Diagram (category theory)

Incategory theory,a branch ofmathematics,adiagramis the categorical analogue of anindexed familyinset theory.The primary difference is that in the categorical setting one hasmorphismsthat also need indexing. An indexedfamily of setsis a collection of sets, indexed by a fixed set; equivalently, afunctionfrom a fixed indexsetto the class ofsets.A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, afunctorfrom a fixed indexcategoryto somecategory.

The universal functor of a diagram is thediagonal functor;itsright adjointis thelimitof the diagram and its left adjoint is the colimit.^[1]Thenatural transformationfrom the diagonal functor to some arbitrary diagram is called acone.

Definition

Formally, adiagramof typeJin acategoryCis a (covariant)functor

D:J→C.

The categoryJis called theindex categoryor theschemeof the diagramD;the functor is sometimes called aJ-shaped diagram.^[2]The actual objects and morphisms inJare largely irrelevant; only the way in which they are interrelated matters. The diagramDis thought of as indexing a collection of objects and morphisms inCpatterned onJ.

Although, technically, there is no difference between an individualdiagramand afunctoror between aschemeand acategory,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.

One is most often interested in the case where the schemeJis asmallor evenfinitecategory. A diagram is said to besmallorfinitewheneverJis.

A morphism of diagrams of typeJin a categoryCis anatural transformationbetween functors. One can then interpret thecategory of diagramsof typeJinCas thefunctor categoryC^J,and a diagram is then an object in this category.

Examples

Given any objectAinC,one has theconstant diagram,which is the diagram that maps all objects inJtoA,and all morphisms ofJto the identity morphism onA.Notationally, one often uses an underbar to denote the constant diagram: thus, for any object $A$ inC,one has the constant diagram ${\underline {A}}$ .
IfJis a (small)discrete category,then a diagram of typeJis essentially just anindexed familyof objects inC(indexed byJ). When used in the construction of thelimit,the result is theproduct;for the colimit, one gets thecoproduct.So, for example, whenJis the discrete category with two objects, the resulting limit is just the binary product.
IfJ= −1 ← 0 → +1, then a diagram of typeJ(A←B→C) is aspan,and its colimit is apushout.If one were to "forget" that the diagram had objectBand the two arrowsB→A,B→C,the resulting diagram would simply be the discrete category with the two objectsAandC,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 theindex setin set theory: by including the morphismsB→A,B→C,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.
Dualto the above, ifJ= −1 → 0 ← +1, then a diagram of typeJ(A→B←C) is acospan,and its limit is apullback.
The index $J=0\rightrightarrows 1$ is called "two parallel morphisms", or sometimes thefree quiveror thewalking quiver.A diagram of type $J$ $(f,g\colon X\to Y)$ is then aquiver;its limit is anequalizer,and its colimit is acoequalizer.
IfJis aposet category,then a diagram of typeJis a family of objectsD_itogether with a unique morphismf_ij:D_i→D_jwheneveri≤j.IfJisdirectedthen a diagram of typeJis called adirect systemof objects and morphisms. If the diagram iscontravariantthen it is called aninverse system.

Cones and limits

Aconewith vertexNof a diagramD:J→Cis a morphism from the constant diagram Δ(N) toD.The constant diagram is the diagram which sends every object ofJto an objectNofCand every morphism to the identity morphism onN.

Thelimitof a diagramDis auniversal conetoD.That is, a cone through which all other cones uniquely factor. If the limit exists in a categoryCfor all diagrams of typeJone obtains a functor

lim:C^J→C

which sends each diagram to its limit.

Dually, thecolimitof diagramDis a universal cone fromD.If the colimit exists for all diagrams of typeJone has a functor

colim:C^J→C

which sends each diagram to its colimit.

Commutative diagrams

Diagrams and functor categories are often visualized bycommutative diagrams,particularly if the index category is a finiteposet categorywith 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 objects in a poset category. Conversely, every commutative diagram represents a diagram (a functor from a poset index category) in this way.

Not every diagram commutes, as not every index category is a poset category: most simply, the diagram of a single object with an endomorphism( $f\colon X\to X$ ),or with two parallel arrows ( $\bullet \rightrightarrows \bullet$ ; $f,g\colon X\to Y$ ) 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.

References

^Mac Lane, Saunders; Moerdijk, Ieke (1992).Sheaves in geometry and logic a first introduction to topos theory.New York: Springer-Verlag. pp.20–23.ISBN 9780387977102.
^May, J. P. (1999).A Concise Course in Algebraic Topology(PDF).University of Chicago Press. p. 16.ISBN 0-226-51183-9.

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 ofGrundlehren der mathematischen Wissenschaften (278)Springer-Verlag, 1983).
diagramat thenLab

External links

Diagram ChasingatMathWorld
WildCatsis a category theory package forMathematica.Manipulation and visualization of objects,morphisms,commutative diagrams, categories,functors,natural transformations.

[1] Mac Lane, Saunders; Moerdijk, Ieke (1992).Sheaves in geometry and logic a first introduction to topos theory.New York: Springer-Verlag. pp.20–23.ISBN 9780387977102.

[2] May, J. P. (1999).A Concise Course in Algebraic Topology(PDF).University of Chicago Press. p. 16.ISBN 0-226-51183-9.

[1]

[2]