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.

  • $\begingroup$ Related: math.stackexchange.com/questions/2925813/… $\endgroup$ – Musa Al-hassy Sep 28 '18 at 19:36
  • $\begingroup$ 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. $\endgroup$ – tcamps Sep 30 '18 at 23:44

The notion of a "free object" doesn't make sense in an arbitrary category. Note that the universal property of e.g. free groups refers to maps from a set to the underlying set of a group. So you should have something like a forgetful functor to the category of sets in order to talk about "free objects".

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 $T$.

  • $\begingroup$ True, I wanted to specify to talk about concrete category. Yes, I think this is what I'm looking for. Thank you! $\endgroup$ – G. Chiusole Sep 28 '18 at 16:25

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