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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?

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    $\begingroup$ It's not true that no one works with general limits, these just occur naturally in many settings. The other important thing is that you can obtain all limits from products and equalizers (and dually colimits): en.wikipedia.org/wiki/…. $\endgroup$
    – Ennar
    Sep 4, 2023 at 18:46
  • $\begingroup$ The limit of one object is just the object with the identity map. What else would it be? I guess it can be any isomorphic object with an isomorphism. $\endgroup$ Sep 4, 2023 at 18:58
  • $\begingroup$ @Ennar, yes, perhaps I didn't express myself right, I was looking for these other interesting limits that occur naturally besides the ones I cited above in the first paragraph (equalizers and products inclusive), and occur naturally in set theory. $\endgroup$ Sep 4, 2023 at 19:04
  • $\begingroup$ @ThomasAndrews, ahh, ok, that was what I thought. $\endgroup$ Sep 4, 2023 at 19:05

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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 $X_n$.

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    $\begingroup$ One might add that this generalizes to any diagram of shape J if J contains an initial (respectively terminal) object. $\endgroup$
    – Nephry
    Sep 29, 2023 at 16:54
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To the first question, the reason is correct, but as Thomas mentioned, you could replace $b$ with any other isomorphic object to $c$ and $f$ with such an isomorphism.

To the second question, I think that terminal/equalizers/products/pullbacks are the most used because they are very useful in general while still being somewhat intuitive. Limits of the form $a \to \dots $ are also called direct limits and are fruitful in algebra. Other than those I don't know of any other (co)limit that is very useful in set theory.

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