Einige Bemerkungen über Graphen, Kategorien und Gruppoide

Hasse, M.

doi:10.1002/mana.19600220502

Cited by 18 publications

(7 citation statements)

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“…) A (B (4 = a ( 6 ) ) ) = {x € (7:( a (2) = 6 (B( a ) ) ) A (B (x) = a (a ( b ) ) ) ) = H (B(a), @ . ( b ) ) -Bemerkungen: Die nicht-leeren Klassen H (e, e') (e, e' E Go) sind paarweise elementefremd, und ihre Vereinigung stimmt mit G uberein.…”

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Zur algebraischen Begründung der Graphentheorie. I

Hasse

1965

Mathematische Nachrichten

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I n der Theorie der Kategorien spielen Graphen als Erzeugendensysteme allgemeiner Strukturen eine grundlegende Rolle. Die bisherigen Definitionen eines Graphen erweisen sich jedoch als in einer fur die Aufdeckung dieser Zusammenhange ungeeigneten Sprache formuliert. Man wird versuchen, den Begriff des Graphen in enger Anlehnung an denjenigen der Kategorie (im Sinne von EHRESMANN [I]) algebraisch zu fassen. In [ 21 wurden derartige Definitionen fur die verschiedenen Typen von Graphen gegeben und bereits einige Zusammenhange zwischen Graphen und Kategorien dargelegt. Ziel dieser und folgender Arbeiten ist es, zunachst die in [2] angegebenen Definitionen und Begriffe zu erweitern und zu prazisieren und danach, auf diesen aufbauend, eine Reihe von Aussagen uber Graphen herzuleiten, die in unmittelbarer Beziehung zur Theorie der Kategorien stehen. Definitionen gerichteter und ungerichteter GraphenUblicherweise liegt dem Begriff des Graphen der Mengenbegriff zugrunde. Das bedeutet eine Beschrankung im Hinblick auf diejenigen Strukturen, die uber Graphen konstruiert werden. Wir werden daher, um von vornherein die volle Allgemeinheit zu wahren, unter dem Trager eines Graphen stets eine Klasse verstehen. Definition 1 : Ein gerichteter Graph G = ( G ; a , B) ist eine algebraische Struktur vom Typ (1, 1 ) uber dem Triiger G mit den beiden vollstandigen einstelligen VetknZi.pfungen a und b. Babei genugen a und , 3 den folgenden Bedingungen : --c fc (a (4) = fl (a (4) = a (4; a (B (4) = B (B (4) = B (4 (x €GI.Bemerkungen: Jedes Element e aus 01 (G) w ( C ) heil3t ein Eckpulzkt von Nach Definition sind jedem Element x aus G eindeutig zwei Eckpunkte zugeordnet, sein linker Eckpunkt (Anfangspunkt) a (x) und sein i 9*

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Zur algebraischen Begründung der Graphentheorie. I

Hasse

1965

Mathematische Nachrichten

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“…Higgins' 1964 paper [38] applied groupoids to group theory, independently of earlier work of Hasse [35]. One main tool was the notion of covering morphism of groupoids, motivated by the notion of covering map of spaces.…”

Section: Change Of Base For Groupoidsmentioning

confidence: 99%

Homotopy theory, and change of base for groupoids and multiple groupoids

Brown

1996

Appl Categor Struct

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“…., j, = k of vertices we have q € X j , -j f , if a, = + 1, x, E XjVj,-, if a, = -1, and (iii) if x, = z,.-~ then a, = I n addition Fjj contains the "empty word" t i . This construction has been given by M. HASSE in [5].…”

Section: Corollary If V Is a Variety Of Z-algebras Then Fu ( X ) Eximentioning

confidence: 99%

“…I n the more important special cases, however, this difficulty does not appear. For example, the term free category has an obvious meaning; it is the category of composite arrows generated by a scheme of arrows (directed graph) aa described in [4], p. 130, and [5]. One may also impose "relations of commutativity" on the arrows and thus present categories by means of generators and relations.…”

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Algebras with a Scheme of Operators

Higgins

1963

Mathematische Nachrichten

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IntroductionThere are many important algebraic systems which do not fit conveniently into the theory of abstract algebras (that is, algebras whose operations are everywhere defined). In the fist place there are various types of groupoids, categories and ringoids, which have often been treated as partial algebras, that is, as sets with operations not everywhere defined. The general theory of partial algebras has been developed t o some extent, but its content does not compare favourably with that of the theory of "full" abstract algebras. One of its many drawbacks is the obvious difficulty of defining a free partial algebra with given operators on given generators in the absence of some rule for determining the domains of the operators. I n the more important special cases, however, this difficulty does not appear. For example, the term free category has an obvious meaning; it is the category of composite arrows generated by a scheme of arrows (directed graph) aa described in [4], p. 130, and [5]. One may also impose "relations of commutativity" on the arrows and thus present categories by means of generators and relations. The theory of partial algebras seems to be unsuitable for a discussion of such constructions.As a second example we may consider some familiar graded slgebraic structures. A graded abelian group G = @ G, can, of course, be considered as a full abstract algebra, but then the general theory will be concerned with all its subgroups and homomorphisms rather than just the graded ones, and there is no easy way of deducing "graded" theorems from the general ones. It is now becoming more usual to treat a graded group (as in [3]) as a family of groups {GJ. A graded subgroup is then a family of subgroups and a graded homomorphism is a family of group homomorphisms. This is certainly more natural from a general standpoint. In the same way a graded ring becomes a family {RJ of abelian groups together with a collection of multiplications p,, : R, X R, --f R,+, .I n this paper we suggest a general theory which is wide enough in scope to deal with all the systems mentioned above (and many others) but which 8.

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Einige Bemerkungen über Graphen, Kategorien und Gruppoide

Cited by 18 publications

References 3 publications

Zur algebraischen Begründung der Graphentheorie. I

Zur algebraischen Begründung der Graphentheorie. I

Homotopy theory, and change of base for groupoids and multiple groupoids

Algebras with a Scheme of Operators

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