I am very beginner in Category theory, and the question I am asking here is related with some example of category theory coming from simple theorems in ring theory.

Fact: Let $R$ be a commutative ring with unity. If l.c.m. of $a,b$ in $R$ exists then $g.c.d.$ exists.

Now let us move in Category theory theory with a specific example. Let $R$ be a ring with unity. Define a category as follows:

Objects: members of $R$.

Morphisms: if $a$ divides $b$ in $R$, let $k_a^b$ denote a morphism, o.w. no morphism between $a$ and $b$.

Thus, the set of morphisms between $a$ and $b$ is either singleton (when $a$ divides $b$) or empty.

Then the co-product of two objects in this category is l.c.m. and product is g.c.d. Of course, the product or co-product of two objects may not exists in this category, for certain rings $R$.

Now the fact mentioned in the beginning says that

if co-product of two objects in the above category exists, then the product of the same two objects also exists.

My natural question is, whether such phenomena occurs in every category, i.e.

Question: If $\mathcal{C}$ is any category, and if co-product of two objects $A,B$ in $\mathcal{C}$ exists, then is it necessary that the product of $A$ and $B$ also exists?

Please do corrections in the setting if there are mistakes in presenting category theory statements; as said initially, I am very beginner in Category theory.


Genrally, no. Simply take a category with three objects $a,b,c$ and other than identity morphisms only two morphisms $a\to c$ and $b \to c$. Then $c$ is the coproduct of $a$ and $b$ but their product does not exist. Take the opposite category, and you'll products withour co-roducts.

This is related to Birkhoff's theorem in lattice theory: if a lattice has all meets, then it has all joins (and vice versa). When a lattice is viewed as a catgory, that theorem becomes: if a thin category has all limits, then it has all colimts (and vice versa). For arbitrary categories this is also true but you must demand really demand all limits exist, namely not just small ones, but then by Freid's theorem such a category must be thin, so you reduce back to the previous setting. There are of course plenty of examples of complete categories that are not cocomplete, so size issues play a vital role here.


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