In mathematics, it seems that 100 % of the time, we deal with objects that have a number of different structures that are "compatible" with each other. For example, 
a Lie group is a manifold and a group such that these two structures are compatible in the sense that the group operations are smooth. 


So my question is on a metamathematical level : 
 is it actually true that it's always the case ? 


More precisely : are there some cases where the "right" object to study and define is an object with two (or more) structures that are not required to be made compatible, except that they live on the same set ? 



On structures that are not submitted to compatibility conditions
Here's the most clear-cut example I can think of:


* A Galois representation consists of a field extension $K \subseteq L$, the group $G = Gal(L/K)$, and a representation $G \to GL(V)$ on a vector space $V$ *with no compatibility condition* between the field structure and the representation (although I have the sense that in more sophisticated treatments, one at least asks for compatibility between a profinite topology on $G$ and the representation.)


I'd point out that oftentimes, the most obvious notion of "compatibility with structure" is too rigid, and one needs to loosen up the notion of compatibility, if not throw it out altogether. For example:


* There are lots of structures in differential geometry which are not functorial on the category of manifolds and smooth maps (or its opposite), so one needs to loosen up and study constructions that are functorial on the category of manifolds and diffeomorphisms, or perhaps local diffeomorphisms. For example, tensors of mixed variance.
* In algebraic geometry, a scheme $X$ has a rich structure. When studying sheaves on $X$, one can ask the sheaf to to be compatible with all this structure and satisfy Zariski descent. But for many purposes this is too rigid, and instead one asks for compatibility with less structure, for example by asking for etale descent.
* Similarly, in differential topology, one might be studying a smooth manifold, but might need to talk about constructions on the manifold that respect only the topological structure, or PL structure.
* In algebra, the category $\mathrm{Vect}\_k$ of vector spaces over a fixed field $k$ is naturally enriched in, well, itself. But there are interesting functors $\mathrm{Vect}\_k \to \mathrm{Vect}\_k$ which don't respect the enrichment -- for example Schur functors like $V \mapsto V \otimes V$ or $V \mapsto \Lambda^k V$ or $V\mapsto \mathrm{Sym}^k V$. Thus one ends up studying functors which are not compatible with all the structure they could be.
* In algebraic topology, one can consider a nice cohomology theory (or $E\_\infty$-ring spectrum) like, say ordinary cohomology with $\mathbb{Z}/2$ coefficients ($H\mathbb{Z}/2$). It turns out to be very important to study the Steenrod algebra, which is the algebra of stable natural transformations $H^{\bullet}(-,\mathbb{Z}/2) \to H^{\bullet+k}(-,\mathbb{Z}/2)$ (where $k$ can vary from operation to operation). In some sense, there's a missing compatibility condition, because one is talking about maps $H\mathbb{Z}/2 \to \Sigma^k H \mathbb{Z}/2$ which are not required to be $H\mathbb{Z}/2$-linear. But it's important to loosen things up: this actually ends up revealing *more* structure (the action of the Steenrod algebra) which is important to think about.
* An example I'm fond of: Norbert Roby defined a *polynomial law* between modules $M$ and $N$ over a fixed ring $k$ to be a natural transformation $V(M \otimes U-) \Rightarrow V(N \otimes U- )$ where $U$ is the forgetful functor $R$-$\mathrm{Alg} \to k$-$\mathrm{Mod}$ and $V$ is the forgetful functor $k$-$\mathrm{Mod} \to \mathrm{Set}$ -- so the definition involves twice forgetting structure, deliberately leaving out compatibility requirements, and it turns out to yield a loosening of the notion of linear map which is just what one wants.
* If $V,W$ are representations of a group $G$, then when studying homomorphisms from $V$ to $W$, one has basically two things one can look at: the space $Hom\_G(V,W)$ of linear maps $V \to W$ which are compatible with the action of $G$, or rather the space $Hom(V,W)$ of all linear maps $V \to W$ with no compatibility. The latter is interesting, though, because $Hom(V,W)$ itself carries an action of $G$, and so can be analyzed as a $G$-representation. Similarly, between chain complexes $C,D$, one can consider the *space* of homomorphisms which respect the boundary maps, or instead the whole *chain complex* of homomorphisms which are not required to be compatible with the boundary maps.


So in some sense there are a number of different ways to study interacting structure. It depends on what your goals are, and what the interesting examples do.


