4116:
4363:
4383:
4373:
1040:
1339:
647:
1568:
2320:
1658:
909:
344:
1246:
489:
468:
1124:
2959:
1427:
728:
2891:
2175:
is
Cartesian closed if and only if it is equivalent to a category with only one object and one identity morphism. Indeed, if 0 is an initial object and 1 is a final object and we have
1210:
2717:
864:
807:
2811:
3055:
3097:
2986:
2204:
2753:
2360:
3013:
2199:
2451:
3377:
Note however that the above list is not complete; type isomorphism in the free BCCC is not finitely axiomatizable, and its decidability is still an open problem.
3147:
One may ask what other such equations are valid in all
Cartesian closed categories. It turns out that all of them follow logically from the following axioms:
2565:
3760:
1035:{\displaystyle Z^{Y}\times Y^{X}\times X{\xrightarrow {\mathrm {id} \times \mathrm {ev} _{X,Y}}}Z^{Y}\times Y{\xrightarrow {\mathrm {ev} _{Y,Z}}}Z}
1583:
3265:
264:
3440:
2595:
3264:, with products distributing over coproducts. Their equational theory is extended with the following axioms, yielding something similar to
2366:. The tensor product is not a categorical product, so this does not contradict the above. We obtain instead that the category of modules is
2042:
1334:{\displaystyle {\begin{array}{ccc}\Gamma _{Y}(p)&\to &X^{Y}\\\downarrow &&\downarrow \\1&\to &Y^{Y}\end{array}}}
642:{\displaystyle \mathrm {papply} _{X,Y,Z}:Z^{X\times Y}\times X\cong (Z^{Y})^{X}\times X{\xrightarrow {\mathrm {ev} _{X,Z^{Y}}}}Z^{Y}.}
3548:
90:(1596–1650), French philosopher, mathematician, and scientist, whose formulation of analytic geometry gave rise to the concept of
3473:
2034:
410:
3753:
2068:, whose continuous maps do form a Cartesian closed category (that is, the objects are the cpos, and the morphisms are the
1051:
54:
can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in
182:
The first two conditions can be combined to the single requirement that any finite (possibly empty) family of objects of
3957:
3912:
2568:
provides a deep isomorphism between intuitionistic logic, simply-typed lambda calculus and
Cartesian closed categories.
63:
2561:; it has led to the realization that simply-typed lambda calculus can be interpreted in any Cartesian closed category.
1984:-sets can be seen as a special case of functor categories: every group can be considered as a one-object category, and
1563:{\displaystyle \hom _{C/Y}(X\times Y{\xrightarrow {\pi _{2}}}Y,Z{\xrightarrow {p}}Y)\cong \hom _{C}(X,\Gamma _{Y}(p)).}
4386:
4326:
3253:
3727:
368:
Take care to note that a
Cartesian closed category need not have finite limits; only finite products are guaranteed.
2906:
662:
4407:
4376:
4162:
4026:
3934:
2831:
2587:
1803:
4335:
3979:
3917:
138:
47:
4412:
4366:
4322:
3927:
3746:
2367:
67:
51:
1146:
3922:
3904:
2667:
2041:
with smooth maps is
Cartesian closed. Substitute categories have therefore been considered: the category of
813:
756:
4129:
3895:
3875:
3798:
3418:
2057:
1681:
1364:
350:
248:
35:
2761:
4011:
3850:
2363:
2082:
are continuous functions in the Scott topology, and currying, together with apply, provide the adjoint.
3018:
3823:
3818:
2330:
3423:
2315:{\displaystyle \mathrm {Hom} (X,Y)\cong \mathrm {Hom} (1,Y^{X})\cong \mathrm {Hom} (0,Y^{X})\cong 1}
4167:
4115:
4045:
4041:
3845:
3060:
2412:
2150:
1799:
479:
383:
is locally
Cartesian closed, it need not actually be Cartesian closed; that happens if and only if
95:
75:
3520:
4021:
4016:
3998:
3880:
3855:
3694:
3618:
3446:
3408:
2964:
2026:
168:
71:
55:
17:
2726:
2339:
2046:
4330:
4267:
4255:
4157:
4082:
4077:
4035:
4031:
3813:
3808:
3686:
3544:
3479:
3469:
3436:
2991:
2591:
2178:
2030:
1934:
into the category of sets, with natural transformations as morphisms, is
Cartesian closed. If
1677:
91:
87:
59:
4291:
4177:
4152:
4087:
4072:
4067:
4006:
3835:
3803:
3678:
3652:
3610:
3583:
3428:
2554:
2326:
2069:
1904:
236:
2424:
2029:, Cartesian closed categories are particularly easy to work with. Neither the category of
4203:
3769:
3713:
3497:
2090:
2086:
1899:
of all small categories (with functors as morphisms) is
Cartesian closed; the exponential
119:
31:
4240:
1251:
4235:
4219:
4182:
4172:
4092:
3261:
2901:
2334:
2065:
2002:
1923:
372:
111:
3601:
Solov'ev, S.V. (1983). "The category of finite sets and
Cartesian closed categories".
4401:
4230:
4062:
3939:
3865:
3622:
3450:
195:
191:
3698:
1998:
is
Cartesian closed; this is a functor category as explained under functor category.
3984:
3885:
3393:
2053:
1995:
1421:
on the slice category, which is right adjoint to a variant of the product functor:
4245:
3397:
2575:, have been proposed as a general setting for mathematics, instead of traditional
4225:
4097:
3967:
3432:
2583:
2172:
3669:
Seely, R. A. G. (1984). "Locally cartesian closed categories and type theory".
3656:
3637:
1232:, suppose the following pullback square exists, which defines the subobject of
4277:
4215:
3828:
3682:
3116:
2576:
1788:
3690:
3483:
4271:
3962:
3565:"Ct.category theory - is the category commutative monoids cartesian closed?"
3257:
1653:{\displaystyle Z^{Y}\cong \Gamma _{Y}(Z\times Y{\xrightarrow {\pi _{2}}}Y).}
252:
2541:) can always be represented as a "function of one variable" (the morphism λ
1791:
sets, with functions as morphisms, is Cartesian closed for the same reason.
3638:"Remarks on isomorphisms in typed lambda calculi with empty and sum types"
3587:
2525:
In Cartesian closed categories, a "function of two variables" (a morphism
2380:
Every elementary topos is locally Cartesian closed. This example includes
4340:
3972:
3870:
2558:
2098:
2073:
2038:
1729:
43:
404:, the counit of the exponential adjunction is a natural transformation
339:{\displaystyle \mathrm {Hom} (X\times Y,Z)\cong \mathrm {Hom} (X,Z^{Y})}
4310:
4300:
3949:
3860:
3614:
2325:
In particular, any non-trivial category with a zero object, such as an
256:
202:
3407:. Lecture Notes in Physics. Vol. 813. Springer. pp. 95–174.
4305:
3564:
2614:
has all pullbacks, because the pullback of two arrows with codomain
1627:
1497:
1469:
1004:
947:
594:
1712:. The adjointness is expressed by the following fact: the function
4187:
3738:
3413:
3107:
In every Cartesian closed category (using exponential notation), (
2828:
can be expressed in terms of the dependent product by the formula
2572:
2459:
does not have a terminal object, and thus is not Cartesian closed.
2126:
2078:
2019:
3732:
The n-Category Café: A group blog on math, physics and philosophy
3717:
3524:
4127:
3780:
3742:
3671:
Mathematical Proceedings of the Cambridge Philosophical Society
2501:
is locally Cartesian closed, then all of its slice categories
201:
The third condition is equivalent to the requirement that the
2598:
is more consciously modelled on Cartesian closed categories.
2509:
Non-examples of locally Cartesian closed categories include:
2411:
whose objects are topological spaces and whose morphisms are
3398:"Physics, Topology, Logic and Computation: A Rosetta Stone"
1348:
and the arrow on the bottom corresponds to the identity on
2093:. An important example arises from topological spaces. If
1045:
the corresponding arrow under the exponential adjunction
2896:
The reason for these names is because, when interpreting
2376:
Examples of locally Cartesian closed categories include:
738:
Evaluating the exponential in one argument at a morphism
1962:
is given by the set of all natural transformations from
198:
in a category is the terminal object of that category.
463:{\displaystyle \mathrm {ev} _{Y,Z}:Z^{Y}\times Y\to Z}
3063:
3021:
2994:
2967:
2909:
2834:
2764:
2729:
2670:
2427:
2342:
2207:
2181:
1988:-sets are nothing but functors from this category to
1586:
1430:
1249:
1149:
1054:
912:
816:
759:
665:
492:
413:
267:
2490:
given by taking pullbacks has a right adjoint, then
1119:{\displaystyle c_{X,Y,Z}:Z^{Y}\times Y^{X}\to Z^{X}}
4290:
4254:
4202:
4195:
4146:
4055:
3997:
3948:
3903:
3894:
3791:
2590:, which in retrospect bears some similarity to the
869:corresponding to the operation of composition with
3091:
3049:
3007:
2980:
2953:
2885:
2805:
2747:
2711:
2445:
2354:
2314:
2193:
1652:
1562:
1333:
1204:
1118:
1034:
858:
801:
722:
641:
462:
338:
2129:is a Cartesian closed category: the "product" of
1668:Examples of Cartesian closed categories include:
3580:Function level programs as mathematical objects
3256:extend Cartesian closed categories with binary
1684:as morphisms, is Cartesian closed. The product
94:, which was later generalized to the notion of
2954:{\displaystyle y:Y\vdash P(y):\mathrm {Type} }
2333:is not Cartesian closed. However, the functor
2329:, is not Cartesian closed. So the category of
1140:, this is the ordinary composition operation:
723:{\displaystyle \mathrm {ev} _{Y,Z}(f,y)=f(y).}
251:, this can be expressed by the existence of a
3754:
2886:{\displaystyle Q^{P}\cong \Pi _{p}(p^{*}(Q))}
2610:be a locally Cartesian closed category. Then
114:it satisfies the following three properties:
58:and the theory of programming, in that their
8:
2109:) for which there is a unique morphism from
1915:, with natural transformations as morphisms.
371:If a category has the property that all its
3636:Fiore, M.; Cosmo, R. Di; Balat, V. (2006).
3539:Barendregt, H.P. (1984). "Theorem 1.2.16".
2661:which has both a left and a right adjoint.
2586:has advocated a variable-free notation, or
2045:is Cartesian closed, as is the category of
1403:, then it can be assembled into a functor Γ
1236:corresponding to maps whose composite with
656:, these reduce to the ordinary operations:
194:of the categorical product and because the
4382:
4372:
4199:
4143:
4124:
3900:
3788:
3777:
3761:
3747:
3739:
1930:consisting of all covariant functors from
477:map. More generally, we can construct the
3422:
3412:
3068:
3062:
3026:
3020:
2999:
2993:
2972:
2966:
2937:
2908:
2865:
2852:
2839:
2833:
2795:
2781:
2769:
2763:
2728:
2701:
2687:
2675:
2669:
2571:Certain Cartesian closed categories, the
2426:
2419:is equivalent to the category of sheaves
2341:
2297:
2273:
2261:
2237:
2208:
2206:
2180:
1954:is the functor whose value on the object
1632:
1622:
1604:
1591:
1585:
1539:
1517:
1492:
1474:
1464:
1439:
1435:
1429:
1321:
1284:
1258:
1250:
1248:
1154:
1148:
1110:
1097:
1084:
1059:
1053:
1014:
1006:
999:
987:
968:
960:
948:
942:
930:
917:
911:
847:
834:
821:
815:
790:
777:
764:
758:
675:
667:
664:
630:
615:
604:
596:
589:
577:
567:
539:
514:
494:
491:
442:
423:
415:
412:
327:
303:
268:
266:
3466:Categories for the Working Mathematician
1205:{\displaystyle c_{X,Y,Z}(g,f)=g\circ f.}
888:. Alternate notations for the operation
873:. Alternate notations for the operation
375:are Cartesian closed, then it is called
3385:
3200:(here 1 denotes the terminal object of
1577:can be expressed in terms of sections:
1136:In the particular case of the category
652:In the particular case of the category
3582:. New York, New York, USA: ACM Press.
2712:{\displaystyle \Sigma _{p}:C/X\to C/Y}
2018:Even more generally, every elementary
859:{\displaystyle Z^{p}:Z^{Y}\to Z^{X},}
802:{\displaystyle p^{Z}:X^{Z}\to Y^{Z},}
239:, usually denoted –, for all objects
229:and morphisms φ to φ × id
7:
2043:compactly generated Hausdorff spaces
3498:"cartesian closed category in nLab"
2806:{\displaystyle \Pi _{p}:C/X\to C/Y}
2641:denote the corresponding object of
2415:is locally Cartesian closed, since
27:Type of category in category theory
3131:. We write this as the "equation"
3065:
3023:
3015:correspond to the type formations
2996:
2969:
2947:
2944:
2941:
2938:
2849:
2766:
2672:
2566:Curry–Howard–Lambek correspondence
2505:are also locally Cartesian closed.
2466:has pullbacks and for every arrow
2280:
2277:
2274:
2244:
2241:
2238:
2215:
2212:
2209:
1601:
1536:
1255:
1010:
1007:
964:
961:
952:
949:
903:Evaluation maps can be chained as
671:
668:
600:
597:
510:
507:
504:
501:
498:
495:
419:
416:
310:
307:
304:
275:
272:
269:
25:
3521:Locally cartesian closed category
2105:form the objects of a category O(
2097:is a topological space, then the
1825:is the set of all functions from
1728:is naturally identified with the
1704:is the set of all functions from
18:Locally cartesian closed category
4381:
4371:
4362:
4361:
4114:
3645:Annals of Pure and Applied Logic
3050:{\displaystyle \Sigma _{x:P(y)}}
2594:of Cartesian closed categories.
2516:is not locally Cartesian closed.
2362:with a fixed module does have a
2125:and no morphism otherwise. This
2089:is a Cartesian closed (bounded)
2064:s) have a natural topology, the
1907:consisting of all functors from
1344:where the arrow on the right is
2557:applications, this is known as
2001:In particular, the category of
1892:-sets is also Cartesian closed.
3084:
3078:
3042:
3036:
2931:
2925:
2880:
2877:
2871:
2858:
2789:
2742:
2736:
2695:
2440:
2434:
2303:
2284:
2267:
2248:
2231:
2219:
1644:
1610:
1554:
1551:
1545:
1526:
1507:
1452:
1372:. It is often abbreviated as Γ
1312:
1300:
1294:
1275:
1270:
1264:
1184:
1172:
1103:
840:
783:
714:
708:
699:
687:
574:
560:
454:
333:
314:
297:
279:
1:
3254:Bicartesian closed categories
3249:Bicartesian closed categories
3092:{\displaystyle \Pi _{x:P(y)}}
2723:and is given by composition
2494:is locally Cartesian closed.
2322:which has only one element.
1926:, then the functor category
1692:is the Cartesian product of
1395:) exists for every morphism
64:simply typed lambda calculus
4056:Constructions on categories
3464:Saunders, Mac Lane (1978).
3433:10.1007/978-3-642-12821-9_2
3266:Tarski's high school axioms
2981:{\displaystyle \Sigma _{p}}
2618:is given by the product in
1802:, then the category of all
110:is called Cartesian closed
78:and classical computation.
70:, whose internal language,
4429:
4163:Higher-dimensional algebra
3657:10.1016/j.apal.2005.09.001
3468:(2nd ed.). Springer.
3405:New Structures for Physics
2748:{\displaystyle p\circ (-)}
2588:Function-level programming
2355:{\displaystyle -\otimes M}
68:closed monoidal categories
66:. They are generalized by
42:if, roughly speaking, any
4357:
4136:
4123:
4112:
3787:
3776:
3728:"CCCs and the λ-calculus"
3714:Cartesian closed category
3683:10.1017/S0305004100061284
2645:. Taking pullbacks along
2602:Dependent sum and product
2037:maps nor the category of
1129:is called the (internal)
190:, because of the natural
3403:. In Coecke, Bob (ed.).
3008:{\displaystyle \Pi _{p}}
2194:{\displaystyle 0\cong 1}
1809:is Cartesian closed. If
377:locally cartesian closed
249:locally small categories
74:, are suitable for both
3973:Cokernels and quotients
3896:Universal constructions
2137:is the intersection of
2058:complete partial orders
2009: : Δ →
1980:The earlier example of
1950:, then the exponential
1888:The category of finite
387:has a terminal object.
209:(i.e. the functor from
4130:Higher category theory
3876:Natural transformation
3734:. University of Texas.
3093:
3051:
3009:
2982:
2955:
2887:
2807:
2749:
2713:
2447:
2356:
2316:
2195:
2013:) is Cartesian closed.
1942:are two functors from
1654:
1564:
1335:
1206:
1120:
1036:
860:
803:
724:
643:
473:called the (internal)
464:
340:
3588:10.1145/800223.806757
3578:Backus, John (1981).
3396:; Stay, Mike (2011).
3094:
3052:
3010:
2983:
2956:
2888:
2808:
2750:
2714:
2448:
2446:{\displaystyle Sh(X)}
2400:for small categories
2357:
2317:
2196:
1655:
1565:
1336:
1207:
1121:
1037:
861:
804:
725:
644:
483:map as the composite
465:
341:
3999:Algebraic categories
3061:
3019:
2992:
2965:
2907:
2832:
2762:
2727:
2668:
2425:
2413:local homeomorphisms
2340:
2205:
2179:
2145:and the exponential
2022:is Cartesian closed.
2005:(which are functors
1994:The category of all
1584:
1428:
1247:
1147:
1052:
910:
814:
757:
663:
490:
411:
265:
4168:Homotopy hypothesis
3846:Commutative diagram
3726:Baez, John (2006).
3541:The Lambda Calculus
2820:The exponential by
2582:Computer scientist
2331:modules over a ring
1837:action defined by (
1638:
1573:The exponential by
1501:
1480:
1026:
980:
623:
480:partial application
391:Basic constructions
186:admit a product in
96:categorical product
72:linear type systems
3881:Universal property
3615:10.1007/BF01084396
3089:
3047:
3005:
2978:
2951:
2883:
2803:
2758:The right adjoint
2745:
2709:
2443:
2392:-sets for a group
2352:
2312:
2191:
2171:A category with a
2031:topological spaces
2027:algebraic topology
1650:
1560:
1365:object of sections
1331:
1329:
1202:
1116:
1032:
856:
799:
720:
639:
460:
336:
217:that maps objects
56:mathematical logic
4408:Closed categories
4395:
4394:
4353:
4352:
4349:
4348:
4331:monoidal category
4286:
4285:
4158:Enriched category
4110:
4109:
4106:
4105:
4083:Quotient category
4078:Opposite category
3993:
3992:
3543:. North-Holland.
3442:978-3-642-12821-9
3268:but with a zero:
3103:Equational theory
2815:dependent product
2664:The left adjoint
2592:internal language
1639:
1502:
1481:
1240:is the identity:
1027:
981:
624:
92:Cartesian product
60:internal language
16:(Redirected from
4420:
4385:
4384:
4375:
4374:
4365:
4364:
4200:
4178:Simplex category
4153:Categorification
4144:
4125:
4118:
4088:Product category
4073:Kleisli category
4068:Functor category
3913:Terminal objects
3901:
3836:Adjoint functors
3789:
3778:
3763:
3756:
3749:
3740:
3735:
3702:
3661:
3660:
3642:
3633:
3627:
3626:
3609:(3): 1387–1400.
3598:
3592:
3591:
3575:
3569:
3568:
3561:
3555:
3554:
3536:
3530:
3518:
3512:
3511:
3509:
3508:
3494:
3488:
3487:
3461:
3455:
3454:
3426:
3416:
3402:
3390:
3119:for all objects
3098:
3096:
3095:
3090:
3088:
3087:
3056:
3054:
3053:
3048:
3046:
3045:
3014:
3012:
3011:
3006:
3004:
3003:
2987:
2985:
2984:
2979:
2977:
2976:
2960:
2958:
2957:
2952:
2950:
2892:
2890:
2889:
2884:
2870:
2869:
2857:
2856:
2844:
2843:
2812:
2810:
2809:
2804:
2799:
2785:
2774:
2773:
2754:
2752:
2751:
2746:
2718:
2716:
2715:
2710:
2705:
2691:
2680:
2679:
2649:gives a functor
2625:For every arrow
2555:computer science
2454:
2452:
2450:
2449:
2444:
2361:
2359:
2358:
2353:
2327:abelian category
2321:
2319:
2318:
2313:
2302:
2301:
2283:
2266:
2265:
2247:
2218:
2200:
2198:
2197:
2192:
2167:
2070:Scott continuous
2047:Frölicher spaces
2039:smooth manifolds
1972:
1968:,−) ×
1905:functor category
1903:is given by the
1787:The category of
1659:
1657:
1656:
1651:
1640:
1637:
1636:
1623:
1609:
1608:
1596:
1595:
1569:
1567:
1566:
1561:
1544:
1543:
1522:
1521:
1503:
1493:
1482:
1479:
1478:
1465:
1448:
1447:
1443:
1362:) is called the
1340:
1338:
1337:
1332:
1330:
1326:
1325:
1298:
1289:
1288:
1263:
1262:
1211:
1209:
1208:
1203:
1171:
1170:
1125:
1123:
1122:
1117:
1115:
1114:
1102:
1101:
1089:
1088:
1076:
1075:
1041:
1039:
1038:
1033:
1028:
1025:
1024:
1013:
1000:
992:
991:
982:
979:
978:
967:
955:
943:
935:
934:
922:
921:
865:
863:
862:
857:
852:
851:
839:
838:
826:
825:
808:
806:
805:
800:
795:
794:
782:
781:
769:
768:
750:gives morphisms
729:
727:
726:
721:
686:
685:
674:
648:
646:
645:
640:
635:
634:
625:
622:
621:
620:
619:
603:
590:
582:
581:
572:
571:
550:
549:
531:
530:
513:
469:
467:
466:
461:
447:
446:
434:
433:
422:
400:For each object
373:slice categories
345:
343:
342:
337:
332:
331:
313:
278:
155:Any two objects
125:Any two objects
40:Cartesian closed
21:
4428:
4427:
4423:
4422:
4421:
4419:
4418:
4417:
4413:Lambda calculus
4398:
4397:
4396:
4391:
4345:
4315:
4282:
4259:
4250:
4207:
4191:
4142:
4132:
4119:
4102:
4051:
3989:
3958:Initial objects
3944:
3890:
3783:
3772:
3770:Category theory
3767:
3725:
3710:
3705:
3668:
3664:
3640:
3635:
3634:
3630:
3600:
3599:
3595:
3577:
3576:
3572:
3563:
3562:
3558:
3551:
3538:
3537:
3533:
3519:
3515:
3506:
3504:
3496:
3495:
3491:
3476:
3463:
3462:
3458:
3443:
3424:10.1.1.296.1044
3400:
3392:
3391:
3387:
3383:
3251:
3105:
3064:
3059:
3058:
3022:
3017:
3016:
2995:
2990:
2989:
2968:
2963:
2962:
2961:, the functors
2905:
2904:
2861:
2848:
2835:
2830:
2829:
2765:
2760:
2759:
2725:
2724:
2671:
2666:
2665:
2604:
2523:
2423:
2422:
2420:
2368:monoidal closed
2338:
2337:
2293:
2257:
2203:
2202:
2177:
2176:
2154:
2121:is a subset of
2087:Heyting algebra
2003:simplicial sets
1996:directed graphs
1963:
1666:
1628:
1600:
1587:
1582:
1581:
1535:
1513:
1470:
1431:
1426:
1425:
1408:
1390:
1377:
1357:
1328:
1327:
1317:
1315:
1310:
1304:
1303:
1297:
1291:
1290:
1280:
1278:
1273:
1254:
1245:
1244:
1220:For a morphism
1218:
1150:
1145:
1144:
1106:
1093:
1080:
1055:
1050:
1049:
1005:
983:
959:
926:
913:
908:
907:
883:
843:
830:
817:
812:
811:
786:
773:
760:
755:
754:
736:
666:
661:
660:
626:
611:
595:
573:
563:
535:
493:
488:
487:
438:
414:
409:
408:
398:
393:
379:. Note that if
323:
263:
262:
234:
120:terminal object
104:
84:
32:category theory
28:
23:
22:
15:
12:
11:
5:
4426:
4424:
4416:
4415:
4410:
4400:
4399:
4393:
4392:
4390:
4389:
4379:
4369:
4358:
4355:
4354:
4351:
4350:
4347:
4346:
4344:
4343:
4338:
4333:
4319:
4313:
4308:
4303:
4297:
4295:
4288:
4287:
4284:
4283:
4281:
4280:
4275:
4264:
4262:
4257:
4252:
4251:
4249:
4248:
4243:
4238:
4233:
4228:
4223:
4212:
4210:
4205:
4197:
4193:
4192:
4190:
4185:
4183:String diagram
4180:
4175:
4173:Model category
4170:
4165:
4160:
4155:
4150:
4148:
4141:
4140:
4137:
4134:
4133:
4128:
4121:
4120:
4113:
4111:
4108:
4107:
4104:
4103:
4101:
4100:
4095:
4093:Comma category
4090:
4085:
4080:
4075:
4070:
4065:
4059:
4057:
4053:
4052:
4050:
4049:
4039:
4029:
4027:Abelian groups
4024:
4019:
4014:
4009:
4003:
4001:
3995:
3994:
3991:
3990:
3988:
3987:
3982:
3977:
3976:
3975:
3965:
3960:
3954:
3952:
3946:
3945:
3943:
3942:
3937:
3932:
3931:
3930:
3920:
3915:
3909:
3907:
3898:
3892:
3891:
3889:
3888:
3883:
3878:
3873:
3868:
3863:
3858:
3853:
3848:
3843:
3838:
3833:
3832:
3831:
3826:
3821:
3816:
3811:
3806:
3795:
3793:
3785:
3784:
3781:
3774:
3773:
3768:
3766:
3765:
3758:
3751:
3743:
3737:
3736:
3723:
3709:
3708:External links
3706:
3704:
3703:
3665:
3663:
3662:
3651:(1–2): 35–50.
3628:
3593:
3570:
3556:
3549:
3531:
3513:
3489:
3474:
3456:
3441:
3384:
3382:
3379:
3375:
3374:
3368:
3362:
3352:
3343:
3314:
3287:
3262:initial object
3250:
3247:
3246:
3245:
3235:
3217:
3208:
3205:
3191:
3174:
3145:
3144:
3104:
3101:
3099:respectively.
3086:
3083:
3080:
3077:
3074:
3071:
3067:
3044:
3041:
3038:
3035:
3032:
3029:
3025:
3002:
2998:
2975:
2971:
2949:
2946:
2943:
2940:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2902:dependent type
2882:
2879:
2876:
2873:
2868:
2864:
2860:
2855:
2851:
2847:
2842:
2838:
2813:is called the
2802:
2798:
2794:
2791:
2788:
2784:
2780:
2777:
2772:
2768:
2744:
2741:
2738:
2735:
2732:
2719:is called the
2708:
2704:
2700:
2697:
2694:
2690:
2686:
2683:
2678:
2674:
2603:
2600:
2522:
2519:
2518:
2517:
2507:
2506:
2495:
2478:, the functor
2460:
2442:
2439:
2436:
2433:
2430:
2405:
2374:
2373:
2372:
2371:
2351:
2348:
2345:
2335:tensor product
2311:
2308:
2305:
2300:
2296:
2292:
2289:
2286:
2282:
2279:
2276:
2272:
2269:
2264:
2260:
2256:
2253:
2250:
2246:
2243:
2240:
2236:
2233:
2230:
2227:
2224:
2221:
2217:
2214:
2211:
2190:
2187:
2184:
2169:
2083:
2066:Scott topology
2050:
2023:
2016:
2015:
2014:
1999:
1992:
1924:small category
1916:
1893:
1886:
1792:
1785:
1665:
1662:
1661:
1660:
1649:
1646:
1643:
1635:
1631:
1626:
1621:
1618:
1615:
1612:
1607:
1603:
1599:
1594:
1590:
1571:
1570:
1559:
1556:
1553:
1550:
1547:
1542:
1538:
1534:
1531:
1528:
1525:
1520:
1516:
1512:
1509:
1506:
1500:
1496:
1491:
1488:
1485:
1477:
1473:
1468:
1463:
1460:
1457:
1454:
1451:
1446:
1442:
1438:
1434:
1404:
1399:with codomain
1386:
1373:
1353:
1342:
1341:
1324:
1320:
1316:
1314:
1311:
1309:
1306:
1305:
1302:
1299:
1296:
1293:
1292:
1287:
1283:
1279:
1277:
1274:
1272:
1269:
1266:
1261:
1257:
1253:
1252:
1217:
1214:
1213:
1212:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1169:
1166:
1163:
1160:
1157:
1153:
1127:
1126:
1113:
1109:
1105:
1100:
1096:
1092:
1087:
1083:
1079:
1074:
1071:
1068:
1065:
1062:
1058:
1043:
1042:
1031:
1023:
1020:
1017:
1012:
1009:
1003:
998:
995:
990:
986:
977:
974:
971:
966:
963:
958:
954:
951:
946:
941:
938:
933:
929:
925:
920:
916:
881:
867:
866:
855:
850:
846:
842:
837:
833:
829:
824:
820:
809:
798:
793:
789:
785:
780:
776:
772:
767:
763:
735:
732:
731:
730:
719:
716:
713:
710:
707:
704:
701:
698:
695:
692:
689:
684:
681:
678:
673:
670:
650:
649:
638:
633:
629:
618:
614:
610:
607:
602:
599:
593:
588:
585:
580:
576:
570:
566:
562:
559:
556:
553:
548:
545:
542:
538:
534:
529:
526:
523:
520:
517:
512:
509:
506:
503:
500:
497:
471:
470:
459:
456:
453:
450:
445:
441:
437:
432:
429:
426:
421:
418:
397:
394:
392:
389:
347:
346:
335:
330:
326:
322:
319:
316:
312:
309:
306:
302:
299:
296:
293:
290:
287:
284:
281:
277:
274:
271:
230:
180:
179:
153:
123:
112:if and only if
103:
100:
88:René Descartes
83:
80:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4425:
4414:
4411:
4409:
4406:
4405:
4403:
4388:
4380:
4378:
4370:
4368:
4360:
4359:
4356:
4342:
4339:
4337:
4334:
4332:
4328:
4324:
4320:
4318:
4316:
4309:
4307:
4304:
4302:
4299:
4298:
4296:
4293:
4289:
4279:
4276:
4273:
4269:
4266:
4265:
4263:
4261:
4253:
4247:
4244:
4242:
4239:
4237:
4234:
4232:
4231:Tetracategory
4229:
4227:
4224:
4221:
4220:pseudofunctor
4217:
4214:
4213:
4211:
4209:
4201:
4198:
4194:
4189:
4186:
4184:
4181:
4179:
4176:
4174:
4171:
4169:
4166:
4164:
4161:
4159:
4156:
4154:
4151:
4149:
4145:
4139:
4138:
4135:
4131:
4126:
4122:
4117:
4099:
4096:
4094:
4091:
4089:
4086:
4084:
4081:
4079:
4076:
4074:
4071:
4069:
4066:
4064:
4063:Free category
4061:
4060:
4058:
4054:
4047:
4046:Vector spaces
4043:
4040:
4037:
4033:
4030:
4028:
4025:
4023:
4020:
4018:
4015:
4013:
4010:
4008:
4005:
4004:
4002:
4000:
3996:
3986:
3983:
3981:
3978:
3974:
3971:
3970:
3969:
3966:
3964:
3961:
3959:
3956:
3955:
3953:
3951:
3947:
3941:
3940:Inverse limit
3938:
3936:
3933:
3929:
3926:
3925:
3924:
3921:
3919:
3916:
3914:
3911:
3910:
3908:
3906:
3902:
3899:
3897:
3893:
3887:
3884:
3882:
3879:
3877:
3874:
3872:
3869:
3867:
3866:Kan extension
3864:
3862:
3859:
3857:
3854:
3852:
3849:
3847:
3844:
3842:
3839:
3837:
3834:
3830:
3827:
3825:
3822:
3820:
3817:
3815:
3812:
3810:
3807:
3805:
3802:
3801:
3800:
3797:
3796:
3794:
3790:
3786:
3779:
3775:
3771:
3764:
3759:
3757:
3752:
3750:
3745:
3744:
3741:
3733:
3729:
3724:
3722:
3720:
3715:
3712:
3711:
3707:
3700:
3696:
3692:
3688:
3684:
3680:
3676:
3672:
3667:
3666:
3658:
3654:
3650:
3646:
3639:
3632:
3629:
3624:
3620:
3616:
3612:
3608:
3604:
3597:
3594:
3589:
3585:
3581:
3574:
3571:
3566:
3560:
3557:
3552:
3550:0-444-87508-5
3546:
3542:
3535:
3532:
3529:
3527:
3522:
3517:
3514:
3503:
3499:
3493:
3490:
3485:
3481:
3477:
3471:
3467:
3460:
3457:
3452:
3448:
3444:
3438:
3434:
3430:
3425:
3420:
3415:
3410:
3406:
3399:
3395:
3394:Baez, John C.
3389:
3386:
3380:
3378:
3372:
3369:
3366:
3363:
3361:
3357:
3353:
3351:
3347:
3344:
3342:
3338:
3334:
3330:
3326:
3322:
3318:
3315:
3312:
3308:
3304:
3300:
3296:
3292:
3288:
3286:
3282:
3278:
3274:
3271:
3270:
3269:
3267:
3263:
3259:
3255:
3248:
3244:
3240:
3236:
3234:
3230:
3226:
3222:
3218:
3216:
3212:
3209:
3206:
3203:
3199:
3195:
3192:
3190:
3186:
3182:
3178:
3175:
3173:
3169:
3165:
3161:
3157:
3153:
3150:
3149:
3148:
3142:
3138:
3134:
3133:
3132:
3130:
3126:
3122:
3118:
3114:
3110:
3102:
3100:
3081:
3075:
3072:
3069:
3039:
3033:
3030:
3027:
3000:
2973:
2934:
2928:
2922:
2919:
2916:
2913:
2910:
2903:
2899:
2894:
2874:
2866:
2862:
2853:
2845:
2840:
2836:
2827:
2823:
2818:
2816:
2800:
2796:
2792:
2786:
2782:
2778:
2775:
2770:
2756:
2739:
2733:
2730:
2722:
2721:dependent sum
2706:
2702:
2698:
2692:
2688:
2684:
2681:
2676:
2662:
2660:
2656:
2652:
2648:
2644:
2640:
2636:
2632:
2628:
2623:
2621:
2617:
2613:
2609:
2601:
2599:
2597:
2593:
2589:
2585:
2580:
2578:
2574:
2569:
2567:
2562:
2560:
2556:
2552:
2548:
2544:
2540:
2536:
2532:
2528:
2520:
2515:
2512:
2511:
2510:
2504:
2500:
2496:
2493:
2489:
2485:
2481:
2477:
2473:
2469:
2465:
2461:
2458:
2437:
2431:
2428:
2418:
2414:
2410:
2407:The category
2406:
2403:
2399:
2396:, as well as
2395:
2391:
2387:
2383:
2379:
2378:
2377:
2369:
2365:
2364:right adjoint
2349:
2346:
2343:
2336:
2332:
2328:
2324:
2323:
2309:
2306:
2298:
2294:
2290:
2287:
2270:
2262:
2258:
2254:
2251:
2234:
2228:
2225:
2222:
2188:
2185:
2182:
2174:
2170:
2165:
2161:
2157:
2152:
2148:
2144:
2140:
2136:
2132:
2128:
2124:
2120:
2116:
2112:
2108:
2104:
2100:
2096:
2092:
2088:
2084:
2081:
2080:
2075:
2071:
2067:
2063:
2059:
2055:
2051:
2048:
2044:
2040:
2036:
2032:
2028:
2024:
2021:
2017:
2012:
2008:
2004:
2000:
1997:
1993:
1991:
1987:
1983:
1979:
1978:
1976:
1971:
1967:
1961:
1957:
1953:
1949:
1945:
1941:
1937:
1933:
1929:
1925:
1921:
1917:
1914:
1910:
1906:
1902:
1898:
1895:The category
1894:
1891:
1887:
1884:
1880:
1876:
1872:
1868:
1864:
1860:
1856:
1852:
1848:
1844:
1840:
1836:
1832:
1828:
1824:
1820:
1816:
1812:
1808:
1806:
1801:
1797:
1793:
1790:
1786:
1783:
1779:
1775:
1771:
1767:
1763:
1759:
1755:
1751:
1747:
1743:
1739:
1735:
1731:
1727:
1723:
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237:right adjoint
233:
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196:empty product
193:
192:associativity
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101:
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46:defined on a
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4292:Categorified
4196:n-categories
4147:Key concepts
3985:Direct limit
3968:Coequalizers
3886:Yoneda lemma
3840:
3792:Key concepts
3782:Key concepts
3731:
3718:
3677:(1): 33–48.
3674:
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2416:
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2072:maps). Both
2061:
2054:order theory
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1854:
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86:Named after
85:
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4260:-categories
4236:Kan complex
4226:Tricategory
4208:-categories
4098:Subcategory
3856:Exponential
3824:Preadditive
3819:Pre-abelian
3502:ncatlab.org
2584:John Backus
2173:zero object
1744:defined by
1131:composition
734:Composition
169:exponential
4402:Categories
4278:3-category
4268:2-category
4241:∞-groupoid
4216:Bicategory
3963:Coproducts
3923:Equalizers
3829:Bicategory
3603:J Math Sci
3507:2017-09-17
3475:1441931236
3381:References
3258:coproducts
3117:isomorphic
2577:set theory
2035:continuous
1768:) for all
475:evaluation
396:Evaluation
102:Definition
4327:Symmetric
4272:2-functor
4012:Relations
3935:Pullbacks
3691:1469-8064
3623:122693163
3484:851741862
3451:115169297
3419:CiteSeerX
3414:0903.0340
3066:Π
3024:Σ
2997:Π
2970:Σ
2920:⊢
2867:∗
2850:Π
2846:≅
2790:→
2767:Π
2740:−
2734:∘
2696:→
2673:Σ
2347:⊗
2344:−
2307:≅
2271:≅
2235:≅
2186:≅
2099:open sets
1732:function
1682:functions
1630:π
1617:×
1602:Γ
1598:≅
1537:Γ
1524:
1511:≅
1472:π
1459:×
1450:
1313:→
1301:↓
1295:↓
1276:→
1256:Γ
1194:∘
1104:→
1091:×
994:×
957:×
937:×
924:×
841:→
784:→
584:×
558:≅
552:×
544:×
455:→
449:×
349:which is
301:≅
286:×
253:bijection
205:– ×
118:It has a
82:Etymology
4387:Glossary
4367:Category
4341:n-monoid
4294:concepts
3950:Colimits
3918:Products
3871:Morphism
3814:Concrete
3809:Additive
3799:Category
3699:15115721
2653: :
2629: :
2559:currying
2545: :
2529: :
2482: :
2470: :
2151:interior
2074:currying
1817:are two
1736: :
1716: :
1664:Examples
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1467:→
1409: :
1352:. Then Γ
1216:Sections
1002:→
945:→
892:include
877:include
742: :
592:→
257:hom-sets
235:) has a
225: ×
167:have an
144: ×
44:morphism
36:category
4377:Outline
4336:n-group
4301:2-group
4256:Strict
4246:∞-topos
4042:Modules
3980:Pushout
3928:Kernels
3861:Functor
3804:Abelian
3716:at the
3523:at the
3260:and an
3111:) and (
2453:
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2149:is the
2091:lattice
1730:curried
1680:, with
1676:of all
351:natural
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203:functor
139:product
137:have a
76:quantum
62:is the
52:objects
50:of two
48:product
4323:Traced
4306:2-ring
4036:Fields
4022:Groups
4017:Magmas
3905:Limits
3697:
3689:
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3439:
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3367:×0 = 0
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2386:FinSet
1789:finite
1700:, and
361:, and
4317:-ring
4204:Weak
4188:Topos
4032:Rings
3695:S2CID
3641:(PDF)
3619:S2CID
3447:S2CID
3409:arXiv
3401:(PDF)
3207:1 = 1
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2900:as a
2573:topoi
2127:poset
2079:apply
2033:with
2020:topos
1922:is a
1833:with
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1800:group
1798:is a
1133:map.
4007:Sets
3687:ISSN
3545:ISBN
3480:OCLC
3470:ISBN
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3127:and
3057:and
2988:and
2606:Let
2596:CAML
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159:and
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3851:End
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