Monoid objects

As an example of the microcosm princple , one can define a monoid (or monoid object) in any monoidal category. Similarly, commutative monoid objects can be defined in any symmetric monoidal category. These are useful special cases of algebras of a PRO in a monoidal category and of algebras of a PROP in a symmetric monoidal category, respectively.

Examples

Table 1: My favorite examples of monoids in monoidal categories
Monoidal category V\mathcal{V} Monoid object in V\mathcal{V}
(Set,×,1)(\mathbf{Set},\times,1) monoid
(Mon,×,1)(\mathbf{Mon},\times,1) commutative monoid, by Eckmann-Hilton
(Proset,×,1)(\mathbf{Proset},\times,1) preordered monoid, aka monoidal proset
(Poset,×,1)(\mathbf{Poset},\times,1) partially ordered monoid, aka monoidal poset
(SupLat,⊗,I)(\mathbf{SupLat},\otimes,I) (unital) quantale
(Cat,×,1)(\mathbf{Cat},\times,1) strict monoidal category
(CMon,⊗,N)(\mathbf{CMon},\otimes,\mathbb{N}) rig
(Ab,⊗,Z)(\mathbf{Ab},\otimes,\mathbb{Z}) (unital) ring
(ModR,⊗,R)(\mathbf{Mod}_R,\otimes,R) (associative, unital) algebra over ring RR
(Vectk,⊗,k)(\mathbf{Vect}_k,\otimes,k) algebra over field kk
(Bank,⊗,k)(\mathbf{Ban}_k,\otimes,k) (unital) Banach algebra
(Species,∘,I)(\mathbf{Species},\circ,I) (symmetric) operad
(EndC,∘,1C)(\mathbf{End}_C,\circ,1_C) monad on category CC

Generalizing the last example, a monad in a bicategory B\mathcal{B} is an object B∈BB \in \mathcal{B} together with a monoid in (EndB,∘,1B)(\mathbf{End}_B,\circ,1_B). So, ordinary monads (monads on a category) are monads in Cat\mathbf{Cat}, the 2-category of categories. As another example, a monad in the bicategory of spans is a category! Specifically, for any set XX, a monoid in Span(Set)(X,X)\mathbf{Span}(\mathbf{Set})(X,X) is a category with object set XX.

Literature

  • Street, 2007: Quantum groups: A path to current algebra, Ch. 15: Monoids in tensor categories (doi)
    • Definition and theory of the 2-category Mon(V)\mathbf{Mon}(\mathcal{V}) of monoids in a monoidal category V\mathcal{V}