Herbrand universe

Given a signature (usually first-order), its Herbrand universe is the set of syntactic terms over this signature.

Categorically-theoretically, the Herbrand univerise is essentially the initial algebra for some functor.

References

Wikipedia: Herbrand structure
The first utterance of the evident connection between Herbrand universes that we know of is in

Goguen, J. A., J. W. Thatcher, E. G. Wagner, and J. B. Wright. 1977. ‘Initial Algebra Semantics and Continuous Algebras’. Journal of the ACM 24 (1): 68–95.

@article{goguen_1977,
	title = {Initial {Algebra} {Semantics} and {Continuous} {Algebras}},
	volume = {24},
	number = {1},
	journal = {Journal of the ACM},
	author = {Goguen, J. A. and Thatcher, J. W. and Wagner, E. G. and Wright, J. B.},
	year = {1977},
	pages = {68--95}
}