Categories which are not monoidal

Question

I am reading about monoidal categories and I am not able to think of categories which are non-monoidal. Am I thinking in the wrong direction?
Is being monoidal, an additional property like topology?
For example, given a set you define a topology on it making it a topological space. Similarly, given a category, you define a tensor product map making it into a monoidal category?
Also, given a category does product of two objects from that category always exist? ( I know about the uniqueness but I am not sure about existence). If yes, can a map sending two objects to their product be considered as a tensor product map?

Answer 1

As already mentioned in the previous answer, a monoidal category is an ordinary category with a "monoidal structure". It mimics the notion of a Monoid in the sense that a monoidal category categorifies the algebraic notion of a Monoid, much as an ordinary category categorifies the notion of a set.

Certainly, if a category $\mathcal{C}$ has finite products (respectively, coproducts), then it has a monoidal structure $(\times, 1)$ (respectively, $(\sqcup, 0)$), called a Cartesian (respectively, Cocartesian) monoidal structure.

However, while most categories are usually equipped with a monoidal structure (viz. Set, Top, Ab, k-Mod, etc), it is certainly not true that every category should have a monoidal structure. Let me give an example of the same :

An interesting property of a monoidal category $(C,\otimes,1)$ is that $End(1)$ is a commutative monoid under composition.(Try to prove this). Now, suppose you take any non-commutative monoid $M$. You can think of $M$ as a category $\mathcal{M}$ with 1 object $•$ and with $Hom_{\mathcal{M}}(•,•) = M$, with composition being the monoid operation and the identity arrow on $•$ being the identity element of the monoid $M$. By definition, $End(•) =M$ , which is not a commutative monoid. Since $\mathcal{M}$ has exactly 1 object, namely $•$, hence using the above property of a monoidal category, $\mathcal{M}$ cannot have a monoidal structure.

If you need an example of a category with many objects such that it doesn't admit a monoidal structure, just take a disjoint union of non-commutative monoids (considered as categories as above). This will be a category that does not admit a monoidal structure.

Answer 2

You are right that being monoidal is like being a topological space, in the sense that it is an extra structure (rather than a property) that is added on the object. Some categories have many natural monoidal structures, just like some sets have many different products making them into monoids (as a simple example, every ring has two monoidal structures, coming from addition and multiplication).

You are also right that is a category has (finite) products, then the direct product defines a monoidal structure on the category, called its cartesian monoidal structure. But not all categories admit finite products: for instance, the category of sets with less than $10$ elements does not have arbitrary finite products.

Using this, an example of a category with two natural monoidal structures is the category of abelian group: it has its cartesian monoidal structure, where the product is the direct product, but it also has the tensor product of abelian groups.