Usually in category theory, we work with terminal and initial objects, products and coproducts, pullbacks and pushous, equalizers and coequalizers, exponential and coexponential objects. This limits and colimits are respectively build from the cones and co-cones of the null diagram, the 2-object diagram, the diagram of $a \rightarrow b \leftarrow c$, the diagram of $ a \rightrightarrows b$ and a adjunction from the product.
We can proof that a category with terminal object, pullbacks is finitely complete. And if we add exponential and subobject classifiers we get a topos.
It seems this limits are useful to construct a topos, and the relationship with usual concepts in set theory (empty set, singleton etc…) and logic established them in practice.
But, what about the other limits? I tried to construct the limit of the diagram with only one object in set theory, and I don't know if I am right. Here follows my thoughts. I know that a limit to a diagram with the object $c$ is a object b and a arrow $f: b \to c$, such that for any other $g:a \to c$ there is an unique $h: a \to b$ such that $f \circ h = g$. And I would guess that this object $b$ is a copy of $c$ and $f$ is the identity function, for then there is only one function $h$, that is the one equal to $g$, i.e. the one that would make $1_b \circ h = h =g$.
First question: Can anyone confirm if this reasoning is correct? If not, can someone correct me?
Second question: why don't the books work with other finite limits and colimits besides the ones I cited above? For example the limits for $a \rightarrow b$, $a \rightarrow b \rightarrow c$, etc…? Is there other interesting limit that worth working in relation with set theory, or for proving completeness or co-completeness, cartesian completeness or other interesting concept?
Best Answer
For your second question, the particular finite limits you mentioned are trivial. In particular, given a diagram of the form $X_1 \to X_2 \to \ldots \to X_n$, the limit of the diagram is just $X_1$. Dually, the colimit is $X_n$.