A monoidal category is a category where morphisms can be composed in parallel.

1 Monoidal category

A monoidal category is a category \(\cC\) equipped with

  • monoidal product: a bifunctor \(\otimes: \cC \times \cC \to \cC\)
  • monoidal unit: a \(\cC\)-object \(I\)
  • associator: a natural isomorphism \(\alpha_{A, B, C}: (A \otimes B) \otimes C \to A \otimes (B \otimes C)\)
  • left unitor: a natural isomorphism \(\lambda_A: I \otimes A \to A\)
  • right unitor: a natural isomorphism \(\rho_A: A \otimes I \to A\)

such that the triangle and pentagon diagrams commute.

A monoidal category is strict if \(\alpha\), \(\lambda\), and \(\rho\) are all identity morphisms.

1.1 Braided monoidal category

A braided monoidal category is a monoidal category \((\cC, \otimes, I, \alpha, \lambda, \rho)\) equipped with

  • braiding: a natural isomorphism \(\beta_{A, B}: A \otimes B \to B \otimes A\)

such that the hexagon diagram commutes.

1.2 Symmetric monoidal category

A symmetric monoidal category is a braided monoidal category \((\cC, \otimes, I, \alpha, \lambda, \rho, \beta)\) whose braiding \(\beta\) is symmetric: \(\beta_{B, A} \compL \beta_{A, B} = \id_{A \otimes B}\).

2 Monoidal functor

For monoidal categories \(\cC\) and \(\cD\), a lax monoidal functor is a functor \(F: \cC \to \cD\) equipped with

  • a natural transformation \(\mu_{A, B}: FA \otimes_\cD FB \to F(A \otimes_\cC B)\)
  • a \(\cD\)-morphism \(\eta: I_\cD \to FI_\cC\)

subject to

  • associativity: \((F\alpha_\cC) \compL \mu \compL (\mu \otimes \id) = \mu \compL (\id \otimes \mu) \compL \alpha_\cD\)
  • identity: \((F \lambda_\cC) \compL \mu \compL (\eta \otimes \id) = \lambda_\cD\) and \((F \rho_\cC) \compL \mu \compL (\id \otimes \eta) = \rho_\cD\)
-- applicative function application
(<*>) :: Applicative f => f (a -> b) -> f a -> f b
(<*)  :: Applicative f => f a -> f b -> f a
 (*>) :: Applicative f => f a -> f b -> f b

2.1 Properties

  • Lax: morphisms
  • Strong: isomorphisms
  • Strict: identity morphisms

2.2 Braided monoidal functor

A braided monoidal functor is a monoidal functor that is compatible with

  • braiding: \((F\beta_\cC) \compL \mu = \mu \compL \beta_\cD\)

2.3 Symmetric monoidal functor

A symmetric monoidal functor is a braided monoidal functor between symmetric monoidal categories.

3 Monoidal natural transformation

A monoidal natural transformation \(\alpha: F \to G\) between two monoidal functors is a natural transformation that is compatible with

  • monoidal product: \(\alpha \compL \mu_F = \mu_G \compL (\alpha \otimes \alpha)\)
  • monoidal unit: \(\alpha \compL \eta_F = \eta_G\)

\(\cMon\cCat\): 2-category of monoidal categories, monoidal functors, and monoidal natural transformations

4 Universal construction

4.1 Internal hom

Left internal hom:

\[A \otimes (-) \adj [A, -]\]

\[\Hom(A \otimes C, B) \iso \Hom(C, [A, B])\]

\[A \otimes [A, B] \xto{\epsilon_A} B \toot [A, B] \xto{\id_{[A, B]}} [A, B]\]

Right internal hom:

\[(-) \otimes A \adj [A, -]\]

\[\Hom(C \otimes A, B) \iso \Hom(C, [A, B])\]

\[[A, B] \otimes A \xto{\epsilon_A} B \toot [A, B] \xto{\id_{[A, B]}} [A, B]\]

A (left/right/bi) closed monoidal category has all (left/right/both side) internal homs.

Every right closed braided monoidal category becomes left closed in a canonical way, and vice versa.

4.2 Distributivity

Left distributive monoidal category:

\[A \otimes \coprod_i B_i \iso \coprod_i (A \otimes B_i)\]

Right distributive monoidal category:

\[\coprod_i A_i \otimes B \iso \coprod_i (A_i \otimes B)\]

Cocartesian and (left/right) closed monoidal category is (left/right) distributive.

5 Examples

5.1 Monoidal preorder

  • \((\N, \leq, +, 0)\)
  • \((\N, \mid, \times, 1)\)
  • \(\cBool = (\set{\lfal, \ltru}, \lfal \lproves \ltru, \lcon, \ltru)\)
  • \(\cCost = ([0, \infty], \geq, +, 0)\)

\(\exp(-x): ([0, \infty], \geq, +, 0) \iso ([0, 1], \leq, \times, 1): -\log(x)\)

5.2 Quantale

A (unital) quantale is a closed monoidal suplattice:

  • preorder \(\leq\) (category)
  • join \(\join\), bottom \(\bot\) (suplattice, cocartesian category)
  • monoidal product \(\otimes\), monoidal unit \(I\) (monoidal category)
  • internal hom \(\ihom\) (closed monoidal category)

By the adjoint functor theorem, a quantale is automatically a complete lattice (bicartesian category):

  • meet \(\meet\), top \(\top\) (inflattice, cartesian category)

Because a quantale is cocartesian and (left/right) closed, it is (left/right) distributive:

\[A \otimes \bigjoin_i B_i = \bigjoin_i (A \otimes B_i)\]

\[\bigjoin_i A_i \otimes B = \bigjoin_i (A_i \otimes B)\]

Additionally, a quantale can be:

  • commutative \(A \otimes B \iso B \otimes A\) (symmetric monoidal category)
  • affineness \(I = \top\)(semicartesian category)

A quantale homomorphism is a cocontinuous monoidal functor.

6 Monoid object

A monoid object in a monoidal category \((\cC, \otimes, I, \alpha, \lambda, \rho)\) is an object \(M\) equipped with two morphisms:

  • product \(\mu: M \otimes M \to M\)
  • unit \(\eta: I \to M\)

subject to

  • associativity: \(\mu \compL (\mu \otimes \id) = \mu \compL (\id \otimes \mu) \compL \alpha\)
  • identity: \(\mu \compL (\eta \otimes \id) = \lambda\) and \(\mu \compL (\id \otimes \eta) = \rho\)
-- mappend
(<>)   :: Monoid m => m -> m -> m
-- unit
mempty :: Monoid a => a
-- associativity
 x <> (y <> z) = (x <> y) <> z
-- left identity
 mempty <> x = x
-- right identity
 x <> mempty = x

6.1 Homomorphism

A monoid homomorphism between two monoid objects \(M\) and \(N\) is a morphism \(f: M \to N\) that preserves

  • monoidal product: \(f \compL \mu_M = \mu_N \compL (f \otimes f)\)
  • monoidal unit: \(f \compL \eta_M = \eta_N\)
  • a monoid object = a lax monoidal functor \(M: (\cOne, \otimes, *) \to (\cC, \otimes, I)\)
  • a monoid homomorphism = a monoidal natural transformation \(f: M \nat N\)

6.2 Examples

  • monoid: \(\cSet\)
  • ring: \(\cAb\)

A commutative monoid object in a symmetric monoidal category is a monoid object subject to commutativity.

A group object \(G\) in a cartesian category is a monoid object \((G, \mu, \eta)\) equipped with an inverse morphism \(\iota: G \to G\) subject to invertibility.

A comonoid object in \(\cC\) is a monoid object in \(\cC\op\).

  • coproduct \(\delta: M \to M \otimes M\)
  • counit \(\epsilon: M \to I\)

A monad is a monoid in the category of endofunctors \([\cC, \cC]\).