Definition of limit in category theory

Question

I am studying by myself the definition of limit in category theory and have this doubt.

In the standard definition one follows the strategy:

1. start from a diagram $D : I \rightarrow C$.
2. Than takes an object $X \in C$ with some morphisms $f_i : X \rightarrow D(i), i \in I$.
3. The morphisms chosen in step 2 should be compatible with the set of morphisms in $C$ given by the images of the morphisms of the diagram $S=\{g=D(f), f: i \rightarrow j, i,j \in I \}$, in the sense that $f_i=uf_j$ for every $u \in S$ for which the equation makes sense.
4. Since definitions 1,2,3 do not univocally (up to isomorphisms) identify an object we also impose an other condition that is not related to my question.

My question is:

• Why in Step 1 we needed a diagram. Could we not define the notion of limit directly inside the category $C$? We could do something like:

1. start from a subcategory $C'\subset C$.
2. Than takes an object $X \in C$ with some morphisms $f_i : X \rightarrow i, i \in Obj(C')$ (considering natural inclusion).
3. The morphisms chosen in step 2 should be compatible with the set of morphismsof $C'$ (we have a natural inclusion), in the sense that $f_i=uf_j$ for every $u \in C'$ for which the equation makes sense.
4. Since definitions 1,2,3 do not univocally (up to isomorphisms) identify an object we also impose an other condition that is not related to my question.

Would this be a working alternative definition or are there issues? If not, why the first way is followed?

Answer 1

This is a very good question. Consider the limit of two objects $A, B$. For simplicity let's say we are working in the category of sets. Then you have no doubt read that this corresponds to the Cartesian product $A \times B$. And we can take the Cartesian product of a set with itself, getting $A \times A$. If $A$ has two elements, then $A \times A$ has four.

Using your definition this is not the case. Since an object is either in a subcategory or it is not, there is no such thing as "limit of $A$ and $A$", because we can't talk about an object in a subcategory "multiple times". So with your definition, $A \times A$ would be just equal to $A$ again.

The same thing happens with morphisms, and this is much more subtle. In many "large" constructions of limits, the same morphisms and objects will appear in a limit millions of times, but each time they play different roles. So a diagram to differentiate these roles is vital. You will see one example when you get to the adjoint functor theorem.