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
| Monoidal category \(\mathcal{V}\) | Monoid object in \(\mathcal{V}\) |
|---|---|
| \((\mathbf{Set},\times,1)\) | monoid |
| \((\mathbf{Mon},\times,1)\) | commutative monoid, by Eckmann-Hilton |
| \((\mathbf{Proset},\times,1)\) | preordered monoid, aka monoidal proset |
| \((\mathbf{Poset},\times,1)\) | partially ordered monoid, aka monoidal poset |
| \((\mathbf{SupLat},\otimes,I)\) | (unital) quantale |
| \((\mathbf{Cat},\times,1)\) | strict monoidal category |
| \((\mathbf{CMon},\otimes,\mathbb{N})\) | rig |
| \((\mathbf{Ab},\otimes,\mathbb{Z})\) | (unital) ring |
| \((\mathbf{Mod}_R,\otimes,R)\) | (associative, unital) algebra over ring \(R\) |
| \((\mathbf{Vect}_k,\otimes,k)\) | algebra over field \(k\) |
| \((\mathbf{Ban}_k,\otimes,k)\) | (unital) Banach algebra |
| \((\mathbf{Species},\circ,I)\) | (symmetric) operad |
| \((\mathbf{End}_C,\circ,1_C)\) | monad on category \(C\) |
Generalizing the last example, a monad in a bicategory \(\mathcal{B}\) is an object \(B \in \mathcal{B}\) together with a monoid in \((\mathbf{End}_B,\circ,1_B)\). So, ordinary monads (monads on a category) are monads in \(\mathbf{Cat}\), the 2-category of categories. As another example, a monad in the bicategory of spans is a category! Specifically, for any set \(X\), a monoid in \(\mathbf{Span}(\mathbf{Set})(X,X)\) is a category with object set \(X\).
Literature
- Street, 2007: Quantum groups: A path to current algebra, Ch. 15: Monoids in
tensor categories (doi)
- Definition and theory of the 2-category \(\mathbf{Mon}(\mathcal{V})\) of monoids in a monoidal category \(\mathcal{V}\)