"The dog goes towards the cat."
Translation:Téann an madra chuig an gcat.
an chait is the genitive, but chuig does not take the genitive, but rather eclipsis (or lenition).
Initial Mutations after chuig
- without an article: no lenition or eclipsis e.g. chuig cailín = to a girl
- with the singular-article: eclipsis (except d,t) e.g. chuig an gcailín = to the girl
- in Munster, d,t also eclipses
- in Connacht, t-prefix precedes femin. nouns with s-: chuig an tsúil
- in Ulster, always lenition e.g.: chuig an chailín
cat isn't lenited because of an. After certain (simple prepositions + an), the noun is eclipsed, except in Ulster Irish, where they lenite it. So "the cat" is an cat, but "on the cat" is ar an gcat (or ar an chat in Ulster Irish), "with the cat" is leis an gcat (or leis an chat in Ulster Irish) and "towards the cat" is chuig an gcat (or chuig an chat in Ulster Irish).
There is no preposition before an madra, so this rule doesn't apply to it. And if there was a preposition, it would only effect Ulster Irish, because you can't eclipse m.
No, "simple preposition" are prepositions like ar, ag, chuig and roimh. There are also genitive prepositions like trasna and tar éis, and "derived prepositions" that use a simple preposition and a noun, like ar feadh, i measc, le haghaidh, etc.
Not all "simple prepositions" cause eclipsis after an, but you should always pay attention to the preposition, and be aware of the effect that it has on the following noun.
When a preposition is combined with a pronoun, it becomes a prepositional pronoun. In a preposition pronoun like chugam or chuige, the pronoun is playing the same role as an gcat in chuig an gcat.
chuig is a preposition. When the object of a preposition is a pronoun ("me", "you", "him", "her", "it"), etc, it combines with the preposition to form a prepositional pronoun (chugam, chugatm etc). an cat is not a pronoun, it's a noun, so it doesn't combine with the preposition. Gender has nothing to do with it.