A Trace on FinSet Sean Sanford Abstract In these notes we calculate the coend Z X2FinSet FinSet(X; X); along with its natural map Z a X2FinSet τ : FinSet(X; X) ! FinSet(X; X); X2FinSet which we interpret as a kind of trace. It turns out that this object is conve- niently described as the set P of all partitions of all natural numbers. Sections 1 through 5 are dedicated to proving the main theorem, and section 6 follows up with some elementary applications. In section 7 we describe a semiring structure on P that is related to τ. We go on to use this structure in section 8 to analyze the category of finite G-sets in close analogy with classical representation theory. 1 The Cowedge Condition and the Relation ∼ Here we give a specific interpretation of the idea of coends that is relevant to our situation. Let C and D be categories, and let K : Cop × C ! D be a functor. 1 Definition. A morphism of the form a η : K(C; C) ! D C2C is called a cowedge under K if for all f : C ! C0 the following square commutes: K(f;C) K(C0;C) K(C; C) K(C0;f) η η K(C0;C0) D In these notes, we will be concerned with the situtation where C = FinSet, D = Set and K( ; ) = FinSet( ; ). In this situation, FinSet(f; C) is usually writ- ten as f ∗ and is precomposition with f, while FinSet(C0; f) is usually written f∗ and is postcompostion with f. By making the appropriate substitutions to the above diagram, we arrive at f ∗ FinSet(B; A) FinSet(A; A) f∗ η η FinSet(B; B) S Note that if g 2 FinSet(B; A), then commutativity of the diagram implies η(g ◦ f) = η(f ◦ g). This will be an important formula for us, so we give it a name. Definition. For the sake of these notes, a morphism a η : FinSet(X; X) ! S X2FinSet is said to satisfy the cowedge condition if η(g ◦ f) = η(f ◦ g) whenever both compositions are defined. Thus we find that such a morphism out of the coproduct is acowedge under FinSet( ; ) if and only if it satisfies the cowedge condition. 2 Definition. A coend is an initial cowedge. This means that τ is a coend of K if for every cowedge η under K there is a unique map ζ such that η = ζ ◦τ. By abuse of notation the term coend also refers to the object at the codomain of the cowedge, and for this object we use the notation Z C2C K(C; C); where the symbol C acts as an index or ‘dummy variable’ similar to the notation used in products and coproducts. The conclusion of these notes is that Z X2FinSet ∼ FinSet(X; X) = P := fall partitions of all natural numbersg: First we begin by examining the consequences of the cowedge condition. If η satisfies the cowedge condition, then η maps f ◦ g and g ◦ f to the same element. This suggests investigation of the relation ∼0 where ϕ ∼0 if there exist f; g such that ϕ = f ◦ g and = g ◦ f. This relation is reflexive and symmetric, but it is not transitive, and thus not an equivalence relation. However there is a canonical way to fixing this issue. ` Definition. The relation ∼ on FinSet(X; X) is defined to be the X2FinSet transitive closure of ∼0. This means that ϕ ∼ if there exists a sequence of k ∼ ∼ ∼ morphisms (fi)i=1 such that ϕ 0 f1, fi 0 fi+1 for i < k and fk 0 . with this definition, it is not difficult to see that forany η satisfying the cowedge condition, f ∼ g implies η(f) = η(g). 2 Reduction to Isomorphisms Using the epi-monic factorization in FinSet, we can write any map f 2 FinSet(A; A) as f = m ◦ e where e : A ↠ im(f) and m : im(f) ,! A. Knowing that f can be decomposed in this way tells us that f ∼ e ◦ m, and this map e ◦ m 2 FinSet(im(f); im(f)) is fjim(f). This allows us to prove our first proposition. Proposition 2.1. For every endomorphism f in FinSet, there is an isomor- phism f^ such that f ∼ f^. 3 The proof of this proposition will first require a lemma: Lemma 2.1.1. For every endomorphism f 2 FinSet(A; A) in FinSet there is a natural number N 2 N such that fjim(f n) = fjim(f N ) for all n ≥ N. Moreover, this common map is an isomorphism, and if A =6 ;, then im(f N ) =6 ;. Proof of lemma. Firstly, for f : A ! A, we have that im(f) ⊆ A. This implies that im(f 2) = f(im(f)) ⊆ f(A) = im(f), and by induction, we have a descending sequence of subsets: · · · ⊆ im(f k+1) ⊆ im(f k) ⊆ · · · ⊆ im(f 2) ⊆ im(f) ⊆ A If A = ;, then f was an isomorphism to begin with, and the sequence above is constant. Assume then that A =6 ;. Since the image of a nonempty set is nonempty, every term in this sequence will always have at least one element. N+1 N If it happens that fjim(f N ) is an isomorphism, then im(f ) = f(im(f )) = im(f N ), so by induction im(f n) = im(f N ) for all n ≥ N. For the sake of contradiction, suppose that for all n, fjim(f n) is not an isomorphism. Since A is finite, fjim(f n) is not surjective for any n. Thus we have that #(im(f)) ≤ #(A) − 1, and by induction #(im(f n)) ≤ #(A) − n. If #(A) = N, then #(im(f N )) ≤ #(A) − N = 0, which contradicts our previous observation that 1 ≤ #(im(f N )) . We can now proceed to prove Proposition 2.1. Proof of Proposition. Given f 2 FinSet(A; A), define ( ) ρ(f) := fjim(f) 2 FinSet im(f); im(f) : We have already seen that f ∼ ρ(f), and this implies that ρ(f) ∼ ρ(ρ(f)) =: ρ2(f). Since ∼ is transitive, f ∼ ρ2(f) and by iduction f ∼ ρn(f) n for all n ≥ 1. By the lemma, it will suffice to prove that ρ (f) = fjim(f n) for all n ≥ 1. 4 Note that ρ(f) := fjim(f 1), so our base case is covered by definition. k Suppose that ρ (f) = fjim(f k) for all k ≤ n. We calculate ρn+1(f) : = ρ (ρn(f)) ( ) = ρ fj n ( im(f )) j = f im(f n) j ( ) im(f im(fn)) j = f im(f n) j n ( ) f im(fn)(im(f )) = fj n ( im(f )) f(im(f n)) j n = f im(f ) im(f n+1) = fjim(f n+1): n Thus by induction ρ (f) = fjim(f n) for all n. Using the lemma, define ^ N f := ρ (f) where N is the first number such that fjim(f N ) is an isomorphism. 3 The Maps τ0 and τ Here we describe some important morphisms between relevent objects, and examine their properties. For any f 2 AutFinSet(A), the subgroup hfi gener- ated by f is a cyclic group that acts on A. Consider the orbit space A⧸ hfi: By ordering the sizes of the orbits from greatest to least, we obtain a partition of the number jAj. Let the partition thus obtained from f be denoted τ0(f). We have just described a map a τ0 : Aut(X) ! P := fall partitions of all natural numbersg: X2FinSet By the universal property of the coproduct, the inclusions iX : Aut(X) ,! FinSet(X; X) form a cocone over the summands, and determine a unique map a a i : Aut(X) ! FinSet(X; X); X2FinSet X2FinSet 5 which is iX on the X component. The construction of Proposition 2.1 shows that we have a surjection r : f 7! f^ which goes in the other direction, and satisfies r ◦ i = id. We now make what is possibly the most important definition in these notes. Definition. The trace map τ is the composition τ0 ◦ r. Given any map f 2 FinSet(A; A) the partition τ(f) 2 P is called the trace of f. Proposition 3.1. The trace map τ is a cowedge under the bifunctor FinSet( ; ). Proof of Proposition. It will suffice to prove that τ0 ◦ r satisfies the cowedge condition. Suppose then that f : A ! B and g : B ! A are any two functions of the finite sets A and B. By Lemma 2.1.1 there are integers N1 and N2 such that ◦ ◦ j r(f g) = (f g) im((f◦g)N1 ); & ◦ ◦ j r(g f) = (g f) im((g◦f)N2 ): Now set N : = maxfN ;N g ( 1 2) A : = im (g ◦ f)N 0 ( ) N B0 : = im (f ◦ g) : We calculate: N B0 = (f ◦ g) (B) = (f ◦ g)N+1(B) ( ) = f ◦ (g ◦ f)N ◦ g (B) ( )( ) = f ◦ (g ◦ f)N g(B) ( ) ⊆ f ◦ (g ◦ f)N (A) ( ) = (f ◦ g)N ◦ f (A) ( ) = (f ◦ g)N f(A) ( ) N ⊆ (f ◦ g) B = B0 =) ( ) N B0 = f ◦ (g ◦ f) (A) = f(A0); 6 j ! which shows that f A0 : A0 B0 is surjective. We also have that j ◦ j ◦ j ◦ g B0 f A0 = (g f) A0 = r(g f) is iso, j and this forces f A0 to be injective and hence an isomorphism. Similar argu- j ments show that g B0 is also an isomorphism. For brevity let us denote these maps f0 and g0.