# Philosophical Equivariance – Sewing Holonomies Towards Equal Trace Endomorphisms.

In d-dimensional topological field theory one begins with a category S whose objects are oriented (d − 1)-manifolds and whose morphisms are oriented cobordisms. Physicists say that a theory admits a group G as a global symmetry group if G acts on the vector space associated to each (d−1)-manifold, and the linear operator associated to each cobordism is a G-equivariant map. When we have such a “global” symmetry group G we can ask whether the symmetry can be “gauged”, i.e., whether elements of G can be applied “independently” – in some sense – at each point of space-time. Mathematically the process of “gauging” has a very elegant description: it amounts to extending the field theory functor from the category S to the category SG whose objects are (d − 1)-manifolds equipped with a principal G-bundle, and whose morphisms are cobordisms with a G-bundle. We regard S as a subcategory of SG by equipping each (d − 1)-manifold S with the trivial G-bundle S × G. In SG the group of automorphisms of the trivial bundle S × G contains G, and so in a gauged theory G acts on the state space H(S): this should be the original “global” action of G. But the gauged theory has a state space H(S,P) for each G-bundle P on S: if P is non-trivial one calls H(S,P) a “twisted sector” of the theory. In the case d = 2, when S = S1 we have the bundle Pg → S1 obtained by attaching the ends of [0,2π] × G via multiplication by g. Any bundle is isomorphic to one of these, and Pg is isomorphic to Pg iff g′ is conjugate to g. But note that the state space depends on the bundle and not just its isomorphism class, so we have a twisted sector state space Cg = H(S,Pg) labelled by a group element g rather than by a conjugacy class.

We shall call a theory defined on the category SG a G-equivariant Topological Field Theory (TFT). It is important to distinguish the equivariant theory from the corresponding “gauged theory”. In physics, the equivariant theory is obtained by coupling to nondynamical background gauge fields, while the gauged theory is obtained by “summing” over those gauge fields in the path integral.

An alternative and equivalent viewpoint which is especially useful in the two-dimensional case is that SG is the category whose objects are oriented (d − 1)-manifolds S equipped with a map p : S → BG, where BG is the classifying space of G. In this viewpoint we have a bundle over the space Map(S,BG) whose fibre at p is Hp. To say that Hp depends only on the G-bundle pEG on S pulled back from the universal G-bundle EG on BG by p is the same as to say that the bundle on Map(S,BG) is equipped with a flat connection allowing us to identify the fibres at points in the same connected component by parallel transport; for the set of bundle isomorphisms p0EG → p1EG is the same as the set of homotopy classes of paths from p0 to p1. When S = S1 the connected components of the space of maps correspond to the conjugacy classes in G: each bundle Pg corresponds to a specific point pg in the mapping space, and a group element h defines a specific path from pg to phgh−1 .

G-equivariant topological field theories are examples of “homotopy topological field theories”. Using Vladimir Turaev‘s two main results: first, an attractive generalization of the theorem that a two-dimensional TFT “is” a commutative Frobenius algebra, and, secondly, a classification of the ways of gauging a given global G-symmetry of a semisimple TFT.

Definition of the product in the G-equivariant closed theory. The heavy dot is the basepoint on S1. To specify the morphism unambiguously we must indicate consistent holonomies along a set of curves whose complement consists of simply connected pieces. These holonomies are always along paths between points where by definition the fibre is G. This means that the product is not commutative. We need to fix a convention for holonomies of a composition of curves, i.e., whether we are using left or right path-ordering. We will take h(γ1 ◦ γ2) = h(γ1) · h(γ2).

A G-equivariant TFT gives us for each element g ∈ G a vector space Cg, associated to the circle equipped with the bundle pg whose holonomy is g. The usual pair-of-pants cobordism, equipped with the evident G-bundle which restricts to pg1 and pg2 on the two incoming circles, and to pg1g2 on the outgoing circle, induces a product

Cg1 ⊗ Cg2 → Cg1g2 —– (1)

making C := ⊕g∈GCg into a G-graded algebra. Also there is a trace θ: C1  → C defined by the disk diagram with one ingoing circle. The holonomy around the boundary of the disk must be 1. Making the standard assumption that the cylinder corresponds to the unit operator we obtain a non-degenerate pairing

Cg ⊗ Cg−1 → C

A new element in the equivariant theory is that G acts as an automorphism group on C. That is, there is a homomorphism α : G → Aut(C) such that

αh : Cg → Chgh−1 —– (2)

Diagramatically, αh is defined by the surface in the immediately above figure. Now let us note some properties of α. First, if φ ∈ Ch then αh(φ) = φ. The reason for this is diagrammatically in the below figure.

If the holonomy along path P2 is h then the holonomy along path P1 is 1. However, a Dehn twist around the inner circle maps P1 into P2. Therefore, αh(φ) = α1(φ) = φ, if φ ∈ Ch.

Next, while C is not commutative, it is “twisted-commutative” in the following sense. If φ1 ∈ Cg1 and φ2 ∈ Cg2 then

αg212 = φ2φ1 —– (3)

The necessity of this condition is illustrated in the figure below.

The trace of the identity map of Cg is the partition function of the theory on a torus with the bundle with holonomy (g,1). Cutting the torus the other way, we see that this is the trace of αg on C1. Similarly, by considering the torus with a bundle with holonomy (g,h), where g and h are two commuting elements of G, we see that the trace of αg on Ch is the trace of αh on Cg−1. But we need a strengthening of this property. Even when g and h do not commute we can form a bundle with holonomy (g,h) on a torus with one hole, around which the holonomy will be c = hgh−1g−1. We can cut this torus along either of its generating circles to get a cobordism operator from Cc ⊗ Ch to Ch or from Cg−1 ⊗ Cc to Cg−1. If ψ ∈ Chgh−1g−1. Let us introduce two linear transformations Lψ, Rψ associated to left- and right-multiplication by ψ. On the one hand, Lψαg : φ􏰀 ↦ ψαg(φ) is a map Ch → Ch. On the other hand Rψαh : φ ↦ αh(φ)ψ is a map Cg−1 → Cg−1. The last sewing condition states that these two endomorphisms must have equal traces:

TrCh 􏰌Lψαg􏰍 = TrCg−1 􏰌Rψαh􏰍 —– (4)

(4) was taken by Turaev as one of his axioms. It can, however, be reexpressed in a way that we shall find more convenient. Let ∆g ∈ Cg ⊗ Cg−1 be the “duality” element corresponding to the identity cobordism of (S1,Pg) with both ends regarded as outgoing. We have ∆g = ∑ξi ⊗ ξi, where ξi and ξi ru􏰟n through dual bases of Cg and Cg−1. Let us also write

h = ∑ηi ⊗ ηi ∈ Ch ⊗ Ch−1. Then (4) is easily seen to be equivalent to

∑αhii = 􏰟 ∑ηiαgi) —– (5)

in which both sides are elements of Chgh−1g−1.

# Conjuncted: Affine Schemes: How Would Functors Carry the Same Information?

If we go to the generality of schemes, the extra structure overshadows the topological points and leaves out crucial details so that we have little information, without the full knowledge of the sheaf. For example the evaluation of odd functions on topological points is always zero. This implies that the structure sheaf of a supermanifold cannot be reconstructed from its underlying topological space.

The functor of points is a categorical device to bring back our attention to the points of a scheme; however the notion of point needs to be suitably generalized to go beyond the points of the topological space underlying the scheme.

Grothendieck’s idea behind the definition of the functor of points associated to a scheme is the following. If X is a scheme, for each commutative ring A, we can define the set of the A-points of X in analogy to the way the classical geometers used to define the rational or integral points on a variety. The crucial difference is that we do not focus on just one commutative ring A, but we consider the A-points for all commutative rings A. In fact, the scheme we start from is completely recaptured only by the collection of the A-points for every commutative ring A, together with the admissible morphisms.

Let (rings) denote the category of commutative rings and (schemes) the category of schemes.

Let (|X|, OX) be a scheme and let T ∈ (schemes). We call the T-points of X, the set of all scheme morphisms {T → X}, that we denote by Hom(T, X). We then define the functor of points hX of the scheme X as the representable functor defined on the objects as

hX: (schemes)op → (sets), haX(A) = Hom(Spec A, X) = A-points of X

Notice that when X is affine, X ≅ Spec O(X) and we have

haX(A) = Hom(Spec A, O(X)) = Hom(O(X), A)

In this case the functor haX is again representable.

Consider the affine schemes X = Spec O(X) and Y = Spec O(Y). There is a one-to-one correspondence between the scheme morphisms X → Y and the ring morphisms O(X) → O(Y). Both hX and haare defined on morphisms in the natural way. If φ: T → S is a morphism and ƒ ∈ Hom(S, X), we define hX(φ)(ƒ) = ƒ ○ φ. Similarly, if ψ: A → Bis a ring morphism and g ∈ Hom(O(X), A), we define haX(ψ)(g) = ψ ○ g. The functors hX and haare for a given scheme X not really different but carry the same information. The functor of points hof a scheme X is completely determined by its restriction to the category of affine schemes, or equivalently by the functor

haX: (rings) → (sets), haX(A) = Hom(Spec A, X)

Let M = (|M|, OM) be a locally ringed space and let (rspaces) denote the category of locally ringed spaces. We define the functor of points of locally ringed spaces M as the representable functor

hM: (rspaces)op → (sets), hM(T) = Hom(T, M)

hM is defined on the manifold as

hM(φ)(g) = g ○ φ

If the locally ringed space M is a differentiable manifold, then

Hom(M, N) ≅ Hom(C(N), C(M))

This takes us to the theory of Yoneda’s Lemma.

Let C be a category, and let X, Y be objects in C and let hX: Cop → (sets) be the representable functors defined on the objects as hX(T) = Hom(T, X), and on the arrows as hX(φ)(ƒ) = ƒ . φ, for φ: T → S, ƒ ∈ Hom(T, X)

If F: Cop → (sets), then we have a one-to-one correspondence between sets:

{hX → F} ⇔ F(X)

The functor

h: C → Fun(Cop, (sets)), X ↦ hX,

is an equivalence of C with a full subcategory of functors. In particular, hX ≅ hY iff X ≅ Y and the natural transformations hX → hY are in one-to-one correspondence with the morphisms X → Y.

Two schemes (manifolds) are isomorphic iff their functors of points are isomorphic.

The advantages of using the functorial language are many. Morphisms of schemes are just maps between the sets of their A-points, respecting functorial properties. This often simplifies matters, allowing allowing for leaving the sheaves machinery in the background. The problem with such an approach, however, is that not all the functors from (schemes) to (sets) are the functors of points of a scheme, i.e., they are representable.

A functor F: (rings) → (sets) is of the form F(A) = Hom(Spec A, X) for a scheme X iff:

F is local or is a sheaf in Zariski Topology. This means that for each ring R and for every collection αi ∈ F(Rƒi), with (ƒi, i ∈ I) = R, so that αi and αj map to the same element in F(Rƒiƒj) ∀ i and j ∃ a unique element α ∈ F(R) mapping to each αi, and

F admits a cover by open affine subfunctors, which means that ∃ a family Ui of subfunctors of F, i.e. Ui(R) ⊂ F(R) ∀ R ∈ (rings), Ui = hSpec Ui, with the property that ∀ natural transformations ƒ: hSpec A  → F, the functors ƒ-1(Ui), defined as ƒ-1(Ui)(R) = ƒ-1(Ui(R)), are all representable, i.e. ƒ-1(Ui) = hVi, and the Vi form an open covering for Spec A.

This states the conditions we expect for F to be the functor of points of a scheme. Namely, locally, F must look like the functor of points of a scheme, moreover F must be a sheaf, i.e. F must have a gluing property that allows us to patch together the open affine cover.

# Functoriality in Low Dimensions. Note Quote.

Let CW be the category of CW-complexes and cellular maps, let CW0 be the full subcategory of path connected CW-complexes and let CW1 be the full subcategory of simply connected CW-complexes. Let HoCW denote the category of CW-complexes and homotopy classes of cellular maps. Let HoCWn denote the category of CW-complexes and rel n-skeleton homotopy classes of cellular maps. Dimension n = 1: It is straightforward to define a covariant truncation functor

t<n = t<1 : CW0 → HoCW together with a natural transformation

emb1 : t<1 → t<∞,

where t<∞ : CW0 → HoCW is the natural “inclusion-followed-by-quotient” functor given by t<∞(K) = K for objects K and t<∞(f) = [f] for morphisms f, such that for all objects K, emb1∗ : H0(t<1K) → H0(t<∞K) is an isomorphism and Hr(t<1K) = 0 for r ≥ 1. The details are as follows: For a path connected CW-complex K, set t<1(K) = k0, where k0 is a 0-cell of K. Let emb1(K) : t<1(K) = k0 → t<∞(K) = K be the inclusion of k0 in K. Then emb1∗ is an isomorphism on H0 as K is path connected. Clearly Hr(t<1K) = 0 for r ≥ 1. Let f : K → L be a cellular map between objects of CW0. The morphism t<1(f) : t<1(K) = k0 → l0 = t<1(L) is the homotopy class of the unique map from a point to a point. In particular, t<1(idK) = [idk0] and for a cellular map g : L → P we have t<1(gf) = t<1(g) ◦ t<1(f), so that t<1 is indeed a functor. To show that emb1 is a natural transformation, we need to see that

that is

commutes in HoCW. This is where we need the functor t<1 to have values only in HoCW, not in CW, because the square need certainly not commute in CW. (The points k0 and l0 do not know anything about f, so l0 need not be the image of k0 under f.) Since L is path connected, there is a path ω : I → L from l0 = ω(0) to f (k0) = ω(1). Then H : {k0} × I → L, H(k0, t) = ω(t), defines a homotopy from

k0 → l0 → L to k0 → K →f L.

Dimension n = 2: We will define a covariant truncation functor t<n = t<2 : CW1 → HoCW

together with a natural transformation
emb2 : t<2 → t<∞,

where t<∞ : CW1 → HoCW is as above (only restricted to simply connected spaces), such that for all objects K, emb2∗ : Hr(t<2K) → Hr(t<∞K) is an isomorphism for r = 0, 1, and Hr(t<2K) = 0 for r ≥ 2. For a simply connected CW-complex K, set t<2(K) = k0, where k0 is a 0-cell of K. Let emb2(K) : t<2(K) = k0 → t<∞(K) = K be the inclusion as in the case n = 1. It follows that emb2∗ is an isomorphism both on H0 as K is path connected and on H1 as H1(k0) = 0 = H1(K), while trivially Hr(t<2K) = 0 for r ≥ 2. On a cellular map f, t<2(f) is defined as in the case n = 1. As in the case n = 1, this yields a functor and emb2 is a natural transformation.

# Grothendieck’s Abstract Homotopy Theory

Let E be a Grothendieck topos (think of E as the category, Sh(X), of set valued sheaves on a space X). Within E, we can pick out a subcategory, C, of locally finite, locally constant objects in E. (If X is a space with E = Sh(X), C corresponds to those sheaves whose espace étale is a finite covering space of X.) Picking a base point in X generalises to picking a ‘fibre functor’ F : C → Setsfin, a functor satisfying various conditions implying that it is pro-representable. (If x0 ∈ X is a base point {x0} → X induces a ‘fibre functor’ Sh(X) → Sh{x0} ≅ Sets, by pullback.)

If F is pro-representable by P, then π1(E, F) is defined to be Aut(P), which is a profinite group. Grothendieck proves there is an equivalence of categories C ≃ π1(E) − Setsfin, the category of finite π1(E)-sets. If X is a locally nicely behaved space such as a CW-complex and E = Sh(X), then π1(E) is the profinite completion of π1(X). This profinite completion occurs only because Grothendieck considers locally finite objects. Without this restriction, a covering space Y of X would correspond to a π1(X) – set, Y′, but if Y is a finite covering of X then the homomorphism from π1(X) to the finite group of transformations of Y factors through the profinite completion of π1(X). This is defined by : if G is a group, Gˆ = lim(G/H : H ◅ G, H of finite index) is its profinite completion. This idea of using covering spaces or their analogue in E raises several important points:

a) These are homotopy theoretic results, but no paths are used. The argument involving sheaf theory, the theory of (pro)representable functors, etc., is of a purely categorical nature. This means it is applicable to spaces where the use of paths, and other homotopies is impossible because of bad (or unknown) local properties. Such spaces have been studied within Shape Theory and Strong Shape Theory, although not by using Grothendieck’s fundamental group, nor using sheaf theory.

b) As no paths are used, these methods can also be applied to non-spaces, e.g. locales and possibly to their non-commutative analogues, quantales. For instance, classically one could consider a field k and an algebraic closure K of k and then choose C to be a category of étale algebras over k, in such a way that π1(E) ≅ Gal(K/k), the Galois group of k. It, in fact, leads to a classification theorem for Grothendieck toposes. From this viewpoint, low dimensional homotopy theory is ssen as being part of Galois theory, or vice versa.

c) This underlines the fact that π1(X) classifies covering spaces – but for i > 1, πi(X) does not seem to classify anything other than maps from Si into X!

This is abstract homotopy theory par excellence.

Algebraic constructs (A,U), such as Vec, Grp, Mon, and Lat, can be fully described by the following data, called the monad associated with (A,U):

1. the functor T : Set → Set, where T = U ◦ F and F : Set → A is the associated free functor,

2. the natural transformation η : idSet → T formed by universal arrows, and

3. the natural transformation μ : T ◦ T → T given by the unique homomorphism μX : T(TX) → TX that extends idTX.

In these cases, there is a canonical concrete isomorphism K between (A,U) and the full concrete subcategory of Alg(T) consisting of those T-algebras TX →x X that satisfy the equations x ◦ ηX = idX and x ◦ Tx = x ◦ μX. The latter subcategory is called the Eilenberg-Moore category of the monad (T, η, μ). The above observation makes it possible, in the following four steps, to express the “degree of algebraic character” of arbitrary concrete categories that have free objects:

Step 1: With every concrete category (A,U) over X that has free objects (or, more generally, with every adjoint functor A →U X) one can associate, in an essentially unique way, an adjoint situation (η, ε) : F -|U : A → X.

Step 2: With every adjoint situation (η, ε) : F -|U : A → X one can associate a monad T = (T, η, μ) on X, where T = U ◦ F : X → X.

Step 3: With every monad T = (T, η, μ) on X one can associate a concrete subcategory of Alg(T) denoted by (XT, UT) and called the category of T-algebras.

Step 4:  With every concrete category (A,U) over X that has free objects one can associate a distinguished concrete functor (A,U) →K (XT , UT) into the associated category of T-algebras called the comparison functor for (A, U).

Concrete categories that are concretely isomorphic to a category of T-algebras for some monad T have a distinct “algebraic flavor”. Such categories (A,U) and their forgetful functors U are called monadic. It turns out that a concrete category (A, U ) is monadic iff it has free objects and its associated comparison functor (A,U) →K (XT , UT) is an isomorphism. Thus, for concrete categories (A,U) that have free objects, the associated comparison functor can be considered as a means of measuring the “algebraic character” of (A,U); and the associated category of T-algebras can be considered to be the “algebraic part” of (A,U). In particular,

(a) every finitary variety is monadic,

(b) the category TopGrp, considered as a concrete category

2. over Set, is not monadic; the associated comparison functor is the forgetful functor TopGrp → Grp, so that the construct Grp may be considered as the “algebraic part” of the construct TopGrp,

(c) the construct Top is not monadic; the associated comparison functor is the forgetful functor Top → Set itself, so that the construct Set may be considered as the “algebraic part” of the construct Top; hence the construct Top may be considered as having a trivial “algebraic part”.

Among constructs, monadicity captures the idea of “algebraicness” rather well. Unfortunately, however, the behavior of monadic categories in general is far from satisfactory. Monadic functors can fail badly to reflect properties of the base category (e.g., the existence of colimits or of suitable factorization structures), and they are not closed under composition.

# Marching Along Categories, Groups and Rings. Part 2

A category C consists of the following data:

A collection Obj(C) of objects. We will write “x ∈ C” to mean that “x ∈ Obj(C)

For each ordered pair x, y ∈ C there is a collection HomC (x, y) of arrows. We will write α∶x→y to mean that α ∈ HomC(x,y). Each collection HomC(x,x) has a special element called the identity arrow idx ∶ x → x. We let Arr(C) denote the collection of all arrows in C.

For each ordered triple of objects x, y, z ∈ C there is a function

○ ∶ HomC (x, y) × HomC(y, z) → HomC (x, z), which is called composition of  arrows. If  α ∶ x → y and β ∶ y → z then we denote the composite arrow by β ○ α ∶ x → z.

If each collection of arrows HomC(x,y) is a set then we say that the category C is locally small. If in addition the collection Obj(C) is a set then we say that C is small.

Identitiy: For each arrow α ∶ x → y the following diagram commutes:

Associative: For all arrows α ∶ x → y, β ∶ y → z, γ ∶ z → w, the following diagram commutes:

We say that C′ ⊆ C is a subcategory if Obj(C′) ⊆ Obj(C) and if ∀ x,y ∈ Obj(C′) we have HomC′(x,y) ⊆ HomC(x,y). We say that the subcategory is full if each inclusion of hom sets is an equality.

Let C be a category. A diagram D ⊆ C is a collection of objects in C with some arrows between them. Repetition of objects and arrows is allowed. OR. Let I be any small category, which we think of as an “index category”. Then any functor D ∶ I → C is called a diagram of shape I in C. In either case, we say that the diagram D commutes if for all pairs of objects x,y in D, any two directed paths in D from x to y yield the same arrow under composition.

Identity arrows generalize the reflexive property of posets, and composition of arrows generalizes the transitive property of posets. But whatever happened to the antisymmetric property? Well, it’s the same issue we had before: we should really define equivalence of objects in terms of antisymmetry.

Isomorphism: Let C be a category. We say that two objects x,y ∈ C are isomorphic in C if there exist arrows α ∶ x → y and β ∶ y → x such that the following diagram commutes:

In this case we write x ≅C y, or just x ≅ y if the category is understood.

If γ ∶ y → x is any other arrow satisfying the same diagram as β, then by the axioms of identity and associativity we must have

γ = γ ○ idy = γ ○ (α ○ β) = (γ ○ α) ○ β = idx ○ β = β

This allows us to refer to β as the inverse of the arrow α. We use the notations β = α−1 and

β−1 = α.

A category with one object is called a monoid. A monoid in which each arrow is invertible is called a group. A small category in which each arrow is invertible is called a groupoid.

Subcategories of Set are called concrete categories. Given a concrete category C ⊆ Set we can think of its objects as special kinds of sets and its arrows as special kinds of functions. Some famous examples of conrete categories are:

• Grp = groups & homomorphisms
• Ab = abelian groups & homomorphisms
• Rng = rings & homomorphisms
• CRng = commutative rings & homomorphisms

Note that Ab ⊆ Grp and CRng ⊆ Rng are both full subcategories. In general, the arrows of a concrete category are called morphisms or homomorphisms. This explains our notation of HomC.

Homotopy: The most famous example of a non-concrete category is the fundamental groupoid π1(X) of a topological space X. Here the objects are points and the arrows are homotopy classes of continuous directed paths. The skeleton is the set π0(X) of path components (really a discrete category, i.e., in which the only arrows are the identities). Categories like this are the reason we prefer the name “arrow” instead of “morphism”.

Limit/Colimit: Let D ∶ I → C be a diagram in a category C (thus D is a functor and I is a small “index” category). A cone under D consists of

• an object c ∈ C,

• a collection of arrows αi ∶ x → D(i), one for each index i ∈ I,

such that for each arrow δ ∶ i → j in I we have αj = D(δ) ○ α

In visualizing this:

The cone (c,(αi)i∈I) is called a limit of the diagram D if, for any cone (z,(βi)i∈I) under D, the following picture holds:

[This picture means that there exists a unique arrow υ ∶ z → c such that, for each arrow δ ∶ i → j in I (including the identity arrows), the following diagram commutes:

When δ = idi this diagram just says that βi = αi ○ υ. We do not assume that D itself is commutative. Dually, a cone over D consists of an object c ∈ C and a set of arrows αi ∶ D(i) → c satisfying αi = αj ○ D(δ) for each arrow δ ∶ i → j in I. This cone is called a colimit of the diagram D if, for any cone (z,(βi)i∈I) over D, the following picture holds:

When the (unique) limit or colimit of the diagram D ∶ I → C exists, we denote it by (limI D, (φi)i∈I) or (colimI D, (φi)i∈I), respectively. Sometimes we omit the canonical arrows φi from the notation and refer to the object limID ∈ C as “the limit of D”. However, we should not forget that the arrows are part of the structure, i.e., the limit is really a cone.

Posets: Let P be a poset. We have already seen that the product/coproduct in P (if they exist) are the meet/join, respectively, and that the final/initial objects in P (if they exist) are the top/bottom elements, respectively. The only poset with a zero object is the one element poset.

Sets: The empty set ∅ ∈ Set is an initial object and the one point set ∗ ∈ Set is a final object. Note that two sets are isomorphic in Set precisely when there is a bijection between them, i.e., when they have the same cardinality. Since initial/final objects are unique up to isomorphism, we can identify the initial object with the cardinal number 0 and the final object with the cardinal number 1. There is no zero object in Set.

Products and coproducts exist in Set. The product of S,T ∈ Set consists of the Cartesian product S × T together with the canonical projections πS ∶ S × T → S and πT ∶ S × T → T. The coproduct of S, T ∈ Set consists of the disjoint union S ∐ T together with the canonical injections ιS ∶ S → S ∐ T and ιT ∶ T → S ∐ T. After passing to the skeleton, the product and coproduct of sets become the product and sum of cardinal numbers.

[Note: The “external disjoint union” S ∐ T is a formal concept. The familiar “internal disjoint union” S ⊔ T is only defined when there exists a set U containing both S and T as subsets. Then the union S ∪ T is the join operation in the Boolean lattice 2U ; we call the union “disjoint” when S ∩ T = ∅.]

Groups: The trivial group 1 ∈ Grp is a zero object, and for any groups G, H ∈ Grp the zero homomorphism 1 ∶ G → H sends all elements of G to the identity element 1H ∈ H. The product of groups G, H ∈ Grp is their direct product G × H and the coproduct is their free product G ∗ H, along with the usual canonical morphisms.

Let Ab ⊆ Grp be the full subcategory of abelian groups. The zero object and product are inherited from Grp, but we give them new names: we denote the zero object by 0 ∈ Ab and for any A, B ∈ Ab we denote the zero arrow by 0 ∶ A → B. We denote the Cartesian product by A ⊕ B and we rename it the direct sum. The big difference between Grp and Ab appears when we consider coproducts: it turns out that the product group A ⊕ B is also the coproduct group. We emphasize this fact by calling A ⊕ B the biproduct in Ab. It comes equipped with four canonical homomorphisms πA, πB, ιA, ιB satisfying the usual properties, as well as the following commutative diagram:

This diagram is the ultimate reason for matrix notation. The universal properties of product and coproduct tell us that each endomorphism φ ∶ A ⊕ B → A ⊕ B is uniquely determined by its four components φij ∶= πi ○ φ ○ ιj for i, j ∈ {A,B},so we can represent it as a matrix:

Then the composition of endomorphisms becomes matrix multiplication.

Rings. We let Rng denote the category of rings with unity, together with their homomorphisms. The initial object is the ring of integers Z ∈ Rng and the final object is the zero ring 0 ∈ Rng, i.e., the unique ring in which 0R = 1R. There is no zero object. The product of two rings R, S ∈ Rng is the direct product R × S ∈ Rng with component wise addition and multiplication. Let CRng ⊆ Rng be the full subcategory of commutative rings. The initial/final objects and product in CRng are inherited from Rng. The difference between Rng and CRng again appears when considering coproducts. The coproduct of R,S ∈ CRng is denoted by R ⊗Z S and is called the tensor product over Z…..