MATH 101A: ALGEBRA I PART A: GROUP THEORY 3911. Categorical limits(Two lectures by Ivan Horozov. Notes by Andrew Gainer and RogerLipsett. [Comments by Kiyoshi].)We first note that what topologists call the “limit”, algebraists callthe “inverse limit” and denote by lim←. Likewise, what topologists callthe “colimit”, algebraists call the direct limit and denote lim→.Example 11.1. Take the inverse system· · · → C[x ]/(xn) → · · · → C[x ]/(x3) → C[x]/(x2) → C[x]/(x)Let f ∈ lim←C[x]/(xn) = C[[x]]. Then f =!n≥0anxnis a formalpower series in x over C.We note that Zp= lim←−nZ/pn[is the inverse limit of]· · · → Z/pn+1→ Z/pn→ · · · → Z/p11.1. colimits in the category of sets.Definition 11.2. Let Xαbe sets indexed by α ∈ I and let fα,β: Xα→Xβbe functions with α, β ∈ I. [Only for pairs (α, β) so that α < βin some partial ordering of I.] Then {fα.β, Xα}α∈Iis a directed systemof sets if, for every pair of composable morphisms fα,β: Xα→ Xβ,fβ,γ: Xβ→ Xγ[i.e., wherever α < β < γ in I], the following diagramcommutesXαfα,β!!fα,γ""!!!!!!!!Xβfβ,γ##Xγand, for every α, β ∈ I there exists a γ ∈ I for which there are mapsfα,γand fβ,γ[i.e., α, β ≤ γ] as such:Xαfα,γ$$""""""XγXβfβ,γ%%######.One can think of a directed system as a graph with sets as pointsand arrows as edges.We can now define the direct limit on directed systems of sets bylim−→α∈IXα="αXα/ ∼where, for each fα,βand for all x ∈ Xα, we set x ∼ fα,β(x). Informallythen, the direct limit is the set of equivalence classes induced by allfunctions fα,β. That ∼ is an equivalence relation follows from the40 MATH 101A: ALGEBRA I PART A: GROUP THEORYtwo properties illustrated diagramatically ab ove. [The colimit of anydiagram of sets exists. The assumption of being “directed” implies thatany two elements of the colimit, represented by say x ∈ Xα, y ∈ Xβ,are equivalent to elements of the same set Xγ.]11.2. pull-back. In the following diagram, ∗ is a “pull-back”:∗!!##Gα##Hβ!!KThe pull-back here is a subgroup (or subset) of G × H given byG ×KH = {(g, h) ∈ G × H | α(g) = β(h)}.11.2.1. universal property of pull-back. If G&is a group such thatG&$$&&''$$$$$$$$$$$$(g, h) ∈ G ×KHh##!!Gα##Hβ!!Kcommutes then there exists a unique map G&→ G ×KH such thatG&(())%%%%%%%%%""G ×KH!!##G##H!!Kcommutes.11.3. push-forward [push-out] of groups.K!!##G##**H!!++G ∗KH&&&&&&&&&&&GMATH 101A: ALGEBRA I PART A: GROUP THEORY 41In this diagram, if K = {e, } then G∗H is the amalgamated free productof G and H given byG ∗ H = {g1h1g2h2· · · gnhn| gi∈ G, hi∈ H}.Note that (g1hg2)−1= g−12h−1g−11. [So, the set of such products isclosed under the operation of taking inverse. So, G ∗ H is a group.]More generally, G ∗KH is the quotient group G ∗ H/ ∼ where(gα(k)) ·#β(k)−1h$∼ ghandhg ∼ (hβ(k) ·#α(k)−1g$.Exercise. Compute (Z/2Z) ∗ (Z/2Z) and explain the computation.11.4. direct limit of groups. In order to take the direct limit ofgroups, we require a directed system of groups:Definition 11.3. {Gα, fα,β} is a directed system of groups ifi) fα,β: Gα→ Gβ, fβ,γ: Gβ→ Gγare homomorphism then [there isa homomorphism fα,γ: Gα→ Gγin the system and]Gαfαβ!!fαβ""''''''''Gβfβγ##Gγ[i.e. the diagram commutes.]ii) For every α, β ∈ I there exists γ ∈ I such that fα,γ, fβ.γaredefined andGαfαγ$$""""""GγGβfβγ%%######We then define a new object [the weak product]%&α∈IGα⊆%α∈IGα:Definition 11.4. For x = {xα}αwe let x ∈%&α∈IGαif x has onlyfinitely many xαcoordinates which are not eα∈ Gα.[The direct limit of a directed system of groups is then the same setas the direct limit of sets:lim−→α∈IGα="α∈IGα/ ∼where, xα∈ Gαis equivalent to xβ∈ Gβif and only iffα,γ(xα) = fβ,γ(xβ)42 MATH 101A: ALGEBRA I PART A: GROUP THEORYfor some γ ∈ I, with the additional structure that the product ofxα∈ Gα, xβ∈ Gβis defined to be the product of their images in Gγ.]In practice, one works with lim→αGαin the following way: [HereHorozov says that each element of the direct limit is represented by asingle element of a single group Gα. I wrote that as the definition.]11.5. universal property of the direct limit of groups. If {Gα, fα,β}is a directed system of groups and gα: Gα→ H are homomorphismssuch thatGαgα!!fαβ##HGβgβ,,((((((((commutes, then there exists a unique homomorphism g : lim→Gα→ Hsuch thatGαhα!!gα--)))))))))))))))))))))lim→Gα∃!g))%%%%%%%%%Hcommutes for all α ∈ I.[Any elementx of the direct limit is represented by some elementof some group xα∈ Gα. Then we let g(x) = gα(xα). If xβ∈ Gβisanother representative of the same equivalence class x then, by def-inition, fα,γ(xα) = fβ,γ(xβ) = xγ∈ Gγfor some γ ∈ I. But thengα(xα) = gγ(xγ) = gβ(xβ). So, g is well-defined.]11.6. free groups. Let X be a set. The free group on X, F (X), isdefined by the following universal property: given any group G andset map f : X → G, there is a unique g : F (X) → G that is a grouphomomorphism such that the diagramXf!!i..********GF (X)∃!g//++++++++commutes.It remains to define F (X) and say what the map i is. Suppose X ={x1, x2, · · · } (note that X need not be countable; we use this subscriptnotation simply for ease of use). Then the words in X, w, are all finitesequences chosen from the set X ∪ X−1, where X−1= {x−11, x−12, · · · }MATH 101A: ALGEBRA I PART A: GROUP THEORY 43(here the−1notation is purely formal). If W = {w} is the set of suchword thenF (X) = W/ ∼where ∼ is the smallest possible relation so that we get a group: i.e.,that for all i, both xix−1iand x−1ixiare trivial.Then each word in F (X) has a unique reduced form, in which nofurther simplification induced by the above relation are possible, andF (X) thus consists of all the reduced words.11.7. an important example. P SL2(Z) = SL2(Z)/ ± I. This groupacts on the upper half-plane H = {z ∈ C | )z > 0}: A ∈ SL2(Z) givesa mapz *→az + bcz + d=−az − b−cz − d.Thus, for example,&1 10 1'translates by 1, while&0 −11 0'is inversion.It turns out thatP SL2(Z) = Z/2Z ∗ Z/3ZThis has something to do with fixed points in C under this action.Similarly,SL2(Z) = Z/4Z ∗Z/2ZZ/6Z44 MATH 101A: ALGEBRA I PART A: GROUP THEORY12. More about free productsI decided to explain