91:
36:
1072:) need not commute. Further, diagrams may be impossible to draw (because they are infinite) or simply messy (because there are too many objects or morphisms); however, schematic commutative diagrams (for subcategories of the index category, or with ellipses, such as for a directed system) are used to clarify such complex diagrams.
968:
with few elements: one draws a commutative diagram with a node for every object in the index category, and an arrow for a generating set of morphisms, omitting identity maps and morphisms that can be expressed as compositions. The commutativity corresponds to the uniqueness of a map between two
300:, the change in terminology reflects a change in perspective, just as in the set theoretic case: one fixes the index category, and allows the functor (and, secondarily, the target category) to vary.
571:, one discovers additional structure in constructions built from the diagram, structure that would not be evident if one only had an index set with no relations between the objects in the index.
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638:
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447:
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551:, and the colimit would simply be the binary coproduct. Thus, this example shows an important way in which the idea of the diagram generalizes that of the
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54:
972:
Not every diagram commutes, as not every index category is a poset category: most simply, the diagram of a single object with an endomorphism
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objects in a poset category. Conversely, every commutative diagram represents a diagram (a functor from a poset index category) in this way.
220:
116:; the article must starts with this, and explain why the content of the article is a formalization of this representation.
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878:. That is, a cone through which all other cones uniquely factor. If the limit exists in a category
184:. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a
105:
50:
1150:
112:, especially: For most readers, diagrams are graphical representations such as those presented in
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398:. Notationally, one often uses an underbar to denote the constant diagram: thus, for any object
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is the discrete category with two objects, the resulting limit is just the binary product.
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are largely irrelevant; only the way in which they are interrelated matters. The diagram
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is the limit and its left adjoint is the colimit. A cone can be thought of as a
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Although, technically, there is no difference between an individual
164:. The primary difference is that in the categorical setting one has
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is thought of as indexing a collection of objects and morphisms in
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100:
provides insufficient context for those unfamiliar with the subject
1145:
Sheaves in geometry and logic a first introduction to topos theory
843:. The constant diagram is the diagram which sends every object of
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903:
224:
172:
is a collection of sets, indexed by a fixed set; equivalently, a
598:
1238:
303:
One is most often interested in the case where the scheme
1174:
Adámek, Jiří; Horst
Herrlich; George E. Strecker (1990).
27:
Indexed collection of objects and morphisms in a category
960:
Diagrams and functor categories are often visualized by
523:. If one were to "forget" that the diagram had object
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946:
from the diagonal functor to some arbitrary diagram.
673:
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640:
is called "two parallel morphisms", or sometimes the
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428:
404:
361:, and a diagram is then an object in this category.
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45:
may be too technical for most readers to understand
1195:Now available as free on-line edition (4.2MB PDF).
1064:
1027:{\displaystyle \bullet \rightrightarrows \bullet }
1026:
998:
703:
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964:, particularly if the index category is a finite
914:. If the colimit exists for all diagrams of type
1287:
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382:, which is the diagram that maps all objects in
855:and every morphism to the identity morphism on
1270:. Manipulation and visualization of objects,
1227:Revised and corrected free online version of
342:between functors. One can then interpret the
798:of objects and morphisms. If the diagram is
1141:Mac Lane, Saunders; Moerdijk, Ieke (1992).
1120:. University of Chicago Press. p. 16.
934:The universal functor of a diagram is the
835:is a morphism from the constant diagram Δ(
555:in set theory: by including the morphisms
1198:
931:which sends each diagram to its colimit.
476:). When used in the construction of the
132:Learn how and when to remove this message
73:Learn how and when to remove this message
57:, without removing the technical details.
949:
899:which sends each diagram to its limit.
14:
1288:
1149:. New York: Springer-Verlag. pp.
1114:A Concise Course in Algebraic Topology
633:{\displaystyle J=0\rightrightarrows 1}
581:= −1 → 0 ← +1, then a diagram of type
499:= −1 ← 0 → +1, then a diagram of type
265:. The actual objects and morphisms in
110:providing more context for the reader
55:make it understandable to non-experts
1274:, commutative diagrams, categories,
258:; the functor is sometimes called a
168:that also need indexing. An indexed
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29:
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24:
704:{\displaystyle (f,g\colon X\to Y)}
315:category. A diagram is said to be
156:is the categorical analogue of an
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1266:is a category theory package for
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1177:Abstract and Concrete Categories
1065:{\displaystyle f,g\colon X\to Y}
747:together with a unique morphism
484:; for the colimit, one gets the
442:{\displaystyle {\underline {A}}}
89:
34:
422:, one has the constant diagram
330:A morphism of diagrams of type
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999:{\displaystyle f\colon X\to X}
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1209:Toposes, Triples and Theories
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1008:or with two parallel arrows (
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394:to the identity morphism on
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1183:. John Wiley & Sons.
910:is a universal cone from
882:for all diagrams of type
734:, then a diagram of type
460:, then a diagram of type
488:. So, for example, when
1280:natural transformations
1231:Springer-Verlag, 1983).
790:then a diagram of type
738:is a family of objects
719:, and its colimit is a
519:, and its colimit is a
464:is essentially just an
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944:natural transformation
886:one obtains a functor
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802:then it is called an
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962:commutative diagrams
950:Commutative diagrams
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648:. A diagram of type
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480:, the result is the
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344:category of diagrams
1111:May, J. P. (1999).
956:Commutative diagram
527:and the two arrows
188:from a fixed index
176:from a fixed index
114:commutative diagram
106:improve the article
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918:one has a functor
715:; its limit is an
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661:{\displaystyle J}
577:to the above, if
458:discrete category
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468:of objects in
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104:Please help
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1268:Mathematica
906:of diagram
889:lim :
721:coequalizer
642:free quiver
150:mathematics
1098:References
711:is then a
608:The index
200:Definition
162:set theory
1272:morphisms
1259:MathWorld
1057:→
1051::
1022:∙
1019:⇉
1016:∙
991:→
985::
774:whenever
717:equalizer
693:→
687::
625:⇉
553:index set
486:coproduct
435:_
323:whenever
221:covariant
166:morphisms
122:June 2023
63:June 2023
1296:Functors
1290:Category
1276:functors
1264:WildCats
1206:(2002).
1076:See also
925:→
893:→
827: :
788:directed
756: :
603:pullback
365:Examples
346:of type
311:or even
298:category
236:→
232: :
214:category
208:of type
194:category
192:to some
190:category
174:function
1237:at the
1235:diagram
904:colimit
644:or the
597:) is a
521:pushout
515:) is a
482:product
354:as the
290:functor
286:diagram
250:or the
225:functor
206:diagram
186:functor
154:diagram
49:Please
1219:
1187:
1157:
1124:
938:; its
713:quiver
599:cospan
321:finite
313:finite
296:and a
294:scheme
288:and a
252:scheme
219:is a (
1213:(PDF)
1181:(PDF)
1153:–23.
1118:(PDF)
870:is a
864:limit
839:) to
782:. If
730:is a
478:limit
338:is a
317:small
309:small
307:is a
212:in a
1217:ISBN
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1155:ISBN
1122:ISBN
862:The
817:cone
575:Dual
547:and
517:span
327:is.
182:sets
152:, a
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495:If
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418:in
386:to
374:in
350:in
319:or
178:set
160:in
144:In
108:by
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778:≤
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