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702:, if it preserves finite limits and coequalizers of kernel pairs. A functor is regular if and only if it preserves finite limits and exact sequences. For this reason, regular functors are sometimes called
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Up to equivalence any small regular category arises in this way as the classifying category of some regular theory.
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regular categories and regular functors to small categories. It is an important result that for each theory
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that satisfies the interpretations of the conditions for reflexivity, symmetry and transitivity.
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1473:"A note on the exact completion of a regular category, and its infinitary generalizations"
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586:{\displaystyle R\;{\overset {r}{\underset {s}{\rightrightarrows }}}\;X\xrightarrow {f} Y}
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if it is both a coequalizer and a kernel pair. The terminology is a generalization of
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Categorical foundations. Special topics in order, topology, algebra, and sheaf theory
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684:{\displaystyle 0\to R{\xrightarrow {(r,s)}}X\oplus X{\xrightarrow {(f,-f)}}Y\to 0}
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1504:. Encyclopedia of Mathematics and Its Applications. Vol. 97. Cambridge:
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is an epimorphism that appears as a coequalizer of some pair of morphisms.
1307:. Every equivalence relation has a coequalizer, which is found by taking
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conditions. In that way, regular categories recapture many properties of
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1093:{\displaystyle \mathbf {Mod} (T,C)\cong \mathbf {RegCat} (R(T),C)}
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1388:
225:. Being a pullback, the kernel pair is unique up to a unique
18:
706:. Functors that preserve finite limits are often said to be
1131:
is a regular theory. An equivalence relation on an object
1435:. Lecture Notes in Mathematics. Vol. 236. Springer.
16:
Mathematical category with finite limits and coequalizers
1500:
Pedicchio, Maria
Cristina; Tholen, Walter, eds. (2004).
1495:. University of Aarhus. BRICS Lectures Series LS-95-1.
1429:; Grillet, Pierre A.; van Osdol, Donovan H. (2006) .
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1264:. Conversely, an equivalence relation is said to be
442:. The factorization is unique in the sense that if
1303:is exact in this sense, and so is any (elementary)
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942:. This gives for each theory (set of sequents)
133:if it satisfies the following three properties:
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895:{\displaystyle \forall x(\phi (x)\to \psi (x))}
776:{\displaystyle \forall x(\phi (x)\to \psi (x))}
698:A functor between regular categories is called
344:More generally, the category of models of any
1151:of a regular category is a monomorphism into
489:In a regular category, a diagram of the form
8:
354:, with morphisms given by the order relation
1456:. Vol. 2. Cambridge University Press.
1432:Exact Categories and Categories of Sheaves
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514:{\displaystyle R\rightrightarrows X\to Y}
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69:Learn how and when to remove this message
1288:exact categories in the sense of Quillen
722:that can express statements of the form
293:Examples of regular categories include:
32:This article includes a list of general
1364:
1389:"Regular Categories and Regular Logic"
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1257:{\displaystyle R\rightarrow X\times X}
998:, such that for each regular category
922:factors through the interpretation of
596:is exact in this sense if and only if
1477:Theory and Applications of Categories
966:. This construction gives a functor
280:is a regular epimorphism as well. A
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714:Regular logic and regular categories
485:Exact sequences and regular functors
1325:over the category of sets is exact.
446:is another regular epimorphism and
408:In a regular category, the regular-
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450:is another monomorphism such that
38:it lacks sufficient corresponding
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1271:A regular category is said to be
718:Regular logic is the fragment of
424:can be factorized into a regular
313:More generally, every elementary
1391:. BRICS Lectures Series LS-98-2.
1232:defines an equivalence relation
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1542:Categories in category theory
835:, the truth constant, binary
364:The following categories are
1294:Examples of exact categories
1123:Exact (effective) categories
994:there is a regular category
831:i.e. formulae built up from
1414:Pedicchio & Tholen 2004
1402:Pedicchio & Tholen 2004
1372:Pedicchio & Tholen 2004
902:, if the interpretation of
1558:
1506:Cambridge University Press
1343:Allegory (category theory)
1332:is regular, but not exact.
841:existential quantification
189:, then the coequalizer of
117:, known as regular logic.
1488:van Oosten, Jaap (1995).
1450:Borceux, Francis (1994).
1170:{\displaystyle X\times X}
352:bounded meet-semilattice
109:, like the existence of
1490:"Basic Category Theory"
1321:Every category that is
454:, then there exists an
53:more precise citations.
1471:Lack, Stephen (1999).
1387:Butz, Carsten (1998).
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469:. The monomorphism
418:factorization system
383:continuous functions
1309:equivalence classes
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531:homological algebra
330:group homomorphisms
282:regular epimorphism
203:exists. The pair (
89:is a category with
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379:topological spaces
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340:ring homomorphisms
324:, the category of
302:, the category of
246:is a morphism in
107:abelian categories
1442:978-3-540-36999-8
1284:effective regular
1144:{\displaystyle X}
958:,C) of models of
720:first-order logic
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142:finitely complete
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1427:Barr, Michael
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433:monomorphism
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99:kernel pairs
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95:coequalizers
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1483:(3): 70ā80.
1113:classifying
1004:equivalence
950:a category
456:isomorphism
426:epimorphism
274:epimorphism
227:isomorphism
219:kernel pair
125:A category
51:introducing
1524:1034.18001
1359:References
708:left exact
461:such that
438:, so that
129:is called
121:Definition
34:references
1318:is exact.
1266:effective
1249:×
1243:→
1217:→
1162:×
1107:. Here,
1044:≅
930:ψ
910:ϕ
878:ψ
875:→
863:ϕ
854:∀
815:ψ
795:ϕ
759:ψ
756:→
744:ϕ
735:∀
676:→
661:−
640:⊕
607:→
555:⇉
506:→
500:⇉
368:regular:
308:functions
103:exactness
1536:Category
1337:See also
829:formulae
648:→
615:→
572:→
533:: in an
444:e':XāE'
412:and the
398:functors
289:Examples
187:pullback
161:morphism
1323:monadic
845:sequent
700:regular
459:h:EāE'
448:m':E'āY
416:form a
346:variety
276:, then
210:,
131:regular
47:improve
1522:
1512:
1460:
1439:
1314:Every
984:RegCat
976:RegCat
787:where
463:he=e'
452:f=m'e'
357:Every
350:Every
326:groups
111:images
36:, but
1493:(PDF)
1348:Topos
1305:topos
1282:, or
1275:, or
1273:exact
988:small
837:meets
691:is a
475:image
467:m'h=m
436:m:EāY
429:e:XāE
422:f:XāY
336:rings
315:topos
250:, and
185:is a
167:, and
159:is a
1510:ISBN
1458:ISBN
1437:ISBN
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1279:Barr
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465:and
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396:and
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304:sets
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389:Cat
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366:not
321:Grp
299:Set
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144:.
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