Can we define structures like groups or monoids in the context of pure category theory? In a category $\mathcal C$ with terminal object $1$ and objects $A$, $B$, $C$ we have 


*

*$\quad$  $A\times (B \times C) \cong (A\times B) \times C$;

*$\quad$ $1 \times A \cong A \cong A\times 1$;

*$\quad$ $A\times B \cong B\times A $.


where $\times$ is a (binary) product defines on objects of  the category (in the case they exist).
This reminds me somehow the structure of monoids (with 3 reminding abelian ones) with objects being regarded as elements of a monoid, $\times$ as the binary operation of the monoid and $1$ as its unit, and finally $\cong $ used instead of $=$.
Then if we define such structure in general, we can have similar notions from the theory of monoids (or groups in particular cases) and have many theorems as well.
Now, my question is that is there any theory dealing with such structures? Or, if not, is it worth developing one? 
Thanks
 A: Short answer: yes what you describe are called monoidal categories.
The theory of monoidal categories is quite important because there are lots of application and because they are the basis for enriched category theory.
Edit There's also a very interesting difference about monoids and monoidal categories: while the structure of a monoid is specified by the composition and unit operation, together with the proofs for the monoid's axioms, the structure of a monoidal category is specified by monoidal operations (the product and the unit) natural isomorphisms (that replace the role played by the proofs for the monoid axioms) and also proofs of coherence axioms for the natural isomorphism. 
A full explaination of the why is in this answer. Shortly this is due to the fact that coherent monoidal categories are the only ones equivalent (in an appropriate sense) to strict monoidal categories (monoidal categories where the natural isomorphisms are actual identities, hence the coherence trivially hold). 
This restrictive (but not so much strict) structure allows monoidal categories to be a good environment where you can do lot classical mathematics in them, for instance identifying products $(A \times B) \times C$ and $A \times (B \times C)$, which is something that cannot be done safely in non coherent categories.
There could be so much more to say about these important objects.... but I think I've already talked too much so I strongly suggerst you to look on the internet where you could find a lot of interesting fact about them.
A: Yes, not only can you define monoidal and symmetric monoidal categories as above, but you can even give a vaguely analogous notion of what an 'abelian group' is. Recall that the set of homomorphisms between two abelian groups is an abelian group itself. This is essentially the condition that a monoidal category is closed. 
