MATH 47A FALL 2008 INTRO TO MATH RESEARCH 45!!!!!!a!b!d!!!!!!c!!!!""""""For example, in this Hasse drawing, a ≤ b, c ≤ a and c ≤ d but a, dare not comparable.Other examples are given by division:Take positive integers as objects and take one arrow a → b if adivides b. This gives a Hasse diagram like this:1#####$$3##$$6##$$5##$$30 25189##$$Tomorrow we will add zeros. This will allow us to make loops withoutbackward nonzero arrows.46 NOTES7.3. Additive categories. Today I talked about additive categories.I gave the general definition then some particular examples. I explainedthe dotted line notation and we did some calculations.7.3.1. general definition. An additive category is a category with twoproperties.(1) Hom sets are additive groups. For any two objects A, B in anadditive category C,HomC(A, B)={f : A → B}is an additive group. This means it is an abelian group with opera-tion written as addition (instead of the usual multiplication notationfor groups). In other words, given two morphisms (arrows, homomor-phisms) f, g : A → B you have their sum which is another morphismf + g : A → B. You also have the zero morphism 0:A → B which isthe identity of the group. This means that0+f = f = f +0.There is also a negative −f so thatf +(−f) = 0 = (−f)+f.(2) Composition is biadditive. This is also called the distributive law.Give f, k : A → B and g, h : B → C we have:(g + h) ◦ f =(g ◦ f)+(h ◦ f)h ◦ (f + k)=h ◦ f + h ◦ kI pointed out that composition with 0 is 0:Proposition 7.4. In an additive category, composition of 0 with anymorphism gives 0.Proof. Since 0 + 0 = 0,0 ◦ f = (0 + 0) ◦ f =0◦ f +0◦ fBy the cancellation law (for groups), this gives 0 ◦ f = 0. Similarly,f ◦ 0 = 0. !Writing these notes, I realized that some of you may not know thecancellation rule for groups:Lemma 7.5. Cancellation holds in any group. I.e., ax = ay impliesx = y and ax = a implies x = e. Similarly, xb = yb ⇒ x = y.MATH 47A FALL 2008 INTRO TO MATH RESEARCH 47Proof. Multiply on the left with a−1:a−1a!"#$ex = a−1a!"#$eySo, x = ex = ey = y. When there is no y you get x = ex = e. !7.3.2. categories over F2. I made this fancy definition really simple bytaking cluster categories (of type An) over F2. This is the field with 2elements F2= {0, 1}. The hom sets are really simple. HomC(A, B) haseither 1 or 2 elements:HomC(A, B)=%{0} or{0,f} 0 and a nonzero fNotice that:(1) There is always a zero arrow: 0 : A → B.(2) There might be a nonzero arrow f : A → B.Addition is given by 0 + 0 = 0, 0 + f = f = f + 0 and f + f = 0. Thatis because −f = f (since it can’t be anything else).Given that composition is biadditive (distributive), what are thepossible compositions? Well there are only four possibilities for twoarrows being composed:A0−→ B0−→ C gives A0−→ CAf−→ Bg−→ C gives A0 or h−−−→ CAf−→ B0−→ C gives A0−→ CA0−→ Bg−→ C gives A0−→ CWe know that composition with 0 gives zero. But composition of twononzero morphisms f : A → B and g : B → C could give either 0 or anonzero morphism h = g ◦ f : A → C. It depends.7.3.3. cycle example. I returned to the cycle example:Bg!!Af""Ch##Dk$$I gave two possible composition laws.(1) Let g ◦ f =0,h◦ g = 0, etc (all compositions of two nonidentitymorphism are zero).(2) Let h ◦ g ◦ f =0,k◦ h ◦g = 0, etc., i.e. all compositions of threenonidentity morphisms are zero. (And g ◦ f &= 0)48 NOTES7.3.4. dotted line notation. People use dotted lines to indicate whichcompositions are zero. For example:Bg%%!!!!!!!Af&&"""""""#######Cmeans g ◦ f = 0.Bg##Ch''!!!!!!!Af&&"""""""###########Dmeans h ◦ g ◦ f = 0.The dotted line means that there is an equation among the pathsgoing from one point to another. For example:Bg''!!!!!!!Af&&"""""""h%%!!!!!!!#######DCk(("""""""means g ◦ f = k ◦ h.If we wanted to indicate that compositions are zero we would drawtwo dotted lines. (The curved lines are supposed to be dotted lines. Icouldn’t figure out how to type it.)Bg''!!!!!!!Af&&"""""""h%%!!!!!!!DCk(("""""""This means g ◦ f = 0 and k ◦ h = 0.In other words, each dotted line indicates one equation. If you wantonly one zero relation you draw only one dotted curved line:MATH 47A FALL 2008 INTRO TO MATH RESEARCH 49Bg''!!!!!!!Af&&"""""""h%%!!!!!!!DCk(("""""""This means k ◦ h = 0.7.4. cluster categories of type Anover F2. Now that we knowthe notation, I can write down the cluster category. It is a repeatingpattern:•%%!!!!!!!#######•%%!!!!!!!#######•%%!!!!!!!#######•%%!!!!!!!#######•%%!!!!!!!#######••&&"""""""%%!!!!!!!#######•&&"""""""%%!!!!!!!#######•&&"""""""%%!!!!!!!#######•&&"""""""%%!!!!!!!#######•&&"""""""%%!!!!!!!#######•&&"""""""%%!!!!!!!•&&"""""""%%!!!!!!!#######•&&"""""""%%!!!!!!!#######•&&"""""""%%!!!!!!!#######•&&"""""""%%!!!!!!!#######•&&"""""""%%!!!!!!!#######••&&"""""""#######•&&"""""""#######•&&"""""""#######•&&"""""""#######•&&"""""""#######•&&"""""""This is actually on a M¨obius strip. The objects A, B, C, D on theleft are the same as those on the right. This is the cluster category oftype A4.•%%!!!!!!!!#######Df%%$$$$$$$$#######Zm''!!!!!!!!#######A%%!!!!!!!#######•''!!!!!!!!#######••&&""""""""%%!!!!!!!!#######Ch(("""""""k''!!!!!!!!#######•g&&%%%%%%%%%%$$$$$$$$#######X(("""""""''!!!!!!!!#######B&&""""""""%%!!!!!!!#######•&&%%%%%%%%%%$$$$$$$$B&&"""""""%%!!!!!!!!#######•!&&%%%%%%%%%%$$$$$$$$#######Y(("""""""''!!!!!!!!#######•&&""""""""%%!!!!!!!!#######C((""""""""''!!!!!!!#######•A&&"""""""#######•((""""""""#######•&&%%%%%%%%#######•((""""""""#######•&&""""""""#######D&&%%%%%%%%Going step-by-step, I asked students to calculate the following. Moreprecisely, the question is: Are these groups zero or nonzero?(1) HomC(D, Z) =?(2) HomC(C, Z) =?(3) HomC(C, X) =?50 NOTES(4) HomC(C, Y ) =?First, students realized that HomC(D, Z) = 0 since the only pathfrom D to Z is the composition g ◦ f which is equal to zero because ofthe dotted line from D to Z.Next, we looked at HomC(C, Z). There are two paths from C to Z.The first is zero since g ◦ f = 0 by the previous calculation.g ◦ f!"#$0◦h =0The other path isg ◦ ! ◦ k = g ◦ f ◦ h =0which is equal to the previous path and is thus zero by the mesh relationf ◦ h = ! ◦ kStudents got the hang of this and they realized that HomC(C, X) = 0since all paths from C to X are equivalent, by repeated use of meshrelations, to the pathm ◦ g ◦