  Despite several (accepted) standards, core notions typically employed in
information technology or system engineering architectures lack the precise and
exact foundations encountered in logic, algebra, and other branches of
mathematics.
  In this contribution we define the syntactical aspects of the term
architecture in a mathematically rigorous way. We motivate our particular
choice by demonstrating (i) how commonly understood and expected properties of
an architecture--as defined by various standards--can be suitably identified or
derived within our formalization, (ii) how our concept is fully compatible with
real life (business) architectures, and (iii) how our definition complements
recent foundational work in this area (Wilkinson 2018, Dickersen 2020).
  We furthermore develop a rigorous notion of architectural similarity based on
the notion of homomorphisms allowing the class of architectures to be regarded
as a category, Arch. We demonstrate the applicability of our concepts to theory
by deriving theorems on the classification of certain types of architectures.
