# Terminology for a category whose objects are all free

The question is straight forward:

Is there a standard terminology for a category whose objects are all free? (defined by the universal property)

The prime example I had in mind was the category of $$\Bbbk$$-vector spaces, but I think for any algebraic structure, the subcategory consisting of the free objects would be interesting to consider.

• – Musa Al-hassy Sep 28 '18 at 19:36
• I asked this as a side question here once, and got some suggestions (my favorite was "panteleutheric"), but nothing standard. It's a very special property, as I learned in the answer to my question. – tcamps Sep 30 '18 at 23:44

If you're looking at a category of algebraic structures, in the sense of the category of algebras for a monad $$T$$, then the subcategory of free algebras is (equivalent to) the Kleisli category of $$T$$.