The Closed String Cochain Complex C is the String Theory Substitute for the de Rham Complex of Space-Time. Note Quote.

In closed string theory the central object is the vector space C = C_S¹ of states of a single parameterized string. This has an integer grading by the “ghost number”, and an operator Q : C → C called the “BRST operator” which raises the ghost number by 1 and satisfies Q² = 0. In other words, C is a cochain complex. If we think of the string as moving in a space-time M then C is roughly the space of differential forms defined along the orbits of the action of the reparametrization group Diff⁺(S¹) on the free loop space LM (more precisely, square-integrable forms of semi-infinite degree). Similarly, the space C of a topologically-twisted N = 2 supersymmetric theory, is a cochain complex which models the space of semi-infinite differential forms on the loop space of a Kähler manifold – in this case, all square-integrable differential forms, not just those along the orbits of Diff⁺(S¹). In both kinds of example, a cobordism Σ from p circles to q circles gives an operator U_Σ,μ : C^⊗p → C^⊗q which depends on a conformal structure μ on Σ. This operator is a cochain map, but its crucial feature is that changing the conformal structure μ on Σ changes the operator U_Σ,μ only by a cochain homotopy. The cohomology H(C) = ker(Q)/im(Q) – the “space of physical states” in conventional string theory – is therefore the state space of a topological field theory.

A good way to describe how the operator U_Σ,μ varies with μ is as follows:

If M_Σ is the moduli space of conformal structures on the cobordism Σ, modulo diffeomorphisms of Σ which are the identity on the boundary circles, then we have a cochain map

U_Σ : C^⊗p → Ω^∗(M_Σ, C^⊗q)

where the right-hand side is the de Rham complex of forms on M_Σ with values in C^⊗q. The operator U_Σ,μ is obtained from U_Σ by restricting from M_Σ to {μ}. The composition property when two cobordisms Σ₁ and Σ₂ are concatenated is that the diagram

commutes, where the lower horizontal arrow is induced by the map M_Σ1 × M_Σ2 → M_{Σ2 ◦ Σ1} which expresses concatenation of the conformal structures.

For each pair a, b of boundary conditions we shall still have a vector space – indeed a cochain complex – O_ab, but it is no longer the space of morphisms from b to a in a category. Rather, what we have is an A_∞-category. Briefly, this means that instead of a composition law O_ab × O_bc → O_ac we have a family of ways of composing, parametrized by the contractible space of conformal structures on the surface of the figure:

In particular, any two choices of a composition law from the family are cochain homotopic. Composition is associative in the sense that we have a contractible family of triple compositions O_ab × O_bc × O_cd → O_ad, which contains all the maps obtained by choosing a binary composition law from the given family and bracketing the triple in either of the two possible ways.

This is not the usual way of defining an A_∞-structure. According to Stasheff’s original definition, an A_∞-structure on a space X consists of a sequence of choices: first, a composition law m₂ : X × X → X; then, a choice of a map

m₃ : [0, 1] × X × X × X → X which is a homotopy between

(x, y, z) ↦ m₂(m₂(x, y), z) and (x, y, z) ↦ m₂(x, m₂(y, z)); then, a choice of a map

m₄ : S₄ × X₄ → X,

where S₄ is a convex plane polygon whose vertices are indexed by the five ways of bracketing a 4-fold product, and m₄|((∂S₄) × X₄) is determined by m₃; and so on. There is an analogous definition – applying to cochain complexes rather than spaces.

Apart from the composition law, the essential algebraic properties are the non-degenerate inner product, and the commutativity of the closed algebra C. Concerning the latter, when we pass to cochain theories the multiplication in C will of course be commutative up to cochain homotopy, but, the moduli space M_Σ of closed string multiplications i.e., the moduli space of conformal structures on a pair of pants Σ, modulo diffeomorphisms of Σ which are the identity on the boundary circles, is not contractible: it has the homotopy type of the space of ways of embedding two copies of the standard disc D² disjointly in the interior of D² – this space of embeddings is of course a subspace of M_Σ. In particular, it contains a natural circle of multiplications in which one of the embedded discs moves like a planet around the other, and there are two different natural homotopies between the multiplication and the reversed multiplication. This might be a clue to an important difference between stringy and classical space-times. The closed string cochain complex C is the string theory substitute for the de Rham complex of space-time, an algebra whose multiplication is associative and (graded)commutative on the nose. Over the rationals or the real or complex numbers, such cochain algebras model the category of topological spaces up to homotopy, in the sense that to each such algebra C, we can associate a space XC and a homomorphism of cochain algebras from C to the de Rham complex of X^C which is a cochain homotopy equivalence. If we do not want to ignore torsion in the homology of spaces we can no longer encode the homotopy type in a strictly commutative cochain algebra. Instead, we must replace commutative algebras with so-called E_∞-algebras, i.e., roughly, cochain complexes C over the integers equipped with a multiplication which is associative and commutative up to given arbitrarily high-order homotopies. An arbitrary space X has an E_∞-algebra C_X of cochains, and conversely one can associate a space X_C to each E_∞-algebra C. Thus we have a pair of adjoint functors, just as in rational homotopy theory. The cochain algebras of closed string theory have less higher commutativity than do E_∞-algebras, and this may be an indication that we are dealing with non-commutative spaces that fits in well with the interpretation of the B-field of a string background as corresponding to a bundle of matrix algebras on space-time. At the same time, the non-degenerate inner product on C – corresponding to Poincaré duality – seems to show we are concerned with manifolds, rather than more singular spaces.

Let us consider the category K of cochain complexes of finitely generated free abelian groups and cochain homotopy classes of cochain maps. This is called the derived category of the category of finitely generated abelian groups. Passing to cohomology gives us a functor from K to the category of Z-graded finitely generated abelian groups. In fact the subcategory K₀ of K consisting of complexes whose cohomology vanishes except in degree 0 is actually equivalent to the category of finitely generated abelian groups. But the category K inherits from the category of finitely generated free abelian groups a duality functor with properties as ideal as one could wish: each object is isomorphic to its double dual, and dualizing preserves exact sequences. (The dual C^∗ of a complex C is defined by (C^∗)ⁱ = Hom(C⁻ⁱ, Z).) There is no such nice duality in the category of finitely generated abelian groups. Indeed, the subcategory K₀ is not closed under duality, for the dual of the complex C_A corresponding to a group A has in general two non-vanishing cohomology groups: Hom(A,Z) in degree 0, and in degree +1 the finite group Ext¹(A,Z) Pontryagin-dual to the torsion subgroup of A. This follows from the exact sequence:

0 → Hom(A, Z) → Hom(F_A, Z) → Hom(R_A, Z) → Ext¹(A, Z) → 0

derived from an exact sequence

0 → R_A → F_A → A → 0

The category K also has a tensor product with better properties than the tensor product of abelian groups, and, better still, there is a canonical cochain functor from (locally well-behaved) compact spaces to K which takes Cartesian products to tensor products.

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Conjuncted: Of Topos and Torsors

The condition that each stalk F_x be equivalent to a classifying space BG can be summarized by saying that F is a gerbe on X: more precisely, it is a gerbe banded by the constant sheaf G associated to G.

For larger values of n, even the language of stacks is not sufficient to describe the nature of the sheaf F associated to the fibration X^~ → X. To address the situation, Grothendieck proposed (in his infamous letter to Quillen; see [35]) that there should be a theory of n-stacks on X, for every integer n ≥ 0. Moreover, for every sheaf of abelian groups G on X, the cohomology group Hⁿ⁺¹_sheaf(X;G) should have an interpreation as sheaf classifying a special type of n-stack: namely, the class of n-gerbes banded by G. In the special case where the space X is a point (and where we restrict our attention to n-stacks in groupoids), the theory of n-stacks on X should recover the classical homotopy theory of n-types: that is, CW complexes Z such that the homotopy groups π_i(Z, z) vanish for i > n (and every base point z ∈ Z). More generally, we should think of an n-stack (in groupoids) on a general space X as a “sheaf of n-types” on X.

When n = 0, an n-stack on a topological space X simply means a sheaf of sets on X. The collection of all such sheaves can be organized into a category Shv_Set(X), and this category is a prototypical example of a Grothendieck topos. The main goal of this book is to obtain an analogous understanding of the situation for n > 0. More precisely, we would like answers to the following questions:

(Q1) Given a topological space X, what should we mean by a “sheaf of n-types” on X?

(Q2)  Let Shv_≤n(X) denote the collection of all sheaves of n-types on X. What sort of a mathematical object is Shv_≤n(X)?

(Q3)  What special features (if any) does Shv_≤n(X) possess?

Our answers to questions (Q2) and (Q3) may be summarized as follows:

(A2)  The collection Shv_≤n(X) has the structure of an ∞-category.

(A3)  The ∞-category Shv_≤n(X) is an example of an (n+1)-topos: that is, an ∞-category which satisfies higher categorical analogues of Giraud’s axioms for Grothendieck topoi.

Grothendieck’s vision has been realized in various ways, thanks to the work of a number of mathematicians (most notably Jardine), and their work can also be used to provide answers to questions (Q1) and (Q2). Question (Q3) has also been addressed (at least in limiting case n = ∞) by Toën and Vezzosi. 

To provide more complete versions of the answers (A2) and (A3), we will need to develop the language of higher category theory. This is generally regarded as a technical and forbidding subject. More precisely, we will need a theory of (∞, 1)-categories: higher categories C for which the k-morphisms of C are required to be invertible for k > 1.

Classically, category theory is a useful tool not so much because of the light it sheds on any particular mathematical discipline, but instead because categories are so ubiquitous: mathematical objects in many different settings (sets, groups, smooth manifolds, etc.) can be organized into categories. Moreover, many elementary mathematical concepts can be described in purely categorical terms, and therefore make sense in each of these settings. For example, we can form products of sets, groups, and smooth manifolds: each of these notions can simply be described as a Cartesian product in the relevant category. Cartesian products are a special case of the more general notion of limit, which plays a central role in classical category theory. 

Diffeomorphism Diffeology via Leaves of Lie Foliation

The notion of diffeological space is due to Jean-Marie Souriau.

Let M be a set. Any set map α: U ⊂ Rⁿ → M defined on an open set U of some Rⁿ, n ≥ 0, will be called a plot on M. The name plot is chosen instead of chart to avoid some confusion with the usual notion of chart in a manifold. When possible, a plot α with domain U will be simply denoted by α_U.

A diffeology of class C^∞ on the set M is any collection P of plots α: U_α ⊂ R^n_α → M, n_α ≥ 0, verifying the following axioms:

1. (1)  Any constant map c: Rⁿ → M, n ≥ 0, belongs to P;
2. (2)  Let α ∈ P be defined on U ⊂ Rⁿ and let h: V ⊂ R^m → U ⊂ Rⁿ beany C^∞ map; then α ◦ h ∈ P;
3. (3)  Let α: U ⊂ Rⁿ → M be a plot. If any t ∈ U has a neighbourhood U_t such that α_|Ut belongs to P then α ∈ P.

Usually, a diffeology P on the set M is defined by means of a generating set, that is by giving any set G of plots (which is implicitly supposed to contain all constant maps) and taking the least diffeology containing it. Explicitly, the diffeology ⟨G⟩ generated by G is the set of plots α: U → M such that any point t ∈ U has a neighbourhood U_t where α can be written as γ ◦ h for some C^∞ map h and some γ ∈ G.

A finite dimensional manifold M is endowed with the diffeology generated by the charts U ⊂ Rⁿ → M, n = dimM, of any atlas.

Basic constructions. A map F : (M, P) → (N, Q) between diffeological spaces is differentiable if F ◦ α ∈ Q for all α ∈ P. A diffeomorphism is a differentiable map with a differentiable inverse.

Let (M,P) be a diffeological space and F : M → N a map of sets. The final diffeology F_⋆P on N is that generated by the plots F ◦ α, α ∈ P. A particular case is the quotient diffeology associated to an equivalence relation on M.

Analogously, let (N, Q) be a diffeological space and F : M → N a map of sets. The initial diffeology F_⋆Q on M is that generated by the plots α in M such that F ◦ α ∈ Q. A particular case is the induced diffeology on any subset M ⊂ N.

Finally, let D(M,N) be the space of differentiable maps between two diffeological spaces (M,P) and (N,Q). We define the functional diffeology on it by taking as a generating set all plots α: U → D(M,N) such that the associated map α~ : U × M → N given by α~ (t, x) = α(t)(x) is differentiable.

Diffeological groups.

Definition 2.3. A diffeological group is a diffeological space (G,P) endowed with a group structure such that the division map δ : G × G → G, δ(x, y) = xy⁻¹, is differentiable.

A typical example of diffeological group is the diffeomorphism group of a finite dimensional manifold M, endowed with the diffeology induced by D(M,M). It is proven that the diffeomorphism group of the space of leaves of a Lie foliation is a diffeological group too.

Both constructions verify the usual universal properties.

Let (M, P), (N, Q) be two diffeological spaces. We can endow the cartesian product M × N with the product diffeology P × Q generated by the plots α × β, α ∈ P, β ∈ Q.

The Ubiquity of Self-Predicative Universality of Adjoint Functors. Note Quote.

One of the most important and beautiful notions in category theory is the notion of a pair of adjoint functors. The developers of category theory, Saunders MacLane and Samuel Eilenberg, famously said that categories were defined in order to define functors, and functors were defined in order to define natural transformations. Adjoints were defined more than a decade later by Daniel Kan but the realization of their ubiquity (“Adjoint functors arise everywhere” (MacLane) and their foundational importance has steadily increased over time (Lawvere). Now it would perhaps not be too much of an exaggeration to see categories, functors, and natural transformations as the prelude to defining adjoint functors. The notion of adjoint functors includes all the instances of self-predicative universal mapping properties discussed above. As Steven Awodey (179) put it:

The notion of adjoint functor applies everything that we have learned up to now to unify and subsume all the different universal mapping properties that we have encountered, from free groups to limits to exponentials. But more importantly, it also captures an important mathematical phenomenon that is invisible without the lens of category theory. Indeed, I will make the admittedly provocative claim that adjointness is a concept of fundamental logical and mathematical importance that is not captured elsewhere in mathematics.

“The isolation and explication of the notion of adjointness is perhaps the most profound contribution that category theory has made to the history of general mathematical ideas.” (Goldblatt)

How do the ubiquitous and important adjoint functors relate to theme of self- predicative universals? MacLane and Birkhoff succinctly state the idea of the self-predicative universals of category theory and note that adjunctions can be analyzed in terms of those universals. The construction of a new algebraic object will often solve a specific problem in a universal way, in the sense that every other solution of the given problem is obtained from this one by a unique homomorphism. The basic idea of an adjoint functor arises from the analysis of such universals. (MacLane and Birkhoff)

We will use a specific novel treatment of adjunctions (Ellerman) that shows they arise by gluing together in a certain way two universal constructions or self-predicative universals (“semi-adjunctions”). But for illustration, we will stay within the methodological restriction of using examples from partial orders (where adjunctions are called “Galois connections”).

We have been working within the inclusion partial order on the set of subsets ζ(U) of a universe set U. Consider the set of all ordered pairs of subsets <a,b> from the Cartesian product ζ(U) x ζ(U) where the partial order (using the same symbol) is defined by pairwise inclusion. That is, given the two ordered pairs <a’, b’> and <a,b>, we define

<a’,b’> ⊆ <a,b> if a ⊆’  a and b ⊆’  b.

Order-preserving maps can be defined each way between these two partial orders. From ζ(U) to ζ(U) x ζ(U), there is the diagonal map Δ(x) = <x,x>, and from ζ(U) x ζ(U) to ζ(U), there is the meet map ∩(<a,b>)  = a ∩ b. Consider now the following “adjointness relation” between the two partial orders:

Δ(c) ⊆ <a,b> iff c ⊆ ∩ (<a,b>) Adjointness Equivalence

for sets a, b, and c in ζ(U). It has a certain symmetry that can be exploited. If we fix <a,b>, then we have the previous universality condition for the meet of a and b: for any c in ζ(U), c ⊆ a ∩ b iff Δ(c) ⊆ <a,b> Universality Condition for Meet of Sets a and b.

The defining property on elements c of ζ(U) is that Δ(c) ⊆ <a,b>. But using the symmetry, we could fix c and have another universality condition using the reverse inclusion in ζ(U) x ζ(U) as the participation relation: for any <a,b> in ζ(U) x ζ(U), <a,b> ⊇ Δ(c) iff c ⊆ a ∩  b. Universality Condition for Δ(c). Here the defining property on elements <a,b> of ζ(U) x ζ(U) is that “the meet of a and b is a superset of the given set c.” The self-predicative universal for that property is the image of c under the diagonal map Δ(c) = <c,c>, just as the self-predicative universal for the other property defined given <a,b> was the image of <a,b> under the meet map ∩(<a,b>) = a ∩ b.

Thus in this adjoint situation between the two categories ζ(U) and ζ(U) x ζ(U), we have a pair of maps (“adjoint functors”) going each way between the categories such that each element in a category defines a certain property in the other category and the map carries the element to the self-predicative universal for that property.

Δ: ζ(U) → ζ(U) x ζ(U) and ∩: ζ(U) x ζ(U) → ζ(U) Example of Adjoint Functors Between Partial Orders

The notion of a pair of adjoint functors is ubiquitous; it is one of the main tools that highlights self-predicative universals throughout modern mathematics.

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 Hom_C (x, y) of arrows. We will write α∶x→y to mean that α ∈ Hom_C(x,y). Each collection Hom_C(x,x) has a special element called the identity arrow id_x ∶ 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

○ ∶ Hom_C (x, y) × Hom_C(y, z) → Hom_C (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 Hom_C(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 Hom_C′(x,y) ⊆ Hom_C(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 ○ id_y = γ ○ (α ○ β) = (γ ○ α) ○ β = id_x ○ 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 Hom_C.

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 α_i ∶ x → D(i), one for each index i ∈ I,

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

In visualizing this:

The cone (c,(α_i)_i∈I) is called a limit of the diagram D if, for any cone (z,(β_i)_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 = id_i this diagram just says that β_i = α_i ○ 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 α_i ∶ D(i) → c satisfying α_i = α_j ○ 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,(β_i)_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 (lim_I D, (φ_i)_i∈I) or (colim_I D, (φ_i)_i∈I), respectively. Sometimes we omit the canonical arrows φ_i from the notation and refer to the object lim_ID ∈ 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 π_S ∶ S × T → S and π_T ∶ S × T → T. The coproduct of S, T ∈ Set consists of the disjoint union S ∐ T together with the canonical injections ι_S ∶ S → S ∐ T and ι_T ∶ 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 2^U ; 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 1_H ∈ 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 π_A, π_B, ι_A, ι_B 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 φ_ij ∶= π_i ○ φ ○ ι_j 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 0_R = 1_R. 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…..