  We construct microscopical models of one-dimensional non-interacting
topological insulators in all of the chiral universality classes. Specifically,
we start with a deformation of the Su-Schrieffer-Heeger (SSH) model that breaks
time-reversal symmetry, which is in the AIII class. We then couple this model
to its time-reversal counterpart in order to build models in the classes BDI,
CII, DIII and CI. We find that the $\mathbb{Z}$ topological index (the winding
number) in individual chains is defined only up to a sign. This comes from
noticing that changing the sign of the chiral symmetry operator changes the
sign of the winding number. The freedom to choose the sign of the chiral
symmetry operator on each chain independently allows us to construct two
distinct possible chiral symmetry operators when the chains are weakly coupled
-- in one case, the total winding number is given by the sum of the winding
number of individual chains while in the second case, the difference is taken.
We find that the chiral models that belong to $\mathbb{Z}$ classes, AIII, BDI
and CII are topologically equivalent, so they can be adiabatically deformed
into one another so long as the chiral symmetry is preserved. We study the
properties of the edge states in the constructed models and prove that
topologically protected edge states must all be localised on the same
sublattice (on any given edge). We also discuss the role of particle-hole
symmetry on the protection of edge states and explain how it manages to protect
edge states in $\mathbb{Z}_2$ classes, where the integer invariant vanishes and
chiral symmetry alone does not protect the edge states anymore. We discuss
applications of our results to the case of an arbitrary number of coupled
chains, construct possible chiral symmetry operators for the multiple chain
case, and briefly discuss the generalisation to any odd number of dimensions.
