Your question is good, as well as the answers. I'm wondering about a first undergraduate course in General Topology, or Measure Theory, or Linear Algebra, or Group Theory, where one would avoid any reference to sets with structures, and everything would be said in terms of morphisms... can you explain linear algebra with just linear maps and never mention vector spaces? probably, since a vector in E is the same as a linear map R->E...

Your question is good, as well as the answers. I'm wondering about a first undergraduate course in General Topology, or Measure Theory, or Group Theory, where one would avoid any reference to sets with structures, and everything would be said in terms of morphisms... can you explain even Linear Algebra from scratch with just linear maps and never mention vector spaces nor of course vectors themselves? probably, since a vector in E is the same as a linear map R->E, or in other words, a linear map composable on the right with (the identity of) E and on the left with (the identity of) R. Fun!