Is product/coproduct in category with only one object possible? Let's say we have category with one object $N$ and infinite number of arrows, which are named as natural numbers, with the same law of composition, where $id$ arrow 0.
I try to understand 1) if categorical product/coproduct could be defined in such category and 2) if it could have properties of usual binary operations on natural numbers?
 A: The category you describe lacks any interesting products and coproducts. But there is a category with only one object that has very interesting products and coproducts. Namely, let $A$ be a $III_1$ factor, and consider the category with only one object and endomorphisms of that object indexed by elements of $A$ (and composition given by multiplication). It is not obvious, but this category has both products and coproducts.
A: A more abstract answer. In some category, let $X$ be an object with isomorphisms $X\cong X\times X$, $X\cong X\sqcup X$ (for example, in the category of sets an empty set, a one-element set or any infinite set have this property). Then the full subcategory with single object $X$ will have all nonempty finite products and coproducts. That simply because this subcategory is equivalent to the full subcategory on finite products and coproducts of $X$.
PS Just noticed that Sam Gunningham has this in a comment to another answer.
A: With a single object, products are powers; the only power that exists is when the index set is a singleton. Similarly, the coproduct only exists if the index set is a singleton
A product of a family $\{X_i\}$ is supposed to be an object $X$ together with morphism $f_i\colon X\to X_i$ such that for every object $C$ and ever family $\{g_i\colon C\to X_i\}$, there exists a unique $G\colon C\to X$ such that $f_i\circ G = g_i$. In your setting, this would amount to an $I$-tuple $(n_i\colon N\to N)$ of numbers such that given any $I$-tuple $(k_i)$, there exists a unique natural number $K$ such that $n_i+K = k_i$ for all $i$. You can only guarantee this if $I$ is a singleton, and $n_i = 0$.
The coproduct would likewise be an $I$-tuple $(n_i\colon N\to N)$ with the property that given any $I$-tuple $(k_i)$ of natural numbers, there is a number $K$ such that $n_i + K = k_i$ for all $i$. So again this only exists for a singleton with $n_i=0$.
