Normally in the definition of initial/terminal objects, or limits/colimits of a diagram. It seems that the existence of such objects is constructed via the knowledge of the category in context. For example, the product of sets, the disjoint union of sets, the direct product/direct sum of modules, tensor product of modules, all are constructed case per case.
Is there any way to show the existence of limits/colimits in some kind of general way?
Sorry for the question seems kind of fuzzy. Any references or comments are appreciated too. Thanks!
Best Answer
Many categories that arise in practice are related to other categories which we already understand, usually via adjoint functors. Since right adjoints preserve limits and left adjoints preserve colimits, we can understand the limits and colimits in our new category of interest by relating them to limits and colimits on the other side of the adjunction.
For example, let's take the category of groups $\mathsf{Grp}$. This admits an adjunction $F \dashv U$ with $\mathsf{Set}$ where $F : \mathsf{Set} \to \mathsf{Grp}$ is the free group functor and $U : \mathsf{Grp} \to \mathsf{Set}$ is the forgetful functor. Since $U$ preserves limits (it's a right adjoint) this tells us that the "underlying set of the limit" $U (\varprojlim G_n)$ should be the "limit of the underlying sets" $\varprojlim U G_n$. In the special case of products, this tells us that $U(G \times H) = UG \times UH$, and of course once we know the underlying set it's quite easy to guess what the group structure should be.
As an aside, notice that $U$ definitely does not preserve colimits! This is true of algebraic categories more generally, and this is why limits (such as products) of groups, rings, etc. are so easy to understand where as colimits (such as coproducts) of groups, rings, etc. are comparatively tricky to understand (they're free products, tensor products, etc.).
That said, we still have a general theory for computing colimits in algebraic categories. Again, this is based on the close relationship between these categories and $\mathsf{Set}$, and again this is true in more general settings of "algebras over a base category". See here for more.
For another example, consider $\mathsf{Top}$, the category of topological spaces. This category is super nice since we have a chain of adjunctions $\Delta \dashv U \dashv \nabla$, where $\Delta, \nabla : \mathsf{Set} \to \mathsf{Top}$ are the discrete and indiscrete functors, respectively. Again $U : \mathsf{Top} \to \mathsf{Set}$ is the forgetful functor that forgets the topology on a space.
Since $U$ is both a left and right adjoint, it preserves limits and colimits! So we can compute the underlying set of our (co)limiting topological space by just computing the (co)limit in $\mathsf{Set}$. Then all that's left is to give this set the best topology rendering all the maps from its construction continuous.
Again, this idea works in more general settings than $\mathsf{Top}$ as well.
Of course, there's more examples than just these, but hopefully it's clear that by using adjunctions between our new category and old categories that we already understand, we can reduce the problem of (co)limits to a simpler one. Between this and the fact that (co)limits in functor categories are computed pointwise (see here, say), we can quickly understand (co)limits in many categories of interest!
I hope this helps ^_^