As noted above, a syntactic category (more precisely: a sentence-syntactic category) covers words which occur in similar syntactic contexts; traditionally these cover NOUN, ADJECTIVE, ARTICLE, PRONOUN, VERB, ADVERB, PREPOSITION, CONJUNCTION, INTERJECTION.
If only a few, very generally defined contexts are identified, only a few syntactic categories need to be defined. An early set of categories contained simply nNOUN, VERB, PARTICLE, for example. If contexts are specified more precisely, more detailed categories can be identified. For example, VERB can be divided into AUXILIARY VERB, SEMI-AUXILIARY VERB, LEXICAL (MAIN) VERB. The category AUXILIARY VERB may be divided into MODAL AUXILIARY, ASPECTUAL AUXILIARY, TENSE AUXILIARY, and so on, down to the individual verbs, each of which in a sense constitutes its own category at a very finely granular level of analysis.
Task: How is each of these categories actually defined in terms of distributional (contextual) properties?