~고있는 = Is it a thing?
So I found myself wanting to describe something like 'a sitting cat'. Because we call our cats our 'chickens' because whenever they sit down, they look like a cooked chicken on a table or something. Haha I don't know if that makes sense but I haven't learned exactly how to describe a word like a ___ing something.

So I guessed. I typed my guess into a translator and it translated fine but I wanted to come here for clarification.

Sitting cat = 앉고있는 고양이

맞아요? 감사합니다

Question

~고있는 = Is it a thing?
So I found myself wanting to describe something like 'a sitting cat'. Because we call our cats our 'chickens' because whenever they sit down, they look like a cooked chicken on a table or something. Haha I don't know if that makes sense but I haven't learned exactly how to describe a word like a ___ing something.

So I guessed. I typed my guess into a translator and it translated fine but I wanted to come here for clarification.

Sitting cat = 앉고있는 고양이

맞아요? 감사합니다

Answer

I think in this situation, " a sitting cat" should be express "앉아있는 고양이", not a "앉고있는 고양이".

"앉아있는 고양이" means a cat keeping sitting for a while. 
and "앉고있는 고양이" means a cat in the middle of action of sitting.
It means that the cat is moving to sit, not stay sitting yet.

Answer

There is a difference btw -아있다/-고있다.
-아있다 is about '상태' and -고있다 is about behavior.
So, when you say 앉아 있다 , it means it is sitting. On the other hand, 앉고 있다 means it is about to sitting.  Doing the 'verb'.

My cat is sitting on the chair. 
=내 고양이가 의자위에 앉아있어.