Is there a theory of structures different from Bourbaki's theory of structures?

Question

In Bourbaki's "Elements of Mathematics: Theory of sets", after an account of a set theory, a "theory of structures" is introduced. This book is difficult to read, but it seems, Bourbaki treat a structure as an object "defined" in set theory. Are there different approaches in defining what are structures?

Answer 1

General idea of a structure - the core aspect of a structure (as opposed to a theory) is to imply a concept of truth (e.g. using meta language or some internal concept) e.g. to assess whether or not the structure is a model of some theory.

Toposes as structures - this said, topos theory can be regarded as an alternative approach to structures: A topos is a certain type of nice (in a way set-like) category with a(n internal) truth concept (by something called a subobject classifier). Depending on the regarded topos, the truth concept given by its subobject classifier might look very different to the classical true/false and in general induces an intuitionistic logic to the category.

Toposes as alternative - category theory can be developed without reference to set theory, i.e. without a concept of membership relationship, see e.g. Category theory without sets (incl. references for further reading).