  We present a unifying approach to the study of entropies in Mathematics, such
as measure entropy, topological entropy, algebraic entropy, set-theoretic
entropy. We take into account discrete dynamical systems, that is, pairs
$(X,T)$, where $X$ is the underlying space and $T:X\to X$ a transformation. We
see entropies as functions $h:\mathfrak X\to \mathbb R_+$, associating to each
flow $(X,T)$ of a category $\mathfrak X$ either a non negative real or
$\infty$. We introduce the notion of semigroup entropy $h_\mathfrak S:\mathfrak
S\to\mathbb R_+$, which is a numerical invariant attached to endomorphisms of
the category $\mathfrak S$ of normed semigroups. Then, for a functor
$F:\mathfrak X\to\mathfrak S$ from any specific category $\mathfrak X$ to
$\mathfrak S$, we define the functorial entropy $h_F:\mathfrak X\to\mathbb R_+$
as the composition $h_{\mathfrak S}\circ F$. Clearly, $h_F$ inherits many of
the properties of $h_\mathfrak S$, depending also on the properties of $F$.
Such general scheme permits to obtain relevant known entropies as functorial
entropies $h_F$, for appropriate categories $\mathfrak X$ and functors $F$, and
to establish the properties shared by them. In this way we point out their
common nature. Finally, we discuss and deeply analyze through the looking glass
of our unifying approach the relations between pairs of entropies. To this end
we formalize the notion of Bridge Theorem between two entropies $h_i:\mathfrak
X_i\to \mathbb R_+$, $i=1,2$, with respect to a functor $\varepsilon:\mathfrak
X_1\to\mathfrak X_2$. Then, for pairs of functorial entropies we use the above
scheme to introduce the notion and the related scheme of Strong Bridge Theorem,
which allows us to put under the same umbrella various relations between pairs
of entropies.
