Breakdown of Kötturinn er að sofa á gólfinu.
vera
to be
kötturinn
the cat
sofa
to sleep
að
to (infinitive marker)
á
onto / on / in (static location) / on (a day of the week)
gólfið
the floor
Questions & Answers about Kötturinn er að sofa á gólfinu.
How do we know the cat is sleeping right now, rather than just 'the cat sleeps'?
This sentence uses the present continuous pattern: er (is) plus að plus the infinitive verb sofa (sleep). Together, er að sofa translates to 'is sleeping' right at this moment. If you just wanted to say 'the cat sleeps' as a general habit, you would use the simple present tense: Kötturinn sefur.
Wait, doesn't að usually mean 'to'? Why is it used here?
You are right that að often means 'to' before a verb, like 'to sleep'. However, when it is paired with a form of the verb vera (to be) like er, it creates a special grammar formula. The pattern vera að plus an action verb is the standard Icelandic way to express that someone is actively doing something right now, identical to the English '-ing' ending.
Why does the word for floor, gólf, change to gólfinu?
This happens because of the preposition á (on). When á describes a stationary location (where something is resting), it forces the following noun into the dative case. The basic word for 'the floor' is gólfið, but in the dative case, the neuter 'the' ending changes from -ið to -inu, making it gólfinu.
Why is the word for 'the cat' Kötturinn instead of having a separate word for 'the'?
In Icelandic, the definite article (the word for 'the') almost always attaches directly to the end of the noun instead of standing alone. Köttur means 'a cat', and adding the masculine singular ending -inn makes it Kötturinn ('the cat').
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