Categories are either atomic elements or single-argument functions that return a category as a value when provided with a desired category as argument.

Formally, we define $\mathcal{C}$ as a set of categories for a grammar, such that:
- $\mathcal{A} \subseteq \mathcal{C}$, where $\mathcal{A}$ is a given set of atomic elements
- $(X/Y), \; (X\setminus Y) \in \mathcal{C}$, if $X, Y \in \mathcal{C}$

The set of atomic categories is typically very small and includes familiar elements such as sentences and noun phrases. Functional categories include verb phrases and complex noun phrases among others.

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Categorical grammar is an early lexicalized grammar model which consists of three major elements: a set of categories, a lexicon that associates words with categories, and a set of rules that govern how categories combine in context.

Categorical Grammar

An on-going but a helpful book resource about NLP
https://web.stanford.edu/~jurafsky/slp3/

Speech and Language Processing (3rd ed. draft) 

CG Categories

The Lexicon consists of assignments of categories to words. These assignments can either be to atomic or functional categories, and due to lexical ambiguity words can be assigned to multiple categories. 

Words such as nouns and proper nouns are assigned to atomic categories, reflecting their typical role as arguments to functions. On the other hand, words such as transitive verbs are assigned to function categories.

CG Lexicon

Rules specify how functions and their arguments combine. The following two rule templates constitute the basis for all categorial grammars:
- $X/Y \; Y \implies X$
- $Y \; X \setminus Y \implies X$

The first rule, referred to as forward function application, applies a function to its argument on the right, while the second, referred to as backward function application,  looks to the left for its argument.

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