  In this paper we consider the classification problem of extensions of
Yang-Mills-type (YMT) theories. For us, a YMT theory differs from the classical
Yang-Mills theories by allowing an arbitrary pairing on the curvature. The
space of YMT theories with a prescribed gauge group $G$ and instanton sector
$P$ is classified, an upper bound to its rank is given and it is compared with
the space of Yang-Mills theories. We present extensions of YMT theories as a
simple and unified approach to many different notions of deformations and
addition of correction terms previously discussed in the literature. A relation
between these extensions and emergence phenomena in the sense of
arXiv:2004.13144 is presented. We consider the space of all extensions of a
fixed YMT theory $S^G$ and we prove that for every additive group action of
$\mathbb{G}$ in $\mathbb{R}$ and every commutative and unital ring $R$, this
space has an induced structure of $R[\mathbb{G}]$-module bundle. We conjecture
that this bundle can be continuously embedded into a trivial bundle. Morphisms
between extensions of a fixed YMT theory are defined in such a way that they
define a category of extensions. It is proved that this category is a
reflective subcategory of a slice category, reflecting some properties of its
limits and colimits.
