I've been studying operads and something that upsets me is that there are some operads that have the same "meaning" but have a different definition depending on the category we're working on.
For instance, the operad $Ass$ os associative algebras is defined on Top as $Ass(n)=*$ for all $n\geq 1$ (for unitary algebras $Ass(0)=*$, and I assume it is empty otherwise), while on $\bf{Vect}_k$ we have to define $Ass(n)=k$ for all $n\geq 1$ (and $Ass(0)=k$ if we want the algebras to be unitary, $Ass(0)=0$ otherwise).
In this case I observe that we could define $Ass$ in terms of the monoidal structure of these categories, with the cartesian product and the tensor product, respectively. Here, $*$ is the unit for the cartesian product and $k$ is the unit for the tensor product. Also, $\emptyset$ and $0$ are resepectively initial objects in these categories. So we could simply define $Ass(n)=\bf{1}$ for $n\geq 1$, where $\bf{1}$ is the unit of the monoidal structure, and if we want, $Ass(0)=I$ (the initial object) or $Ass(0)=\bf{1}$.
I wonder if this approach is taken for this and other common (possibly symmetric) operads in the literature, because I've mainly seen the authors focus on a particular category when it comes to talk about these operads. Other common examples I haven't checked yet are the operads of commutative algebras and the operad of Lie algebras.
I'd also like to know if this definition I've given of $Ass$ applies nicely to other categories, such as the category of chain complexes.
Other more complex examples probably won't have such a general description but maybe they do in any context were homotopy equivalences are defined. For example the operads $A_n$, $A_\infty$, $E_n$ and $E_\infty$.