Symmetric Monoidal Categories

Table of Contents

• 1. Monoidal Categories
  • 1.1 The Bifunctor and Structural Isomorphisms
  • 1.2 The Pentagon and Triangle Axioms
  • 1.3 Strict Monoidal Categories
• 2. The Coherence Theorem
  • 2.1 Statement and Intuition
  • 2.2 Strictification
  • 2.3 Working Consequences
• 3. Examples
  • 3.1 Cartesian Monoidal Categories
  • 3.2 The Tensor Product of Abelian Groups
  • 3.3 Vector Spaces over a Field
  • 3.4 Chain Complexes
  • 3.5 Endofunctor Categories
  • 3.6 Based Spaces
  • 3.7 Spectra
• 4. Braided and Symmetric Monoidal Categories
  • 4.1 The Braiding
  • 4.2 The Hexagon Axioms
  • 4.3 Symmetric Monoidal Categories
  • 4.4 Braided versus Symmetric: the Braid Group
• 5. Monoidal Functors
  • 5.1 Lax, Strong, and Strict Monoidal Functors
  • 5.2 Monoidal Natural Transformations
  • 5.3 The 2-Category of Monoidal Categories
• 6. Algebras in a Monoidal Category
  • 6.1 Monoid Objects
  • 6.2 Commutative Monoids in Symmetric Monoidal Categories
  • 6.3 Classical Structures as Monoid Objects
• 7. Duals and Compact Closed Categories
  • 7.1 Left and Right Duals
  • 7.2 The Zig-Zag Identities
  • 7.3 Pivotal and Compact Closed Categories
  • 7.4 Graphical Calculus: a Hint
• References

1. Monoidal Categories 📐

A monoidal category is the categorical distillation of the notion of a “tensor product.” Informally, it is a category equipped with a binary operation on objects that is associative and unital — but only up to coherent natural isomorphism, not on the nose. This weakening is essential: the categories that arise in practice (modules, chain complexes, sets under Cartesian product) are almost never strictly associative, and forcing strict equality destroys functoriality.

The definition has six data and two axioms.

1.1 The Bifunctor and Structural Isomorphisms

Definition (Monoidal Category). A monoidal category is a tuple \((\mathcal{C}, \otimes, I, \alpha, \lambda, \rho)\) consisting of:

1. A category \(\mathcal{C}\).
2. A functor \(\otimes : \mathcal{C} \times \mathcal{C} \to \mathcal{C}\), called the tensor product or monoidal product.
3. An object \(I \in \mathcal{C}\), called the unit object.
4. A natural isomorphism \(\alpha\), called the associator, with components

\[\alpha_{A,B,C} : (A \otimes B) \otimes C \xrightarrow{\;\sim\;} A \otimes (B \otimes C),\]

natural in \(A, B, C \in \mathcal{C}\).

5. A natural isomorphism \(\lambda\), called the left unitor, with components

\[\lambda_A : I \otimes A \xrightarrow{\;\sim\;} A,\]

natural in \(A \in \mathcal{C}\).

6. A natural isomorphism \(\rho\), called the right unitor, with components

\[\rho_A : A \otimes I \xrightarrow{\;\sim\;} A,\]

natural in \(A \in \mathcal{C}\).

These six data are subject to two coherence axioms given in the next subsection.

1.2 The Pentagon and Triangle Axioms

The six pieces of data above are not free: they must satisfy two commutative diagrams. These diagrams are the only axioms; Mac Lane’s coherence theorem (§2) then guarantees that every other diagram built from \(\alpha\), \(\lambda\), \(\rho\) automatically commutes.

Axiom 1 (Pentagon Identity). For all \(A, B, C, D \in \mathcal{C}\), the following diagram commutes:

This diagram has five vertices (the five ways to parenthesize \(A \otimes B \otimes C \otimes D\)) and five edges. Its commutativity says that the two paths from \(((A \otimes B) \otimes C) \otimes D\) to \(A \otimes (B \otimes (C \otimes D))\) — one going left-and-down, the other going right-and-down — agree.

Axiom 2 (Triangle Identity). For all \(A, B \in \mathcal{C}\), the following diagram commutes:

This says that the two canonical ways of “deleting” the unit \(I\) from the middle of \(A \otimes I \otimes B\) agree.

1.3 Strict Monoidal Categories

Definition (Strict Monoidal Category). A monoidal category \((\mathcal{C}, \otimes, I, \alpha, \lambda, \rho)\) is strict if \(\alpha\), \(\lambda\), and \(\rho\) are all identity morphisms. Equivalently, \((A \otimes B) \otimes C = A \otimes (B \otimes C)\), \(I \otimes A = A\), and \(A \otimes I = A\) as equalities of objects, not merely isomorphisms.

Strict monoidal categories are conceptually simpler — all parenthesizations of a tensor product are literally equal — but they are rare in nature. The primary examples include:

• The endofunctor category \(\mathrm{End}(\mathcal{C}) = [\mathcal{C}, \mathcal{C}]\) with composition \(\circ\) and unit \(\mathrm{Id}_\mathcal{C}\) (see §3.5).
• Monoids regarded as one-object categories.
• The free strict monoidal category on a set of objects.

2. The Coherence Theorem 🔑

The pentagon and triangle identities look like a small fragment of all the coherence conditions one could impose. The miracle — proved by Mac Lane in 1963 — is that they are sufficient.

2.1 Statement and Intuition

To state the theorem precisely, we need to make “formal diagram built from \(\alpha\), \(\lambda\), \(\rho\)” rigorous.

Definition (Free Monoidal Category). Let \(S\) be a set of generating objects. The free monoidal category \(\mathcal{F}(S)\) on \(S\) has: - Objects: all well-formed words over \(S \cup \{I\}\) built by iterating \(\otimes\). - Morphisms: generated by \(\alpha\), \(\alpha^{-1}\), \(\lambda\), \(\lambda^{-1}\), \(\rho\), \(\rho^{-1}\) and their tensor products and composites, modulo the pentagon and triangle identities.

Theorem (Mac Lane’s Coherence Theorem). In the free monoidal category \(\mathcal{F}(S)\), there is at most one morphism between any two objects. Equivalently, every diagram in a monoidal category that is built entirely from instances of \(\alpha^{\pm 1}\), \(\lambda^{\pm 1}\), \(\rho^{\pm 1}\) and tensor products and composites thereof commutes.

The intuition: think of a parenthesization of \(A_1 \otimes \cdots \otimes A_n\) as a binary tree. The associator rebalances trees. Coherence says: any two paths of rebalancing operations connecting the same two trees yield the same composite. The proof proceeds by reducing every such diagram to a diagram in a free monoidal category and then exhibiting an explicit normal form (the fully left-associated bracketing) for each object, showing the reduction maps are well-defined.

2.2 Strictification

A stronger consequence of coherence is the strictification theorem:

Theorem (Strictification). Every monoidal category \(\mathcal{C}\) is monoidally equivalent to a strict monoidal category \(\mathcal{C}^{\mathrm{st}}\).

The construction of \(\mathcal{C}^{\mathrm{st}}\) is explicit. Objects of \(\mathcal{C}^{\mathrm{st}}\) are finite lists \([A_1, \ldots, A_n]\) of objects of \(\mathcal{C}\) (including the empty list \([]\) as the unit). The tensor product of two lists is concatenation: \([A_1, \ldots, A_m] \otimes [B_1, \ldots, B_n] = [A_1, \ldots, A_m, B_1, \ldots, B_n]\). This is strictly associative because list concatenation is strictly associative. Morphisms from \([A_1, \ldots, A_n]\) to \([B_1, \ldots, B_m]\) are morphisms \(A_1 \otimes \cdots \otimes A_n \to B_1 \otimes \cdots \otimes B_m\) in \(\mathcal{C}\) (where the left and right sides use any fixed parenthesization, with coherence ensuring this is well-defined).

The inclusion functor \(i : \mathcal{C} \to \mathcal{C}^{\mathrm{st}}\) sending \(A \mapsto [A]\) is a strong monoidal equivalence.

2.3 Working Consequences

In practice, Mac Lane’s coherence theorem licenses the following conventions:

1. Drop parentheses: Write \(A \otimes B \otimes C\) unambiguously, since all parenthesizations are canonically isomorphic and the isomorphisms are coherent.
2. Drop unit instances: Write \(A\) for \(I \otimes A\) or \(A \otimes I\), since the unitors provide canonical isomorphisms and these are coherent with the associator.
3. Reorder structural morphisms freely: Any composite of \(\alpha^{\pm 1}\), \(\lambda^{\pm 1}\), \(\rho^{\pm 1}\) connecting the same source and target is equal.

Key conclusion: once the pentagon and triangle are verified, no further coherence work is needed for morphisms built from the structural isomorphisms.

3. Examples 💡

3.1 Cartesian Monoidal Categories

Example (Set, ×, 1). The category \(\mathbf{Set}\) with the Cartesian product \(\times\) and the one-element set \(\mathbf{1} = \{*\}\) is a monoidal category. Concretely:

• \(\otimes = \times\): on objects, \(A \times B = \{(a,b) : a \in A, b \in B\}\); on morphisms, \((f \times g)(a,b) = (f(a),g(b))\).
• Unit object: any singleton \(\mathbf{1}\).
• Associator: \(\alpha_{A,B,C}((a,b),c) = (a,(b,c))\).
• Left unitor: \(\lambda_A(*,a) = a\).
• Right unitor: \(\rho_A(a,*) = a\).

More generally, any category \(\mathcal{C}\) with finite products and a terminal object \(\mathbf{1}\) is a Cartesian monoidal category with \(A \otimes B = A \times B\) and \(I = \mathbf{1}\). Examples: \(\mathbf{Top}\), \(\mathbf{Cat}\), \(\mathbf{Grp}\), any topos.

3.2 The Tensor Product of Abelian Groups

Example (Ab, tensor, Z). The category \(\mathbf{Ab}\) of abelian groups with the tensor product \(\otimes_\mathbb{Z}\) and unit \(\mathbb{Z}\) is a monoidal category. Recall that for abelian groups \(A\) and \(B\), the tensor product \(A \otimes_\mathbb{Z} B\) is the abelian group generated by symbols \(a \otimes b\) subject to bilinearity:

\[a \otimes (b + b') = a \otimes b + a \otimes b', \qquad (a + a') \otimes b = a \otimes b + a' \otimes b.\]

The associator \(\alpha_{A,B,C} : (A \otimes B) \otimes C \to A \otimes (B \otimes C)\) is defined on generators by \((a \otimes b) \otimes c \mapsto a \otimes (b \otimes c)\) and extended by linearity. The left unitor \(\lambda_A : \mathbb{Z} \otimes A \to A\) is \(n \otimes a \mapsto na\), and similarly for \(\rho\).

This monoidal structure is closed: there is an internal hom \([A, B] = \mathrm{Hom}_\mathbb{Z}(A, B)\) satisfying \(\mathrm{Hom}(A \otimes B, C) \cong \mathrm{Hom}(A, [B,C])\) naturally.

3.3 Vector Spaces over a Field

Example (Vect_k, tensor, k). Let \(k\) be a field. The category \(\mathbf{Vect}_k\) of \(k\)-vector spaces with the tensor product \(\otimes_k\) and unit \(k\) is a monoidal category. For finite-dimensional vector spaces, \(\dim(V \otimes W) = \dim(V) \cdot \dim(W)\).

This example is fundamental: it is the home of representation theory (representations of a group \(G\) are monoid objects in a monoidal category of \(G\)-modules), linear algebra (matrix multiplication is composition in the enriched hom), and — with the symmetric monoidal structure added — commutative algebra (commutative algebras over \(k\) are commutative monoid objects in \(\mathbf{Vect}_k\)).

3.4 Chain Complexes

Example (Ch(R), tensor, R[0]). Let \(R\) be a commutative ring. The category \(\mathbf{Ch}(R)\) of chain complexes of \(R\)-modules, with the tensor product of chain complexes and unit \(R[0]\) (the complex with \(R\) in degree \(0\) and \(0\) elsewhere), is a monoidal category.

The tensor product of complexes \((A_\bullet, d^A)\) and \((B_\bullet, d^B)\) is

\[(A \otimes B)_n = \bigoplus_{p+q=n} A_p \otimes_R B_q\]

with differential \(d^{A \otimes B}(a \otimes b) = d^A(a) \otimes b + (-1)^{|a|} a \otimes d^B(b)\), where \(|a|\) is the degree of \(a\). The sign \((-1)^{|a|}\) is the Koszul sign rule, which is necessary for \(d^2 = 0\) and for the associator and unitors to be chain maps.

3.5 Endofunctor Categories

Example (End(C), composition, Id). For any category \(\mathcal{C}\), the functor category \(\mathrm{End}(\mathcal{C}) = [\mathcal{C}, \mathcal{C}]\) of endofunctors is a strict monoidal category with:

• Tensor product: functor composition \(F \otimes G = F \circ G\).
• Unit: the identity functor \(\mathrm{Id}_\mathcal{C}\).
• Associator, unitors: all identity natural transformations (composition of functors is strictly associative and \(\mathrm{Id}\) is strictly a unit).

A monad on \(\mathcal{C}\) is exactly a monoid object in \((\mathrm{End}(\mathcal{C}), \circ, \mathrm{Id})\) (see Adjoint Functor Theorems and Monads).

3.6 Based Spaces

Example (Top, ×, ). The category \(\mathbf{Top}_*\) of based (pointed) topological spaces carries three distinct symmetric monoidal structures:

The Cartesian and coCartesian structures are the categorical product and coproduct respectively, and exist in any category with finite (co)products. The smash product is the genuinely new one.

Why the smash product is the right choice. 💡 The Cartesian and wedge structures exist on \(\mathbf{Top}_*\) for the same reason they exist on every pointed category — they are forced by the universal properties of products and coproducts. They carry no information specific to the based structure. The smash product, by contrast, is the structure that makes \(\mathbf{Top}_*\) behave like a tensor category:

• Closed structure. \((\mathbf{Top}_*, \wedge, S^0)\) is closed symmetric monoidal with internal hom \(\mathbf{Map}_*(A, B)\) (the based mapping space), satisfying \(\mathbf{Map}_*(A \wedge B, C) \cong \mathbf{Map}_*(A, \mathbf{Map}_*(B, C))\). Neither \(\times\) nor \(\vee\) yields a closed structure on \(\mathbf{Top}_*\).
• Suspension and loops. The suspension \(\Sigma A = S^1 \wedge A\) and the loop-suspension adjunction \([\Sigma A, B]_* \cong [A, \Omega B]_*\) both arise directly from the closed structure of \(\wedge\). The formula \(\Sigma^n A = S^n \wedge A\) makes the smash product the natural language for stable homotopy.
• The unit is \(S^0\), not \(*\). The unit for \(\wedge\) is the two-point space \(S^0 = \{0,1\}\) (basepoint at \(0\)). This is not the terminal object — it is the sphere spectrum’s zeroth space. Under stabilization, \(S^0\) becomes the sphere spectrum \(\mathbb{S}\), the unit for the smash product of spectra.
• Distributivity. The smash product distributes over the wedge: \(A \wedge (B \vee C) \cong (A \wedge B) \vee (A \wedge C)\). This is the “bilinearity” that makes the smash product the correct analogue of \(\otimes_k\) for based spaces.

3.7 Spectra

Stabilizing \((\mathbf{Top}_*, \wedge, S^0)\) — inverting the suspension functor \(\Sigma = S^1 \wedge -\) — produces the category of spectra \(\mathbf{Sp}\), the natural home for stable homotopy theory.

Example (Sp, smash, S). The category \(\mathbf{Sp}\) of spectra with the smash product \(\wedge\) and unit the sphere spectrum \(\mathbb{S}\) is a symmetric monoidal category. Achieving this at the point-set level (not merely up to homotopy) was a major technical achievement of 1990s homotopy theory:

All four point-set models are Quillen equivalent — they present the same homotopy theory. The \(\infty\)-categorical formulation is the canonical one: \((\mathbf{Sp}, \wedge, \mathbb{S})\) is the universal presentably symmetric monoidal stable \(\infty\)-category.

🔑 Monoid objects in \((\mathbf{Sp}, \wedge, \mathbb{S})\) are ring spectra:

The last row is the key comparison: the derived category of abelian groups \(D(\mathbb{Z})\) is equivalent to \(H\mathbb{Z}\)-module spectra, with the tensor product \(\otimes^\mathbf{L}_\mathbb{Z}\) corresponding to \(\wedge_{H\mathbb{Z}}\). This is why spectra are the “universal” generalization of homological algebra: \((\mathbf{Sp}, \wedge, \mathbb{S})\) stands above \(D(R)\) for any ring \(R\) in the same way \((\mathbf{Ab}, \otimes_\mathbb{Z}, \mathbb{Z})\) stands above modules over any ring.

4. Braided and Symmetric Monoidal Categories 📐

A monoidal category lets us tensor objects together but gives no canonical way to swap the two factors. A braided monoidal category adds a natural isomorphism \(\beta_{A,B}: A \otimes B \to B \otimes A\) subject to coherence. A symmetric monoidal category imposes the additional condition that swapping twice returns to the identity.

4.1 The Braiding

Definition (Braided Monoidal Category). A braided monoidal category is a monoidal category \((\mathcal{C}, \otimes, I, \alpha, \lambda, \rho)\) equipped with a natural isomorphism \(\beta\), called the braiding, with components

\[\beta_{A,B} : A \otimes B \xrightarrow{\;\sim\;} B \otimes A,\]

natural in \(A, B \in \mathcal{C}\), subject to the two hexagon axioms below.

Naturality of \(\beta\) means that for any morphisms \(f : A \to A'\) and \(g : B \to B'\), the square

commutes.

4.2 The Hexagon Axioms

The braiding must be compatible with the associator via two hexagon identities.

Axiom (First Hexagon Identity). For all \(A, B, C \in \mathcal{C}\):

Axiom (Second Hexagon Identity). For all \(A, B, C \in \mathcal{C}\):

The first hexagon says: braiding \(A\) past \(B \otimes C\) in one step equals braiding \(A\) past \(B\) and then past \(C\) (with appropriate reassociation). The second is the mirror image.

4.3 Symmetric Monoidal Categories

Definition (Symmetric Monoidal Category). A symmetric monoidal category is a braided monoidal category \((\mathcal{C}, \otimes, I, \alpha, \lambda, \rho, \beta)\) such that for all objects \(A, B \in \mathcal{C}\):

\[\beta_{B,A} \circ \beta_{A,B} = \mathrm{id}_{A \otimes B}.\]

This condition says: the braiding is its own inverse. Swapping \(A \otimes B\) to \(B \otimes A\) and then swapping back yields the identity. In a symmetric monoidal category, \(\beta_{A,B}^{-1} = \beta_{B,A}\).

For chain complexes \(\mathbf{Ch}(R)\), the symmetry isomorphism includes a Koszul sign: \(\beta_{A,B}(a \otimes b) = (-1)^{|a||b|} b \otimes a\). The sign is necessary for \(\beta^2 = \mathrm{id}\) at the chain level (since \((-1)^{|a||b|} \cdot (-1)^{|b||a|} = (-1)^{2|a||b|} = 1\)) and for the hexagon axioms.

Coherence for Symmetric Monoidal Categories. Mac Lane and Kelly proved an analogue of the coherence theorem: every symmetric monoidal category is equivalent to a permutative category (a strict symmetric monoidal category). The free symmetric monoidal category on objects \(\{A_1, \ldots, A_n\}\) has morphisms that are permutations in \(\Sigma_n\) (the symmetric group on \(n\) letters), and every diagram built from \(\alpha^{\pm 1}\), \(\lambda^{\pm 1}\), \(\rho^{\pm 1}\), \(\beta^{\pm 1}\) commutes.

4.4 Braided versus Symmetric: the Braid Group

The distinction between braided and symmetric monoidal categories is captured by the action on \(n\)-fold tensor products.

In a braided monoidal category, the group acting on \(A^{\otimes n}\) (via the braidings \(\beta_{A,A}\)) is the braid group \(B_n\), generated by elementary braids \(\sigma_i\) (swapping the \(i\)-th and \((i+1)\)-th factor) subject to: - \(\sigma_i \sigma_j = \sigma_j \sigma_i\) for \(|i - j| \geq 2\). - \(\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}\) (braid relation).

In a symmetric monoidal category, the additional condition \(\beta^2 = \mathrm{id}\) forces \(\sigma_i^2 = \mathrm{id}\), quotienting \(B_n\) to the symmetric group \(\Sigma_n\). The group acting on \(A^{\otimes n}\) in a symmetric monoidal category is therefore \(\Sigma_n\).

5. Monoidal Functors 📐

A functor between monoidal categories should “respect” the monoidal structure. But “respect” can mean several things: strictly preserve it, preserve it up to coherent isomorphism, or merely preserve it up to coherent morphism. These give three levels of monoidal functor.

5.1 Lax, Strong, and Strict Monoidal Functors

Definition (Lax Monoidal Functor). Let \((\mathcal{C}, \otimes, I)\) and \((\mathcal{D}, \otimes', I')\) be monoidal categories. A lax monoidal functor \((F, \varphi, \varphi_0)\) consists of:

1. A functor \(F : \mathcal{C} \to \mathcal{D}\).
2. A natural transformation \(\varphi\) with components \[\varphi_{A,B} : F(A) \otimes' F(B) \to F(A \otimes B),\] natural in \(A, B \in \mathcal{C}\).
3. A morphism \(\varphi_0 : I' \to F(I)\).

These must satisfy three coherence conditions:

Associativity coherence: For all \(A, B, C \in \mathcal{C}\), the following diagram commutes:

Left unit coherence: For all \(A \in \mathcal{C}\):

Right unit coherence: For all \(A \in \mathcal{C}\):

Definition (Strong Monoidal Functor). A lax monoidal functor \((F, \varphi, \varphi_0)\) is strong (or monoidal without further qualification) if \(\varphi_{A,B}\) is an isomorphism for all \(A, B\) and \(\varphi_0\) is an isomorphism.

Definition (Strict Monoidal Functor). A lax monoidal functor is strict if \(\varphi_{A,B} = \mathrm{id}_{FA \otimes' FB}\) and \(\varphi_0 = \mathrm{id}_{I'}\) (which forces \(F(A \otimes B) = F(A) \otimes' F(B)\) and \(F(I) = I'\) on the nose).

5.2 Monoidal Natural Transformations

Definition (Monoidal Natural Transformation). Let \((F, \varphi, \varphi_0)\) and \((G, \psi, \psi_0)\) be lax monoidal functors \(\mathcal{C} \to \mathcal{D}\). A monoidal natural transformation \(\eta : F \Rightarrow G\) is a natural transformation such that:

1. For all \(A, B \in \mathcal{C}\): \(\psi_{A,B} \circ (\eta_A \otimes' \eta_B) = \eta_{A \otimes B} \circ \varphi_{A,B}\).
2. \(\psi_0 = \eta_I \circ \varphi_0\).

In diagrams, condition 1 is:

5.3 The 2-Category of Monoidal Categories

Monoidal categories, lax monoidal functors, and monoidal natural transformations organize into a 2-category \(\mathbf{MonCat}\). This 2-category has: - Objects: monoidal categories. - 1-morphisms: lax monoidal functors. - 2-morphisms: monoidal natural transformations.

One similarly has \(\mathbf{MonCat}_{\mathrm{str}}\) (strong monoidal functors) and \(\mathbf{SymMonCat}\) (symmetric monoidal categories with symmetric monoidal functors).

Proposition. A monoidal equivalence (an equivalence in \(\mathbf{MonCat}_{\mathrm{str}}\), i.e., a strong monoidal functor with a strong monoidal inverse up to monoidal natural isomorphism) is the appropriate notion of “isomorphism” of monoidal categories.

6. Algebras in a Monoidal Category 📐

The notion of an “algebra” or “monoid” can be internalized to any monoidal category. This single abstraction simultaneously captures associative algebras, rings, monoids, operads (of a sort), and monads.

6.1 Monoid Objects

Definition (Monoid Object). A monoid object (or algebra object, or internal monoid) in a monoidal category \((\mathcal{C}, \otimes, I)\) is a triple \((M, \mu, \eta)\) where:

• \(M \in \mathcal{C}\) is an object.
• \(\mu : M \otimes M \to M\) is a morphism, called multiplication.
• \(\eta : I \to M\) is a morphism, called the unit.

These must satisfy:

Associativity: The following diagram commutes:

Left unit law: The following diagram commutes:

Right unit law: The following diagram commutes:

Definition (Monoid Morphism). A morphism of monoid objects \(f : (M, \mu, \eta) \to (M', \mu', \eta')\) is a morphism \(f : M \to M'\) in \(\mathcal{C}\) such that \(f \circ \mu = \mu' \circ (f \otimes f)\) and \(f \circ \eta = \eta'\).

The category of monoid objects in \((\mathcal{C}, \otimes, I)\) is denoted \(\mathbf{Mon}(\mathcal{C})\).

6.2 Commutative Monoids in Symmetric Monoidal Categories

In a symmetric monoidal category, there is a canonical isomorphism \(\beta_{M,M} : M \otimes M \to M \otimes M\). This permits a notion of commutativity.

Definition (Commutative Monoid Object). A monoid object \((M, \mu, \eta)\) in a symmetric monoidal category \((\mathcal{C}, \otimes, I, \beta)\) is commutative if

\[\mu \circ \beta_{M,M} = \mu.\]

This says the multiplication is unchanged by swapping the two inputs.

6.3 Classical Structures as Monoid Objects

The following table records classical algebraic structures as monoid objects:

A DG algebra (differential graded algebra) is an associative algebra internal to \((\mathbf{Ch}(R), \otimes, R[0])\): it is a chain complex \(A_\bullet\) equipped with a chain map \(\mu : A \otimes A \to A\) that is associative and unital. A commutative DG algebra satisfies \(\mu(a \otimes b) = (-1)^{|a||b|}\mu(b \otimes a)\) (the Koszul sign appears because \(\beta_{A,A}\) in \(\mathbf{Ch}(R)\) carries a sign).

7. Duals and Compact Closed Categories 🔑

In a monoidal category, some objects admit “duals” — objects that interact with them via unit and counit morphisms satisfying zig-zag identities. In finite-dimensional linear algebra, every vector space is dualizable, with dual \(V^* = \mathrm{Hom}(V,k)\). Compact closed categories are those where every object is dualizable.

7.1 Left and Right Duals

In a general (non-symmetric) monoidal category, left and right duals are distinct.

Definition (Right Dual). Let \((\mathcal{C}, \otimes, I)\) be a monoidal category and \(A \in \mathcal{C}\). A right dual of \(A\) is an object \(A^*\) together with morphisms

\[\eta : I \to A \otimes A^* \qquad \text{(unit, or coevaluation)}\] \[\varepsilon : A^* \otimes A \to I \qquad \text{(counit, or evaluation)}\]

satisfying the zig-zag identities (see §7.2).

Definition (Left Dual). A left dual of \(A\) is an object \({}^*A\) together with morphisms

\[\eta' : I \to {}^*A \otimes A \qquad \text{(unit)}\] \[\varepsilon' : A \otimes {}^*A \to I \qquad \text{(counit)}\]

satisfying the zig-zag identities.

7.2 The Zig-Zag Identities

The unit \(\eta\) and counit \(\varepsilon\) of a right duality \((A, A^*, \eta, \varepsilon)\) must satisfy the zig-zag identities (also called snake identities or triangle identities for duals):

First zig-zag (for \(A\)):

\[(\mathrm{id}_A \otimes \varepsilon) \circ \alpha_{A,A^*,A}^{-1} \circ (\eta \otimes \mathrm{id}_A) = \mathrm{id}_A\]

using left/right unitors to identify \(I \otimes A \cong A \cong A \otimes I\).

Second zig-zag (for \(A^*\)):

\[(\varepsilon \otimes \mathrm{id}_{A^*}) \circ \alpha_{A^*,A,A^*} \circ (\mathrm{id}_{A^*} \otimes \eta) = \mathrm{id}_{A^*}\]

using unitors similarly.

In a strict monoidal category (dropping the unitors for readability), these become:

\[(\mathrm{id}_A \otimes \varepsilon) \circ (\eta \otimes \mathrm{id}_A) = \mathrm{id}_A, \qquad (\varepsilon \otimes \mathrm{id}_{A^*}) \circ (\mathrm{id}_{A^*} \otimes \eta) = \mathrm{id}_{A^*}.\]

Proposition. If a right dual \((A^*, \eta, \varepsilon)\) exists, it is unique up to unique isomorphism.

Proof sketch: Suppose \((B, \eta_B, \varepsilon_B)\) is another right dual. Define \(f : A^* \to B\) by \(f = \lambda_B \circ (\varepsilon_B \otimes \mathrm{id}_{A^*}) \circ (\mathrm{id}_{A^*} \otimes \eta)\) and \(g : B \to A^*\) similarly. The zig-zag identities imply \(g \circ f = \mathrm{id}_{A^*}\) and \(f \circ g = \mathrm{id}_B\). \(\square\)

7.3 Pivotal and Compact Closed Categories

Definition (Rigid Monoidal Category). A monoidal category is left rigid (resp. right rigid) if every object has a left dual (resp. right dual). It is rigid (or autonomous) if it is both.

Definition (Compact Closed Category). A compact closed category is a symmetric monoidal category in which every object has a dual (equivalently, left and right duals coincide in the symmetric case).

The following categories are compact closed:

Definition (Pivotal Category). A pivotal category is a rigid monoidal category equipped with a monoidal natural isomorphism \(A \cong A^{**}\) (the double dual isomorphism). In a pivotal category, one can define a trace for endomorphisms.

7.4 Graphical Calculus: a Hint

Compact closed categories admit a graphical calculus (also called string diagrams) in which:

• Objects \(A\) are represented by oriented strings (wires), with \(A^*\) by the same string oriented oppositely.
• Morphisms \(f : A \to B\) are represented by boxes with input wire \(A\) at the bottom and output wire \(B\) at the top.
• The tensor product is horizontal juxtaposition of diagrams.
• Composition is vertical stacking.
• The unit \(\eta : I \to A \otimes A^*\) is a cup (a U-shaped wire).
• The counit \(\varepsilon : A^* \otimes A \to I\) is a cap (an inverted U-shaped wire).

The zig-zag identities say exactly that a cup followed by a cap (forming a zig-zag in the string) can be “straightened” to a straight wire. This graphical intuition is not merely heuristic — by a theorem of Joyal and Street, the graphical calculus is sound and complete: a diagram of string diagrams commutes if and only if the corresponding equation of morphisms holds in every compact closed category.

Mathematical Development Exercises (continued)

Algorithmic Applications

import numpy as np

def tensor_of_maps(F: np.ndarray, G: np.ndarray) -> np.ndarray:
    """
    Compute the matrix of f ⊗ g.
    F has shape (dim_V', dim_V); G has shape (dim_W', dim_W).
    Output has shape (dim_V' * dim_W', dim_V * dim_W).
    Basis ordering: (i,j) -> i * dim_W + j (row-major).
    """
    return np.kron(F, G)

def is_commutative_monoid(C, mu, beta_MM):
    """
    C: monoidal category with decidable morphism equality
    mu: morphism M ⊗ M -> M (the multiplication)
    beta_MM: morphism M ⊗ M -> M ⊗ M (the symmetry)
    Returns True iff mu ∘ beta_MM == mu
    """
    lhs = compose(mu, beta_MM)   # mu ∘ beta_{M,M}
    return morphisms_equal(C, lhs, mu)

def simplify_zig_zags(diagram):
    """
    diagram: list of (type, wires) pairs, where type in
             {'box', 'cup', 'cap', 'id'} and wires are wire IDs.
    Returns simplified diagram with all cup-cap (zig-zag) pairs removed.
    """
    changed = True
    while changed:
        changed = False
        for i, node_i in enumerate(diagram):
            if node_i.type == 'cup':
                # Look for a cap that immediately follows on the same wire
                for j, node_j in enumerate(diagram):
                    if node_j.type == 'cap' and forms_zig_zag(node_i, node_j, diagram):
                        diagram = remove_zig_zag(diagram, i, j)
                        changed = True
                        break
    return diagram

References