1165:
503:
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400:, and higher-level generalizations are increasingly harder to define explicitly. Several definitions have been given, and telling when they are equivalent, and in what sense, has become a new object of study in category theory.
308:, where "weak" structures arise in the form of higher categories, strict cubical higher homotopy groupoids have also arisen as giving a new foundation for algebraic topology on the border between
341:, the associativity and identity conditions are no longer strict (that is, they are not given by equalities), but rather are satisfied up to an isomorphism of the next level. An example in
470:
Topologically enriched categories (sometimes simply called topological categories) are categories enriched over some convenient category of topological spaces, e.g. the category of
456:) do not agree with the corresponding notions in the sense of enriched categories. The same for other enriched models like topologically enriched categories.
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Simplicially enriched categories, or simplicial categories, are categories enriched over simplicial sets. However, when we look at them as a model for
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These are models of higher categories introduced by
Hirschowitz and Simpson in 1998, partly inspired by results of Graeme Segal in 1974.
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showed that they are a good foundation for higher category theory. Recently, in 2009, the theory has been systematized further by
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721:, the collective and open wiki notebook project on higher category theory and applications in physics, mathematics and philosophy
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in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in
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who simply calls them infinity categories, though the latter term is also a generic term for all models of (infinity,
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96:. This approach is particularly valuable when dealing with spaces with intricate topological features, such as the
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727:, a wiki dedicated to polished expositions of categorical and higher categorical mathematics with proofs
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Hirschowitz, André; Simpson, Carlos (2001). "Descente pour les n-champs (Descent for n-stacks)".
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Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids
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In higher category theory, the concept of higher categorical structures, such as (
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738:. Tracts in Mathematics. Vol. 15. European Mathematical Society.
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While this concept is too strict for some purposes in for example,
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can be given a monoidal structure. The recursive construction of
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and expressing this is the difficulty in the definition of weak
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that have the same fundamental group but differ in their higher
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775:— a group blog devoted to higher category theory.
658:(2010). "Homotopy theory of higher categories".
418:satisfying a weak version of the Kan condition.
301:-enriched categories has finite products too.
1462:
803:
734:; Higgins, Philip J.; Sivera, Rafael (2011).
414:Weak Kan complexes, or quasi-categories, are
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124:in the context of higher category theory. A
611:; Dolan, James (1998). "Categorification".
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279:as unit. In fact any category with finite
781:"A Perspective on Higher Category Theory"
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84:), allows for a more robust treatment of
234: + 1)-category is a category
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452:, then many categorical notions (e.g.,
369:-isomorphisms must well behave between
297:has finite products, the category of
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149: − 1)-morphisms gives an
130:generalizes this by also including
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1547:List of mathematical logic topics
628:Higher Operads, Higher Categories
460:Topologically enriched categories
293:works fine because if a category
27:Generalization of category theory
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1420:
1411:
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501:
438:Simplicially enriched categories
320:, referenced in the book below.
760:Baez, John (24 February 1996).
672:Alternative PDF with hyperlinks
58:), where one studies algebraic
1716:List of category theory topics
779:Leinster, Tom (8 March 2010).
689:. Princeton University Press.
630:. Cambridge University Press.
450:(infinity, 1)-categories
444:Simplicially enriched category
159:Just as the category known as
1:
318:Nonabelian algebraic topology
197:-categories is actually an (
165:category of small categories
1711:Glossary of category theory
1585:Zermelo–Fraenkel set theory
1537:Mathematical constructivism
1105:Constructions on categories
532:Coherency (homotopy theory)
211:is defined by induction on
201: + 1)-category.
1758:
1737:Foundations of mathematics
1706:Mathematical structuralism
1643:Intuitionistic type theory
1479:Foundations of Mathematics
1212:Higher-dimensional algebra
517:Higher-dimensional algebra
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463:
441:
407:
327:
1610:List of set theory topics
1406:
1185:
1172:
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522:General abstract nonsense
271:is the one given by the
138:. Continuing this up to
104:Strict higher categories
1590:Constructive set theory
1022:Cokernels and quotients
945:Universal constructions
177:natural transformations
98:Eilenberg-MacLane space
1742:Higher category theory
1691:Higher category theory
1595:Descriptive set theory
1500:Mathematical induction
1179:Higher category theory
925:Natural transformation
626:Leinster, Tom (2004).
345:is the composition of
324:Weak higher categories
36:higher category theory
1653:Univalent foundations
1638:Dependent type theory
1628:Axiom of reducibility
571:Baez & Dolan 1998
430:) categories for any
1648:Homotopy type theory
1575:Axiomatic set theory
1048:Algebraic categories
466:Topological category
1217:Homotopy hypothesis
895:Commutative diagram
773:The n-Category Cafe
686:Higher Topos Theory
554:Higher Topos Theory
472:compactly generated
120:, which are called
1633:Simple type theory
1580:Zermelo set theory
1527:Mathematical proof
1487:Mathematical logic
930:Universal property
762:"Week 73: Tale of
509:Mathematics portal
353:, and hence up to
351:reparameterization
316:; see the article
238:over the category
90:topological spaces
52:algebraic topology
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1605:Russell's paradox
1520:Natural deduction
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1380:monoidal category
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1207:Enriched category
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1132:Quotient category
1127:Opposite category
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745:978-3-03719-083-8
706:978-0-691-14048-3
670:Draft of a book.
560:. MIT. p. 4.
390:monoidal category
273:cartesian product
16:(Redirected from
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1686:Category of sets
1658:Girard's paradox
1570:Naive set theory
1510:Axiomatic system
1477:Major topics in
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1122:Kleisli category
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475:Hausdorff spaces
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54:(especially in
40:category theory
38:is the part of
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487:Segal category
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410:Quasi-category
408:Main article:
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1196:Key concepts
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1034:Direct limit
1017:Coequalizers
935:Yoneda lemma
841:Key concepts
831:Key concepts
766:-Categories"
763:
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681:Lurie, Jacob
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618:math/9802029
590:math/9807049
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394:3-categories
386:bicategories
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108:An ordinary
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81:∞-categories
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44:higher order
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29:
1620:Type theory
1600:Determinacy
1542:Modal logic
1309:-categories
1285:Kan complex
1275:Tricategory
1257:-categories
1147:Subcategory
905:Exponential
873:Preadditive
868:Pre-abelian
573:, p. 6
424:Jacob Lurie
420:André Joyal
378:-categories
339:-categories
256:is just a (
193:of (small)
181:2-morphisms
136:1-morphisms
132:2-morphisms
122:1-morphisms
68:fundamental
32:mathematics
1731:Categories
1696:∞-groupoid
1557:Set theory
1327:3-category
1317:2-category
1290:∞-groupoid
1265:Bicategory
1012:Coproducts
972:Equalizers
878:Bicategory
602:References
363:2-category
254:1-category
221:0-category
173:2-category
143:-morphisms
127:2-category
60:invariants
18:3-category
1376:Symmetric
1321:2-functor
1061:Relations
984:Pullbacks
665:1001.4071
361:for this
277:singleton
209:-category
154:-category
145:between (
118:morphisms
1676:Category
1436:Glossary
1416:Category
1390:n-monoid
1343:concepts
999:Colimits
967:Products
920:Morphism
863:Concrete
858:Additive
848:Category
683:(2009).
495:See also
371:hom-sets
365:. These
355:homotopy
343:topology
334:In weak
310:homology
281:products
265:monoidal
236:enriched
169:functors
110:category
1426:Outline
1385:n-group
1350:2-group
1305:Strict
1295:∞-topos
1091:Modules
1029:Pushout
977:Kernels
910:Functor
853:Abelian
380:. Weak
179:as its
114:objects
1372:Traced
1355:2-ring
1085:Fields
1071:Groups
1066:Magmas
954:Limits
742:
703:
644:
454:limits
64:spaces
48:arrows
1366:-ring
1253:Weak
1237:Topos
1081:Rings
691:arXiv
660:arXiv
632:arXiv
613:arXiv
585:arXiv
558:(PDF)
538:Notes
347:paths
252:So a
223:is a
175:with
42:at a
1056:Sets
740:ISBN
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701:ISBN
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312:and
263:The
230:An (
215:by:
167:and
116:and
112:has
1565:Set
900:End
890:CCC
713:PDF
711:As
290:Cat
269:Set
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