  In order to provide a local description of a regular function in a small
neighbourhood of a point $x$, it is sufficient by Taylor's theorem to know the
value of the function as well as all of its derivatives up to the required
order at the point $x$ itself. In other words, one could say that a regular
function is locally modelled by the set of polynomials. The theory of
regularity structures due to Hairer generalizes this observation and provides
an abstract setup, which in the application of singular SPDE extends the set of
polynomials by functionals constructed from, e.g., white noise. In this
context, the notion of Taylor polynomials is lifted to the notion of so-called
modelled distributions. The celebrated reconstruction theorem, which in turn
was inspired by Gubinelli's sewing lemma, is of paramount importance for the
theory. It enables to reconstruct a modelled distribution as a true
distribution on $R^d$ which is locally approximated by this extended set of
models or "monomials". In the original work of Hairer, the error is measured
based on H\"{o}lder norms. This was then generalized to the whole scale of
Besov spaces by Hairer and Labb\'{e} in subsequent papers. It is the aim of
this work to adapt the analytic part of the theory of regularity structures to
the scale of Triebel-Lizorkin spaces.
