  The classical homomorphism preservation theorem, due to {\L}o\'s, Lyndon and
Tarski, states that a first-order sentence $\phi$ is preserved under
homomorphisms between structures if, and only if, it is equivalent to an
existential positive sentence $\psi$. Given a notion of (syntactic) complexity
of sentences, an "equi-resource" homomorphism preservation theorem improves on
the classical result by ensuring that $\psi$ can be chosen so that its
complexity does not exceed that of $\phi$.
  We describe an axiomatic approach to equi-resource homomorphism preservation
theorems based on the notion of arboreal category. This framework is then
employed to establish novel homomorphism preservation results, and improve on
known ones, for various logic fragments, including first-order, guarded and
modal logics.
