# Is monoid the category for untyped lambda calculus?

If cartesian closed categories are the model for simply typed lambda calculus, then can it be said that a monoid is a categorical model for untyped lambda calculus?

$$C$$-monoids and their variants is what you are looking for. You can find further references and an account of what is what in Martin Hyland's Towards a Notion of Lambda Monoid.
• Nope. It's a rather special kind of object in a category whose structure allows you to model $\lambda$-calculus. – Andrej Bauer Oct 8 '19 at 21:07
• $C$-monoids are described in Lambek & Scott's "Introduction to higher-order categorical logic", section 15 – Andrej Bauer Oct 8 '19 at 21:22