715:
6710:
6709:
6707:
6706:
6701:
6696:
6691:
6686:
6681:
6670:
6668:
6663:
6655:
6651:
6650:
6648:
6643:
6641:String diagram
6638:
6633:
6631:Model category
6628:
6623:
6618:
6613:
6608:
6606:
6599:
6598:
6595:
6592:
6591:
6586:
6579:
6578:
6571:
6569:
6566:
6565:
6562:
6561:
6559:
6558:
6553:
6551:Comma category
6548:
6543:
6538:
6533:
6528:
6523:
6517:
6515:
6511:
6510:
6508:
6507:
6497:
6487:
6485:Abelian groups
6482:
6477:
6472:
6467:
6461:
6459:
6453:
6452:
6449:
6448:
6446:
6445:
6440:
6435:
6434:
6433:
6423:
6418:
6412:
6410:
6404:
6403:
6401:
6400:
6395:
6390:
6389:
6388:
6378:
6373:
6367:
6365:
6356:
6350:
6349:
6347:
6346:
6341:
6336:
6331:
6326:
6321:
6316:
6311:
6306:
6301:
6296:
6291:
6290:
6289:
6284:
6279:
6274:
6269:
6264:
6253:
6251:
6243:
6242:
6239:
6232:
6231:
6226:
6224:
6223:
6216:
6209:
6201:
6195:
6194:
6188:
6182:
6176:
6150:
6139:
6134:
6124:
6111:
6109:on 1997-05-03.
6102:10.1.1.24.3264
6088:
6078:
6064:
6044:
6043:External links
6041:
6040:
6039:
6034:
6011:
6005:
5987:
5984:
5981:
5980:
5967:
5954:
5942:
5936:
5903:
5897:
5885:Moerdijk, Ieke
5872:
5858:
5838:
5826:
5822:Simmons (2011)
5814:
5802:
5794:Carnap, Rudolf
5786:
5780:
5758:
5757:
5755:
5752:
5751:
5750:
5745:
5740:
5735:
5730:
5724:
5723:
5707:
5704:
5659:Main article:
5656:
5653:
5630:direct product
5622:tensor product
5586:
5583:
5582:
5581:
5553:
5546:
5539:
5532:
5510:
5503:
5490:
5483:
5432:of objects in
5418:
5415:
5387:
5380:
5373:
5366:
5340:
5333:
5320:
5313:
5260:abelian groups
5250:To every pair
5248:
5245:
5212:
5209:
5148:
5145:
5119:
5116:
5113:
5103:tensor product
5091:
5088:
5077:
5074:
5038:
4993:
4990:
4975:vector bundles
4963:tangent bundle
4955:
4952:
4887:the algebra C(
4873:
4870:
4853:
4838:
4817:
4802:
4792:pointed spaces
4773:
4761:. This is the
4756:
4745:
4739:
4728:
4716:
4709:
4682:is given by a
4677:
4662:
4648:is a point in
4645:
4632:
4616:
4613:
4594:
4587:
4575:
4555:
4535:
4511:
4508:
4505:
4502:
4499:
4496:
4493:
4490:
4470:
4467:
4464:
4461:
4458:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4413:
4410:
4407:
4404:
4401:
4398:
4395:
4392:
4389:
4386:
4383:
4380:
4377:
4374:
4371:
4349:
4346:
4343:
4340:
4337:
4334:
4331:
4328:
4325:
4322:
4319:
4316:
4313:
4310:
4307:
4304:
4301:
4298:
4295:
4292:
4289:
4286:
4264:
4261:
4258:
4255:
4252:
4249:
4246:
4243:
4240:
4237:
4234:
4231:
4228:
4225:
4222:
4219:
4216:
4213:
4210:
4207:
4204:
4201:
4198:
4178:
4175:
4172:
4169:
4166:
4163:
4160:
4157:
4154:
4151:
4148:
4145:
4142:
4122:
4119:
4116:
4113:
4093:
4073:
4070:
4067:
4064:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4023:
4020:
4017:
4014:
3994:
3974:
3954:
3951:
3948:
3945:
3925:
3922:
3919:
3916:
3913:
3910:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3852:
3849:
3846:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3796:
3791:
3788:
3785:
3782:
3779:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3721:
3718:
3715:
3712:
3709:
3706:
3701:
3698:
3694:
3670:
3667:
3664:
3644:
3641:
3638:
3635:
3632:
3608:
3605:
3602:
3597:
3592:
3589:
3586:
3583:
3580:
3560:
3557:
3554:
3549:
3544:
3541:
3521:
3518:
3515:
3512:
3509:
3480:
3477:
3427:index category
3423:
3420:
3401:
3398:
3392:
3386:
3379:
3376:
3374:
3372:
3365:
3362:
3360:
3358:
3317:
3314:
3272:
3269:
3266:
3217:
3214:
3186:
3183:
3180:
3177:
3174:
3147:
3142:
3130:
3127:
3124:
3121:
3118:
3091:
3084:
3081:
3020:
3019:
2992:
2965:
2962:
2910:binary functor
2901:
2898:
2885:
2882:
2876:
2873:
2867:
2861:
2858:
2853:
2849:
2823:
2820:
2814:
2811:
2806:
2782:
2779:
2774:
2770:
2767:
2745:
2741:
2737:
2731:
2728:
2722:
2718:
2714:
2711:
2689:
2685:
2681:
2676:
2672:
2668:
2665:
2645:
2622:
2619:
2614:
2593:
2573:
2550:
2547:
2542:
2521:
2498:
2495:
2490:
2469:
2449:
2422:
2419:
2414:
2390:
2387:
2382:
2358:
2355:
2350:
2346:
2340:
2337:
2332:
2328:
2322:
2319:
2314:
2290:
2287:
2284:
2281:
2278:
2269:Every functor
2266:
2263:
2255:pushed forward
2228:
2223:
2216:
2211:
2207:
2203:
2198:
2193:
2169:
2164:
2157:
2152:
2148:
2144:
2139:
2134:
2103:
2079:
2074:
2070:
2049:
2037:
2031:
2027:
2023:
2019:
1995:
1991:
1985:
1980:
1976:
1972:
1968:
1964:
1960:
1938:
1933:
1929:
1925:
1921:
1897:
1893:
1887:
1882:
1878:
1874:
1869:
1862:
1859:
1832:
1827:
1823:
1796:
1792:
1787:
1783:
1778:
1772:
1748:
1745:
1735:tangent bundle
1722:
1719:
1716:
1713:
1710:
1673:
1670:
1665:
1661:
1658:
1655:
1652:
1643:(or sometimes
1632:
1629:
1623:
1620:
1615:
1611:
1608:
1588:
1585:
1582:
1579:
1576:
1553:
1550:
1545:
1519:
1518:
1517:
1516:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1464:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1420:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1378:
1377:
1376:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1336:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1292:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1241:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1201:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1169:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1123:
1122:
1121:
1120:
1104:
1101:
1098:
1095:
1092:
1072:
1069:
1066:
1063:
1060:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
985:
969:
945:
942:
939:
936:
931:
928:
923:
920:
915:
910:
907:
902:
899:
873:
870:
867:
864:
861:
858:
855:
852:
849:
846:
843:
840:
837:
834:
810:
807:
804:
801:
798:
787:
771:
768:
765:
762:
738:
701:
698:
696:of morphisms.
686:
685:
684:
683:
667:
664:
661:
658:
655:
633:
630:
627:
624:
621:
601:
598:
595:
592:
589:
586:
583:
580:
577:
574:
571:
568:
565:
562:
559:
556:
546:
530:
506:
503:
500:
497:
492:
489:
484:
481:
476:
471:
468:
463:
460:
434:
431:
428:
425:
422:
419:
416:
413:
410:
407:
404:
401:
398:
395:
386:to a morphism
371:
368:
365:
362:
359:
345:
329:
326:
323:
320:
296:
239:
219:
199:
174:
173:
135:
133:
126:
120:
117:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6997:
6986:
6983:
6982:
6980:
6965:
6962:
6960:
6959:Representable
6957:
6955:
6952:
6950:
6947:
6945:
6942:
6940:
6937:
6935:
6932:
6930:
6927:
6925:
6922:
6920:
6917:
6915:
6912:
6910:
6907:
6905:
6902:
6900:
6897:
6896:
6893:
6888:
6881:
6876:
6874:
6869:
6867:
6862:
6861:
6858:
6846:
6838:
6836:
6828:
6826:
6818:
6817:
6814:
6800:
6797:
6795:
6792:
6790:
6786:
6782:
6778:
6776:
6774:
6767:
6765:
6762:
6760:
6757:
6756:
6754:
6751:
6747:
6737:
6734:
6731:
6727:
6724:
6723:
6721:
6719:
6711:
6705:
6702:
6700:
6697:
6695:
6692:
6690:
6689:Tetracategory
6687:
6685:
6682:
6679:
6678:pseudofunctor
6675:
6672:
6671:
6669:
6667:
6659:
6656:
6652:
6647:
6644:
6642:
6639:
6637:
6634:
6632:
6629:
6627:
6624:
6622:
6619:
6617:
6614:
6612:
6609:
6607:
6603:
6597:
6596:
6593:
6589:
6584:
6580:
6575:
6557:
6554:
6552:
6549:
6547:
6544:
6542:
6539:
6537:
6534:
6532:
6529:
6527:
6524:
6522:
6521:Free category
6519:
6518:
6516:
6512:
6505:
6504:Vector spaces
6501:
6498:
6495:
6491:
6488:
6486:
6483:
6481:
6478:
6476:
6473:
6471:
6468:
6466:
6463:
6462:
6460:
6458:
6454:
6444:
6441:
6439:
6436:
6432:
6429:
6428:
6427:
6424:
6422:
6419:
6417:
6414:
6413:
6411:
6409:
6405:
6399:
6398:Inverse limit
6396:
6394:
6391:
6387:
6384:
6383:
6382:
6379:
6377:
6374:
6372:
6369:
6368:
6366:
6364:
6360:
6357:
6355:
6351:
6345:
6342:
6340:
6337:
6335:
6332:
6330:
6327:
6325:
6324:Kan extension
6322:
6320:
6317:
6315:
6312:
6310:
6307:
6305:
6302:
6300:
6297:
6295:
6292:
6288:
6285:
6283:
6280:
6278:
6275:
6273:
6270:
6268:
6265:
6263:
6260:
6259:
6258:
6255:
6254:
6252:
6248:
6244:
6237:
6233:
6229:
6222:
6217:
6215:
6210:
6208:
6203:
6202:
6199:
6192:
6189:
6186:
6185:Video archive
6183:
6180:
6177:
6174:
6170:
6166:
6162:
6158:
6154:
6151:
6148:
6146:
6140:
6138:
6135:
6132:
6128:
6125:
6123:
6119:
6116:
6112:
6108:
6103:
6098:
6094:
6089:
6086:
6082:
6079:
6076:
6074:
6069:
6065:
6061:
6057:
6056:
6051:
6047:
6046:
6042:
6037:
6031:
6027:
6023:
6019:
6018:
6012:
6008:
6002:
5998:
5997:Basic algebra
5994:
5990:
5989:
5985:
5977:
5971:
5968:
5964:
5958:
5955:
5951:
5946:
5943:
5939:
5933:
5929:
5925:
5921:
5917:
5913:
5907:
5904:
5900:
5894:
5890:
5886:
5882:
5876:
5873:
5861:
5859:9789400995505
5855:
5851:
5850:
5842:
5839:
5835:
5830:
5827:
5823:
5818:
5815:
5811:
5806:
5803:
5799:
5795:
5790:
5787:
5783:
5777:
5773:
5769:
5763:
5760:
5753:
5749:
5748:Pseudofunctor
5746:
5744:
5743:Kan extension
5741:
5739:
5736:
5734:
5731:
5729:
5726:
5725:
5721:
5715:
5710:
5705:
5703:
5701:
5697:
5693:
5689:
5685:
5676:
5672:
5668:
5662:
5654:
5652:
5650:
5645:
5643:
5639:
5635:
5631:
5627:
5623:
5619:
5614:
5612:
5608:
5604:
5600:
5596:
5592:
5584:
5579:
5574:
5570:
5566:
5562:
5552:
5545:
5538:
5531:
5528:) : Hom(
5527:
5523:
5517:
5509:
5502:
5498:
5489:
5482:
5478:
5472:
5468:
5464:
5459:
5455:
5451:
5445:
5441:
5435:
5431:
5427:
5423:
5419:
5416:
5413:
5408:
5404:
5400:
5396:
5386:
5379:
5372:
5365:
5357:
5353:
5347:
5339:
5332:
5328:
5319:
5312:
5308:
5303:
5299:
5294:
5290:
5286:
5281:
5277:
5273:
5269:
5265:
5261:
5257:
5253:
5249:
5246:
5243:
5239:
5236:generated by
5235:
5231:
5226:
5222:
5218:
5213:
5211:Free functors
5210:
5207:
5203:
5199:
5195:
5194:abelian group
5191:
5188:which maps a
5186:
5182:
5177:
5176:
5171:
5167:
5164:which maps a
5162:
5158:
5154:
5149:
5146:
5142:
5138:
5134:
5117:
5114:
5111:
5104:
5100:
5096:
5092:
5089:
5086:
5082:
5078:
5075:
5072:
5068:
5064:
5059:
5055:
5050:
5046:
5041:
5037:
5033:
5029:
5025:
5021:
5017:
5013:
5009:
5005:
5001:
4998:
4994:
4991:
4988:
4984:
4980:
4979:tangent space
4976:
4972:
4968:
4964:
4960:
4956:
4953:
4950:
4946:
4941:
4937:
4933:
4929:
4921:
4917:
4913:
4908:
4903:
4899:
4895:
4890:
4886:
4882:
4878:
4874:
4871:
4868:
4867:
4861:
4852:
4848:
4837:
4833:
4827:
4823:
4816:
4812:
4808:
4801:
4797:
4793:
4788:
4784:
4780:
4772:
4768:
4764:
4755:
4751:
4738:
4734:
4727:
4723:
4715:
4708:
4704:
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4694:
4690:
4685:
4676:
4672:
4661:
4657:
4651:
4644:
4640:
4631:
4627:
4621:
4617:
4614:
4611:
4607:
4604:and to every
4603:
4599:
4595:
4588:
4573:
4553:
4533:
4525:
4503:
4500:
4497:
4488:
4468:
4462:
4459:
4447:
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4429:
4423:
4417:
4411:
4402:
4399:
4396:
4387:
4378:
4375:
4372:
4347:
4341:
4335:
4326:
4320:
4314:
4305:
4296:
4287:
4262:
4247:
4238:
4232:
4226:
4217:
4208:
4199:
4170:
4149:
4143:
4117:
4111:
4065:
4050:
4018:
4012:
4005:to its image
3992:
3972:
3949:
3943:
3923:
3920:
3914:
3908:
3882:
3876:
3870:
3847:
3844:
3838:
3832:
3826:
3820:
3808:
3802:
3789:
3783:
3777:
3754:
3751:
3748:
3742:
3739:
3719:
3716:
3713:
3707:
3699:
3696:
3692:
3684:
3683:inverse image
3668:
3665:
3662:
3642:
3636:
3633:
3630:
3622:
3603:
3590:
3584:
3578:
3571:to its image
3555:
3542:
3539:
3519:
3513:
3510:
3507:
3499:
3494:
3490:
3486:
3481:
3478:
3475:
3471:
3467:
3463:
3462:right-adjoint
3458:
3454:
3449:
3445:
3440:
3436:
3431:
3428:
3424:
3422:Limit functor
3421:
3418:
3414:
3410:
3406:
3402:
3399:
3395:
3389:
3384:
3380:
3375:
3373:
3370:
3366:
3361:
3359:
3356:
3352:
3348:
3344:
3340:
3336:
3332:
3327:
3323:
3318:
3315:
3312:
3308:
3304:
3300:
3296:
3292:
3291:
3287:) are called
3286:
3270:
3267:
3264:
3255:
3251:
3246:
3242:
3238:
3234:
3230:
3226:
3222:
3218:
3215:
3212:
3208:
3204:
3200:
3184:
3178:
3175:
3172:
3164:
3161:-presheaf on
3160:
3156:
3152:
3148:
3146:
3143:
3128:
3122:
3119:
3116:
3108:
3104:
3100:
3096:
3092:
3090:
3087:
3086:
3082:
3080:
3078:
3077:homomorphisms
3074:
3069:
3067:
3063:
3059:
3054:
3050:
3045:
3041:
3037:
3033:
3029:
3025:
3017:
3013:
3009:
3005:
3001:
2997:
2993:
2990:
2986:
2982:
2978:
2975:
2974:
2973:
2971:
2963:
2961:
2957:
2952:
2948:
2943:
2941:
2937:
2932:
2928:
2924:
2919:
2915:
2911:
2907:
2899:
2897:
2883:
2880:
2865:
2851:
2847:
2837:
2821:
2818:
2804:
2772:
2768:
2765:
2743:
2739:
2720:
2716:
2712:
2709:
2687:
2683:
2674:
2670:
2666:
2663:
2643:
2612:
2591:
2571:
2540:
2519:
2488:
2467:
2447:
2439:
2412:
2380:
2348:
2330:
2326:
2312:
2304:
2288:
2282:
2279:
2276:
2264:
2262:
2260:
2256:
2252:
2248:
2247:contravariant
2244:
2226:
2214:
2209:
2201:
2196:
2167:
2155:
2150:
2142:
2137:
2077:
2072:
2047:
2025:
2021:
1993:
1989:
1983:
1978:
1970:
1966:
1962:
1958:
1927:
1923:
1895:
1891:
1885:
1880:
1872:
1867:
1860:
1857:
1846:
1830:
1825:
1821:
1813:
1794:
1790:
1785:
1781:
1776:
1762:
1746:
1743:
1736:
1717:
1714:
1700:
1699:vector fields
1695:
1693:
1688:
1663:
1656:
1653:
1650:
1630:
1613:
1609:
1606:
1586:
1580:
1577:
1574:
1543:
1535:
1531:
1527:
1522:
1418:
1382:
1381:
1379:
1290:
1242:
1167:
1131:
1130:
1128:
1127:
1126:
1118:
1102:
1096:
1093:
1090:
1070:
1064:
1061:
1058:
1035:
1029:
1026:
1020:
1014:
1011:
1005:
1002:
999:
993:
986:
983:
967:
940:
934:
921:
913:
897:
890:
889:
887:
868:
862:
853:
847:
844:
838:
832:
824:
808:
802:
799:
796:
788:
785:
766:
760:
752:
736:
728:
727:
726:
724:
720:
716:
713:
707:
699:
697:
695:
691:
681:
665:
659:
656:
653:
631:
625:
622:
619:
596:
590:
587:
581:
575:
572:
566:
563:
560:
554:
547:
544:
528:
501:
495:
482:
474:
458:
451:
450:
448:
429:
423:
414:
408:
405:
399:
393:
385:
369:
363:
360:
357:
350:
346:
343:
324:
318:
311:to an object
310:
294:
287:
283:
282:
281:
279:
275:
271:
268:
264:
260:
256:
237:
217:
197:
188:
180:
170:
167:
159:
156:November 2023
149:
145:
139:
136:This article
134:
125:
124:
118:
116:
114:
113:function word
111:context; see
110:
106:
102:
101:Rudolf Carnap
98:
94:
90:
85:
83:
79:
75:
71:
67:
63:
59:
55:
51:
47:
40:
33:
19:
18:Functoriality
6909:Conservative
6886:
6769:
6750:Categorified
6654:n-categories
6605:Key concepts
6443:Direct limit
6426:Coequalizers
6344:Yoneda lemma
6318:
6250:Key concepts
6240:Key concepts
6179:The catsters
6159:package for
6147:-categories.
6144:
6143:The Tale of
6107:the original
6072:
6053:
6016:
5996:
5970:
5957:
5945:
5930:, Springer,
5927:
5906:
5891:, Springer,
5888:
5875:
5863:. Retrieved
5848:
5841:
5829:
5817:
5805:
5797:
5789:
5771:
5762:
5699:
5695:
5690:used to map
5664:
5646:
5615:
5602:
5598:
5594:
5590:
5588:
5572:
5568:
5564:
5560:
5558:is given by
5550:
5543:
5536:
5529:
5525:
5521:
5515:
5507:
5500:
5496:
5487:
5480:
5476:
5470:
5466:
5462:
5457:
5453:
5449:
5443:
5439:
5433:
5429:
5425:
5421:
5406:
5402:
5398:
5394:
5392:is given by
5384:
5377:
5370:
5363:
5355:
5351:
5345:
5337:
5330:
5326:
5317:
5310:
5306:
5300:denotes the
5297:
5292:
5288:
5284:
5279:
5275:
5267:
5263:
5255:
5251:
5237:
5229:
5224:
5220:
5216:
5205:
5197:
5184:
5180:
5173:
5160:
5156:
5152:
5150:The functor
5140:
5136:
5132:
5094:
5076:Lie algebras
5070:
5066:
5062:
5057:
5053:
5048:
5039:
5035:
5027:
5023:
5019:
5016:group action
5011:
5007:
5003:
4999:
4948:
4944:
4939:
4935:
4931:
4927:
4924:by the rule
4919:
4915:
4911:
4901:
4897:
4893:
4888:
4884:
4864:fundamental
4863:
4850:
4846:
4835:
4831:
4814:
4810:
4806:
4799:
4795:
4786:
4782:
4778:
4770:
4753:
4749:
4736:
4725:
4721:
4713:
4706:
4702:
4696:
4692:
4688:
4674:
4670:
4659:
4655:
4649:
4642:
4638:
4629:
4625:
4598:vector space
3623:which sends
3620:
3492:
3488:
3484:
3456:
3452:
3447:
3438:
3434:
3429:
3425:For a fixed
3416:
3412:
3408:
3393:
3387:
3382:
3381:In category
3354:
3350:
3346:
3342:
3338:
3334:
3330:
3325:
3321:
3319:The functor
3310:
3306:
3298:
3294:
3288:
3284:
3253:
3249:
3244:
3240:
3232:
3220:
3210:
3205:is called a
3202:
3198:
3162:
3158:
3154:
3150:
3106:
3102:
3098:
3094:
3070:
3061:
3057:
3052:
3048:
3043:
3039:
3035:
3031:
3027:
3023:
3021:
3015:
3011:
3007:
3003:
2995:
2988:
2984:
2976:
2967:
2955:
2950:
2947:multifunctor
2946:
2944:
2935:
2930:
2926:
2922:
2909:
2905:
2903:
2302:
2301:induces the
2268:
2254:
2250:
2246:
2242:
1696:
1691:
1689:
1529:
1525:
1523:
1520:
1124:
1116:
981:
885:
822:
783:
750:
722:
718:
714:
711:
709:
687:
679:
542:
446:
383:
341:
308:
277:
273:
269:
266:
258:
254:
252:
162:
153:
137:
104:
92:
88:
86:
84:is applied.
53:
43:
6718:-categories
6694:Kan complex
6684:Tricategory
6666:-categories
6556:Subcategory
6314:Exponential
6282:Preadditive
6277:Pre-abelian
6161:Mathematica
6081:AndrΓ© Joyal
5412:Hom functor
5242:free object
5099:linear maps
5085:Lie algebra
4914:) : C(
4905:induces an
3385:, written 1
3363:Endofunctor
3227:, then the
3000:isomorphism
2940:Hom functor
2918:Hom functor
2257:. See also
1845:expressions
694:composition
46:mathematics
6736:3-category
6726:2-category
6699:β-groupoid
6674:Bicategory
6421:Coproducts
6381:Equalizers
6287:Bicategory
5986:References
5733:Profunctor
5728:Anafunctor
5626:direct sum
5234:free group
4987:dual space
4971:derivative
4967:smooth map
4965:and every
4943:for every
4742:, denoted
4684:continuous
4612:to itself.
4606:linear map
4602:dual space
4481:Note that
3863:. Suppose
3290:presheaves
2964:Properties
1692:cofunctors
704:See also:
263:categories
119:Definition
109:linguistic
87:The words
78:continuous
62:categories
6939:Forgetful
6785:Symmetric
6730:2-functor
6470:Relations
6393:Pullbacks
6165:morphisms
6097:CiteSeerX
6060:EMS Press
6050:"Functor"
5696:morphisms
5692:functions
5115:⊗
5081:Lie group
4735:based at
4385:↦
4294:↦
4206:↦
4092:↦
4048:↦
3714:⊆
3697:−
3666:⊆
3640:→
3634::
3591:∈
3543:∈
3517:→
3511::
3498:power set
3355:selection
3268:⊆
3229:open sets
3182:→
3176::
3126:→
3120::
2906:bifunctor
2819:∘
2769:∘
2736:→
2713::
2680:→
2667::
2345:→
2327::
2286:→
2280::
2251:covariant
2243:pullbacks
2206:Λ
2147:Λ
2102:Λ
2069:Λ
2036:Λ
2030:ω
2018:ω
1990:ω
1975:Λ
1959:ω
1932:Λ
1877:Λ
1826:∗
1786:∗
1771:Γ
1709:Γ
1660:→
1654::
1628:→
1610::
1584:→
1578::
1530:covariant
1463:→
1419:∘
1335:→
1291:∘
1200:→
1168:∘
1100:→
1094::
1068:→
1062::
1027:∘
1003:∘
860:→
845::
806:→
800::
663:→
657::
629:→
623::
588:∘
564:∘
421:→
406::
367:→
361::
97:Aristotle
6985:Functors
6979:Category
6954:Monoidal
6924:Enriched
6919:Diagonal
6899:Additive
6845:Glossary
6825:Category
6799:n-monoid
6752:concepts
6408:Colimits
6376:Products
6329:Morphism
6272:Concrete
6267:Additive
6257:Category
6153:WildCats
6118:Archived
5995:(2009),
5926:(2004),
5887:(1992),
5865:23 April
5796:(1937).
5770:(1971),
5706:See also
5542:) β Hom(
5499: :
5479: :
5376:) β Hom(
5329: :
5309: :
5219: :
5155: :
4896: :
4866:groupoid
4781: :
4767:homotopy
4691: :
4637:, where
4084:, where
3487: :
3357:functor.
3351:constant
3207:presheaf
3083:Examples
2532:. Since
2436:are the
2372:, where
2022:′
1967:′
1924:′
1861:′
1847:such as
349:morphism
190:Functor
89:category
60:between
6949:Logical
6914:Derived
6904:Adjoint
6887:Functor
6835:Outline
6794:n-group
6759:2-group
6714:Strict
6704:β-topos
6500:Modules
6438:Pushout
6386:Kernels
6319:Functor
6262:Abelian
6070:at the
6068:functor
6062:, 2001
5678:Functor
5671:Haskell
5638:inverse
5296:(where
5232:to the
5043:, is a
5030:to the
4969:to its
4961:to its
3936:. Then
3681:to its
3464:to the
3235:form a
3089:Diagram
3006:, then
1761:1-forms
267:functor
142:Please
105:functor
93:functor
58:mapping
54:functor
6964:Smooth
6781:Traced
6764:2-ring
6494:Fields
6480:Groups
6475:Magmas
6363:Limits
6099:
6085:CatLab
6032:
6003:
5934:
5895:
5856:
5778:
5680:where
5673:has a
5634:direct
5624:, the
5410:. See
4995:Every
4989:above.
4918:) β C(
4351:
4266:
3442:has a
3073:monoid
2998:is an
2970:axioms
286:object
6934:Exact
6889:types
6775:-ring
6662:Weak
6646:Topos
6490:Rings
6155:is a
5754:Notes
5686:is a
5675:class
5474:. If
5274:from
5166:group
5069:. If
4997:group
4947:in C(
4828:from
4763:group
4700:with
4610:field
3770:then
3444:limit
3391:or id
3239:Open(
3223:is a
3056:from
2972:are:
2702:with
1810:of a
1733:of a
717:from
272:from
107:in a
56:is a
6465:Sets
6066:see
6030:ISBN
6001:ISBN
5974:See
5932:ISBN
5893:ISBN
5867:2016
5854:ISBN
5776:ISBN
5700:Hask
5683:fmap
5636:and
5628:and
5593:and
5589:Let
5520:Hom(
5494:and
5438:Hom(
5362:Hom(
5350:Hom(
5324:and
5190:ring
5036:Vect
4934:) =
4712:) =
4686:map
4600:its
3901:and
3403:The
3301:the
3157:, a
3153:and
3097:and
3034:and
2460:and
2404:and
2008:for
1910:for
1083:and
692:and
646:and
265:. A
257:and
253:Let
230:and
99:and
91:and
52:, a
6309:End
6299:CCC
6129:: "
6075:Lab
6022:doi
5698:on
5601:to
5471:Set
5458:Set
5452:to
5278:to
5258:of
5225:Grp
5221:Set
5198:Rng
5181:Rng
5161:Set
5157:Grp
5093:If
5047:of
5018:of
5012:Set
5010:to
4843:to
4765:of
4667:to
4526:on
3985:of
3493:Set
3489:Set
3353:or
3337:in
3293:on
3231:in
3219:If
3209:on
3199:Set
3105:in
3060:to
3042:to
3030:to
3002:in
2994:if
2983:in
2958:= 2
2936:two
2931:Set
2796:or
2440:to
1950:or
1115:in
980:in
884:in
821:in
782:in
749:in
721:to
678:in
541:in
445:in
382:in
340:in
307:in
276:to
261:be
146:to
44:In
6981::
6787:)
6783:)(
6171:,
6095:.
6083:,
6058:,
6052:,
6028:,
5922:;
5918:;
5914:;
5883:;
5651:.
5571:β
5567:β
5563:β¦
5549:,
5535:,
5524:,
5506:β
5486:β
5469:β
5465:Γ
5442:,
5428:,
5405:β
5401:β
5397:β¦
5383:,
5369:,
5360::
5354:,
5346:Ab
5336:β
5316:β
5298:Ab
5293:Ab
5291:β
5289:Ab
5287:Γ
5285:Ab
5266:,
5254:,
5223:β
5206:Ab
5185:Ab
5183:β
5159:β
5139:β
5135:Γ
5056:β
5034:,
4951:).
4938:β
4930:)(
4926:C(
4910:C(
4900:β
4849:,
4845:Ο(
4834:,
4830:Ο(
4785:β
4752:,
4695:β
4673:,
4658:,
4628:,
3491:β
3455:β
3437:β
3324:β
3252:β
3068:.
3051:β
2960:.
2945:A
2929:β
2925:Γ
2904:A
2896:.
2838:,
2604:,
2261:.
1694:.
115:.
6879:e
6872:t
6865:v
6779:(
6772:n
6770:E
6732:)
6728:(
6716:n
6680:)
6676:(
6664:n
6506:)
6502:(
6496:)
6492:(
6220:e
6213:t
6206:v
6175:.
6145:n
6073:n
6024::
6010:.
5869:.
5694:(
5603:D
5599:C
5595:D
5591:C
5573:f
5569:Ο
5565:g
5561:Ο
5556:)
5554:2
5551:Y
5547:1
5544:X
5540:1
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5533:2
5530:X
5526:g
5522:f
5516:C
5511:2
5508:Y
5504:1
5501:Y
5497:g
5491:2
5488:X
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5477:f
5467:C
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5454:Y
5450:X
5446:)
5444:Y
5440:X
5434:C
5430:Y
5426:X
5422:C
5414:.
5407:f
5403:Ο
5399:g
5395:Ο
5390:)
5388:2
5385:B
5381:1
5378:A
5374:1
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5364:A
5358:)
5356:g
5352:f
5341:2
5338:B
5334:1
5331:B
5327:g
5321:2
5318:A
5314:1
5311:A
5307:f
5280:B
5276:A
5268:B
5264:A
5256:B
5252:A
5244:.
5238:X
5230:X
5217:F
5200:(
5153:U
5141:C
5137:C
5133:C
5118:W
5112:V
5095:C
5071:C
5067:C
5063:G
5058:C
5054:G
5049:G
5040:K
5028:G
5024:G
5020:G
5008:G
5004:G
5000:G
4949:Y
4945:Ο
4940:f
4936:Ο
4932:Ο
4928:f
4922:)
4920:X
4916:Y
4912:f
4902:Y
4898:X
4894:f
4889:X
4885:X
4856:)
4854:0
4851:y
4847:Y
4841:)
4839:0
4836:x
4832:X
4818:0
4815:y
4811:Y
4807:f
4803:0
4800:x
4796:X
4787:Y
4783:X
4779:f
4774:0
4771:x
4759:)
4757:0
4754:x
4750:X
4748:(
4746:1
4744:Ο
4740:0
4737:x
4729:0
4726:x
4722:X
4717:0
4714:y
4710:0
4707:x
4705:(
4703:f
4697:Y
4693:X
4689:f
4680:)
4678:0
4675:y
4671:Y
4669:(
4665:)
4663:0
4660:x
4656:X
4654:(
4650:X
4646:0
4643:x
4639:X
4635:)
4633:0
4630:x
4626:X
4624:(
4574:X
4554:f
4534:X
4510:)
4507:}
4504:1
4501:,
4498:0
4495:{
4492:(
4489:f
4469:.
4466:}
4463:X
4460:,
4457:}
4454:{
4451:{
4448:=
4445:}
4442:)
4439:1
4436:(
4433:f
4430:,
4427:)
4424:0
4421:(
4418:f
4415:{
4412:=
4409:)
4406:}
4403:1
4400:,
4397:0
4394:{
4391:(
4388:f
4382:}
4379:1
4376:,
4373:0
4370:{
4348:,
4345:}
4342:X
4339:{
4336:=
4333:}
4330:)
4327:1
4324:(
4321:f
4318:{
4315:=
4312:)
4309:}
4306:1
4303:{
4300:(
4297:f
4291:}
4288:1
4285:{
4263:,
4260:}
4257:}
4254:{
4251:{
4248:=
4245:}
4242:)
4239:0
4236:(
4233:f
4230:{
4227:=
4224:)
4221:}
4218:0
4215:{
4212:(
4209:f
4203:}
4200:0
4197:{
4177:}
4174:{
4171:=
4168:)
4165:}
4162:{
4159:(
4156:)
4153:)
4150:f
4147:(
4144:F
4141:(
4121:)
4118:f
4115:(
4112:F
4072:}
4069:{
4066:=
4063:)
4060:}
4057:{
4054:(
4051:f
4045:}
4042:{
4022:)
4019:U
4016:(
4013:f
3993:X
3973:U
3953:)
3950:f
3947:(
3944:F
3924:X
3921:=
3918:)
3915:1
3912:(
3909:f
3889:}
3886:{
3883:=
3880:)
3877:0
3874:(
3871:f
3851:}
3848:X
3845:,
3842:}
3839:1
3836:{
3833:,
3830:}
3827:0
3824:{
3821:,
3818:}
3815:{
3812:{
3809:=
3806:)
3803:X
3800:(
3795:P
3790:=
3787:)
3784:X
3781:(
3778:F
3758:}
3755:1
3752:,
3749:0
3746:{
3743:=
3740:X
3720:.
3717:X
3711:)
3708:V
3705:(
3700:1
3693:f
3669:Y
3663:V
3643:Y
3637:X
3631:f
3607:)
3604:Y
3601:(
3596:P
3588:)
3585:U
3582:(
3579:f
3559:)
3556:X
3553:(
3548:P
3540:U
3520:Y
3514:X
3508:f
3485:P
3457:C
3453:C
3448:C
3439:C
3435:J
3430:J
3417:D
3413:D
3409:D
3394:C
3388:C
3383:C
3371:.
3347:X
3343:C
3339:D
3335:X
3331:C
3326:D
3322:C
3313:.
3311:X
3307:U
3299:U
3295:X
3285:X
3271:V
3265:U
3254:V
3250:U
3245:X
3241:X
3233:X
3221:X
3213:.
3211:C
3203:D
3185:J
3179:C
3173:D
3163:C
3159:J
3155:J
3151:C
3141:.
3129:C
3123:J
3117:D
3107:C
3103:J
3099:J
3095:C
3062:C
3058:A
3053:F
3049:G
3044:C
3040:B
3036:G
3032:B
3028:A
3024:F
3018:.
3016:D
3012:f
3010:(
3008:F
3004:C
2996:f
2991:;
2989:D
2985:C
2977:F
2956:n
2951:n
2927:C
2923:C
2884:F
2881:=
2875:p
2872:o
2866:)
2860:p
2857:o
2852:F
2848:(
2822:F
2813:p
2810:o
2805:G
2781:p
2778:o
2773:F
2766:G
2744:2
2740:C
2730:p
2727:o
2721:1
2717:C
2710:G
2688:1
2684:C
2675:0
2671:C
2664:F
2644:F
2621:p
2618:o
2613:F
2592:D
2572:C
2549:p
2546:o
2541:C
2520:F
2497:p
2494:o
2489:F
2468:D
2448:C
2421:p
2418:o
2413:D
2389:p
2386:o
2381:C
2357:p
2354:o
2349:D
2339:p
2336:o
2331:C
2321:p
2318:o
2313:F
2289:D
2283:C
2277:F
2227:j
2222:e
2215:i
2210:j
2202:=
2197:i
2192:e
2168:j
2163:e
2156:j
2151:i
2143:=
2138:i
2133:e
2108:T
2078:j
2073:i
2048:.
2042:T
2026:=
1994:j
1984:j
1979:i
1971:=
1963:i
1937:x
1928:=
1920:x
1896:j
1892:x
1886:i
1881:j
1873:=
1868:i
1858:x
1831:M
1822:T
1795:)
1791:M
1782:T
1777:(
1747:M
1744:T
1721:)
1718:M
1715:T
1712:(
1672:p
1669:o
1664:D
1657:C
1651:F
1631:D
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1619:o
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1607:F
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1581:C
1575:F
1552:p
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1503:t
1500:n
1497:a
1494:i
1491:r
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1117:C
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1036:g
1033:(
1030:F
1024:)
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1015:F
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1009:)
1006:f
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994:F
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591:F
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576:F
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319:F
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140:.
41:.
34:.
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