Komutativni dijagram

Komutativni dijagram koji se koristi u dokazu pet lema.

U matematici, i posebno u teoriji kategorija, komutativni dijagram je takav dijagram da svi usmereni putevi u dijagramu sa istim početnim i krajnjim tačkama vode do istog rezultata. Komutativni dijagrami igraju ulogu u teoriji kategorija ekvivalentnu ulozi jednačina u algebri.

Opis

A commutative diagram often consists of three parts:

• objects (also known as vertices)
• morphisms (also known as arrows or edges)
• paths or composites

Simboli strelica

In algebra texts, the type of morphism can be denoted with different arrow usages:

• A monomorphism (injective homomorphism) may be labeled with a $\hookrightarrow$.
• An epimorphism (surjective homomorphism) may be labelled with a $\twoheadrightarrow$.
• An isomorphism (bijective homomorphism) may be labelled with a ${\overset {\sim }{\rightarrow }}$.
• The dashed arrow typically represents the claim that the indicated morphism exists (whenever the rest of the diagram holds); the arrow may be optionally labeled as $\exists$.
  • If the morphism is in addition unique, then the dashed arrow may be labeled $!$ or $\exists !$.

These conventions are common enough that texts often do not explain the meanings of the different types of arrow.

Provera komutativnosti

Commutativity makes sense for a polygon of any finite number of sides (including just 1 or 2), and a diagram is commutative if every polygonal subdiagram is commutative.

Note that a diagram may be non-commutative, i.e., the composition of different paths in the diagram may not give the same result.

Primeri

U levom dijagramu, koji izražava prvu teoremu izomorfizma, komutativnost trougla znači da je $f={\tilde {f}}\circ \pi$. U desnom dijagramu komutativnost kvadrata znači $h\circ f=k\circ g$.

Da bi dijagram ispod bio komutativan, moraju biti zadovoljene tri jednakosti:

1. $r\circ h\circ g=H\circ G\circ l$
2. $m\circ g=G\circ l$
3. $r\circ h=H\circ m$

Ovde, pošto prva jednakost sledi iz zadnje dve, dovoljno je pokazati da su (2) i (3) istinite da bi dijagram bio komutativan. Međutim, pošto jednakost (3) generalno ne proizilazi iz druge dve, u opštem slučaju nije dovoljno imati samo jednakosti (1) i (2) da bi se pokazalo da je dijagram komutativan.

Praćenje dijagrama

Praćenje dijagrama (koja se takođe naziva dijagramskom pretragom) je metoda matematičkog dokaza koja se naročito koristi u homološkoj algebri, gde se uspostavlja svojstvo nekog morfizma pronalazeći elemente komutativnog dijagrama. Dokaz pomoću dijagrama obično uključuje formalnu upotrebu svojstava dijagrama, kao što su injektivne ili surjektivne mape, ili tačne sekvence. Konstruiše se silogizam za koji je grafički prikaz dijagrama samo vizuelno pomagalo. Iz toga sledi da se vrši „pretraga” elemenata na dijagramu, sve dok se ne konstruiše ili potvrdi željeni element ili rezultat.

Primeri dokaza pomoću dijagramskog praćenja uključuju one koji se obično daju za pet lema, zmijsku lemu, zig-zag lemu, i devet lema.

U višoj teoriji kategorija

Glavni članak: Viša teorija kategorija

In higher category theory, one considers not only objects and arrows, but arrows between the arrows, arrows between arrows between arrows, and so on ad infinitum. For example, the category of small categories Cat is naturally a 2-category, with functors as its arrows and natural transformations as the arrows between functors. In this setting, commutative diagrams may include these higher arrows as well, which are often depicted in the following style: $\Rightarrow$. For example, the following (somewhat trivial) diagram depicts two categories C and D, together with two functors F, G : C → D and a natural transformation α : F ⇒ G:

There are two kinds of composition in a 2-category (called vertical composition and horizontal composition), and they may also be depicted via pasting diagrams (see 2-category#Definition for examples).

Dijagrami kao funktori

Glavni članak: Dijagram (teorija kategorija)

Komutativni dijagram u kategoriji C can be interpreted as a functor from an index category J to C; one calls the functor a diagram.

More formally, a commutative diagram is a visualization of a diagram indexed by a poset category. Such a diagram typically include:

• a node for every object in the index category,
• an arrow for a generating set of morphisms (omitting identity maps and morphisms that can be expressed as compositions),
• the commutativity of the diagram (the equality of different compositions of maps between two objects), corresponding to the uniqueness of a map between two objects in a poset category.

Conversely, given a commutative diagram, it defines a poset category, where:

• the objects are the nodes,
• there is a morphism between any two objects if and only if there is a (directed) path between the nodes,
• with the relation that this morphism is unique (any composition of maps is defined by its domain and target: this is the commutativity axiom).

However, not every diagram commutes (the notion of diagram strictly generalizes commutative diagram). As a simple example, the diagram of a single object with an endomorphism ($f\colon X\to X$), or with two parallel arrows ($\bullet \rightrightarrows \bullet$, that is, $f,g\colon X\to Y$, sometimes called the free quiver), as used in the definition of equalizer need not commute. Further, diagrams may be messy or impossible to draw, when the number of objects or morphisms is large (or even infinite).

Vidi još

• Matematički dijagram

Literatura

• Adámek, Jiří; Horst Herrlich; George E. Strecker (1990), Abstract and Concrete Categories (PDF), John Wiley & Sons, ISBN 0-471-60922-6  Now available as free on-line edition (4.2MB PDF).
• Barr, Michael; Wells, Charles (2002), Toposes, Triples and Theories (PDF), ISBN 0-387-96115-1  Revised and corrected free online version of Grundlehren der mathematischen Wissenschaften (278) Springer-Verlag, 1983).
• Barker-Plummer, Dave; Bailin, Sidney C. (1997). „The Role of Diagrams in Mathematical Proofs”. Machine Graphics and Vision. 6 (1): 25—56. 10.1.1.49.4712.  (Special Issue on Diagrammatic Representation and Reasoning).
• Barker-Plummer, Dave; Bailin, Sidney C. (2001). „On the practical semantics of mathematical diagrams”. Ур.: Anderson, M. Reasoning with Diagrammatic Representations. Springer Verlag. ISBN 978-1-85233-242-6. CiteSeerX: 10.1.1.30.9246. 
• Kidman, G. (2002). „The Accuracy of mathematical diagrams in curriculum materials”. Ур.: Cockburn, A.; Nardi, E. Proceedings of the PME 26. 3. University of East Anglia. стр. 201—8. 
• Kulpa, Zenon (2004). „On Diagrammatic Representation of Mathematical Knowledge”. Ур.: Andréa Asperti; Bancerek, Grzegorz; Trybulec, Andrzej. Mathematical knowledge management: third international conference, MKM 2004, Białowieża, Poland, September 19–21, 2004 : Proceedings. Springer. стр. 191—204. ISBN 978-3-540-23029-8. 
• Puphaiboon, K.; Woodcock, A.; Scrivener, S. (25. 3. 2005). „Design method for developing mathematical diagrams”. Ур.: Bust, Philip D.; McCabe, P.T. Contemporary ergonomics 2005 Proceedings of the International Conference on Contemporary Ergonomics (CE2005). Taylor & Francis. ISBN 978-0-415-37448-4. CS1 одржавање: Формат датума (веза)

Spoljašnje veze

• Diagram Chasing at MathWorld
• WildCats is a category theory package for Mathematica. Manipulation and visualization of objects, morphisms, categories, functors, natural transformations.
• „Diagrams”. The Stanford Encyclopedia of Philosophy. јесен 2008. CS1 одржавање: Формат датума (веза)
• Kulpa, Zenon. „Diagrammatics: The art of thinking with diagrams”. Архивирано из оригинала 25. 4. 2013. г. CS1 одржавање: Формат датума (веза)
• One of the oldest extant diagrams from Euclid by Otto Neugebauer
• Lomas, Dennis (1998). „Diagrams in Mathematical Education: A Philosophical Appraisal”. Philosophy of Education Society. Архивирано из оригинала 2011. г.