# Are projective limits always subobjects of a product and dualy, inductive limits quotient objects of a coproduct?

This question is motivated by the first few explicit examples I came across, e.g.

In order to answer that question, let's start with the following two examples and an attempt to link the pullback and pushforward to the product and coproduct respectively. In the category $\mathbf{Set}$ with monomorphisms only as arrows, these correspond respectively to the intersection and union:

The explicit (and seemingly general) constructed goes like $$X\cap Y:= \left\{(x,y)\in X\times Y,\ \iota_X(x) = \iota_Y (y) \right\}$$ where $p_{X/Y}$ are the projections on the first (resp. second) element in the pair $(x,y)$. (This construction actually corresponds to $\Delta(X\cap Y)\subset Z\times Z$ where $\Delta$ is the "diagonal map") and $$X\cup Y:= X\coprod Y \Bigg/\sim$$ where $\sim$ is the finest equivalence relation such that $$u\sim v ,\enspace u,v\in X\coprod Y \quad \Longleftrightarrow\quad u=i_X\circ p_X (x,y),\ v=i_Y\circ p_Y (x,y)\ \text{for some}\ (x,y)\in X\cap Y$$

(The attemp to relate intersection-product and union-coprodut by inserting $Q:= X\times Y$ or $P:= X\coprod Y$ in their respective diagrams fails because these don't make the diagrams commute.)

However, one can insert $C:= X\cap Y,\ f/g = p_X/p_Y$ in the following product diagram and $D:= X\cup Y,\ h/l =i_X/i_Y$ in the coproduct diagram:

This gives a maps $\Phi : X\cap Y \rightarrow X\times Y$ and $\Psi: X\coprod Y \rightarrow X\cup Y$. The question can now be formulated as: under what condition are $\Psi,\Phi$ respectively monomorphism, epimorphism?

Same question for more general projective limits and inductive limits.

Remark: In e.g. this post, people emphasized that the diagrams correspond to the set construction only in $\mathbf{Set}$ with monomorphisms as arrows, not will all arrows, so that $\Psi$ is naturally a monomorphism, i.e. $X\cap Y$ a subobject of $X\times Y$.

Feel free to add the relevant tags.

• Yes. Every limit can be written as an equalizer between two products, and equalizers are always monomorphisms. Jun 22, 2015 at 20:19
• Oh nice, I've (vaguely) seen that theorem but I did not make the link. As soon as I have time (probably in a long time), I'll develop an answer Jun 22, 2015 at 20:59
• @Berci: This is only true if products exist. A counterexample to your claim is the dual of the category of finite abelian groups. Jun 23, 2015 at 13:01