What is a C-Monoid?

Question

I am fairly new to category theory and have came across C-Monoids while studying the mapping of $\lambda$-calculus to category theory.

I would appreciate if someone can explain me what a C-Monoid is in simple language?

Answer 1

I'm assuming that what you're after is the definition and sort of the story behind them, rather than any specific theorems about them.

Briefest definition: it's a monoid that, as a one-object category, has binary products and exponentials. Somewhat more concretely, it's a monoid with extra operations $\langle-,-\rangle$ and $(-)^*$ representing pairing and exponential transpose, and special elements $\pi,\pi',\varepsilon$ that are left and right projection, and evaluation, all of which satisfy equalities like

• $\pi\langle a,b\rangle=a$
• $\pi'\langle a,b\rangle=b$
• $\langle\pi a,\pi' a\rangle=a$
• $\varepsilon\langle h^*\pi,\pi'\rangle=h$
• $(\varepsilon\langle k\pi,\pi'\rangle)^*=k$

If we think of a monoid as a category with a single object $A$, the elements of the monoid are generalized elements of $A$, but also generalized functions $A\to A$ and generalized pairs of elements in $A\times A$. Defining things the right way, this is enough machinery to interpret closed lambda expressions (in a certain extended language) as elements of the monoid, and define an application operation (which is not the same as the monoid operation). Conversely, with a model of the untyped lambda calculus one can also define a $C$-monoid. I recommend Lambek & Scott's Introduction to Higher Order Categorical Logic for the details, which are kind of involved.