I am reading Fosco's Coend Calculus and Emily Riehl's Categorical Homotopy Theory, Riehl's book motivates it in the following way,
Abstraction 1: Classical limits in terms of cones: Cones from an object $C$ to a system $F:I\to \mathcal{C}$ are thought of as natural transformations from terminal functor, to the collection of natural transformations from the constant functor to the system.
Instead of terminal functor, when one has a weight functor $W: I\to \mathbf{Sets}$, a natural transformation $\tau$ from $W$ to the functor $Hom_{\mathcal{C}}(C,F-)$ has as components set maps $\tau_i:Wi\to Hom_{\mathcal{C}}(C, Fi)$. So, each natural transformation gives a more complicated cone (there are different sets of connections from $C$ to $Fi$ for each $i$)
$$Hom_{\mathcal{C}}(C,\textstyle\lim^W F)\cong Hom_{\mathbf{Sets}^{I}}(W, Hom_{\mathcal{C}}(C, F-))$$
Each component natural transformation $\tau\in Hom_{\mathbf{Sets}^{I}}(W, Hom_{\mathcal{C}}(C, F-))$ is a set map $\tau_i:Wi\to Hom_{\mathcal{C}}(C, Fi)$, which assigns to elements of the set $Wi$ cones from $C$ to the system $F:I\to \mathcal{C}$. This is 'weighted' in the following sense,
Abstraction 2: I was thinking that $W:I\to \mathbf{Sets}$ assigns some 'generalised weightage' to the objects in the category $I$, weights measured by sets. By this I mean, when we assign weight in other branches of mathematics we are usually thinking of real numbers, and 'more weight' is based on the ordering of reals. Instead of this order relation we generalise 'more' to mean any set map.
Intuition for Limits: The intuitive way I think of limits are as the object closest from a system, closest in terms of morphisms. By that I mean any other object which can be related to the system must be farther than the limit, that is, there must exist a map from limit to this object. How do I think of weighted limits?
So, I feel weighted limit does the following: It is the object closest to the weight, in the sense that, the collection of maps $\{\lim^W F\to Fi\}$ is closest to the set $Wi$ (again in terms of morphisms) or something like this?
This all seems way too abstract for me, it felt like abstraction of things that are abstract generalizations of the intuitive things. 'Higher abstraction' of some sorts. But how does someone come up with such definitions, and why? and is my intuition correct?
In the below diagram, the weighted limit $Lim^W F$ I guess should be such that the collection of maps from $Lim^W F$ to $Fj$ is closest to $Wj$, closest in the sense that given any other object $C$, there should exist a map from $Hom(C, Fj)$ to $Wj$
Best Answer
In enriched category theory, weighted limits may be strictly more general than conical limits, in the sense that an enriched category with all conical limits may fail to have all weighted limits. However, in ordinary category theory (and $(\infty, 1)$-category theory too) weighted limits are the same as limits: any weighted limit can be rewritten as a conical limit in a uniform way.
Here is an analogy which might help you understand the meaning of weighted limits: if you have two elements of a group, say $x$ and $y$, you could certainly multiply them together to get another element $x y$... but you could equally well multiply them in the reverse order and get $y x$, or indeed any number of other combinations, such as $y^{-1} x^{-1}$ or $x y x^{-1} y^{-1}$ or whatever; the free group on two elements parametrises all the possible natural operations on two elements of a group. In the same way, a weight $\mathcal{I} \to \textbf{Set}$ parametrises natural operations on a diagram $\mathcal{I} \to \mathcal{C}$ where $\mathcal{C}$ is a complete category, because the category $[\mathcal{I}, \textbf{Set}]^\textrm{op}$ is the free complete category generated by a diagram of shape $\mathcal{I}$.
Specifically, if $\mathcal{C}$ is complete and $F : \mathcal{I} \to \mathcal{C}$ is a diagram, there is a unique (in the appropriate sense) limit-preserving functor $\{ {-}, F \} : [\mathcal{I}, \textbf{Set}]^\textrm{op} \to \mathcal{C}$ such that the composite with the Yoneda embedding $\mathcal{I} \to [\mathcal{I}, \textbf{Set}]^\textrm{op}$ is the given $F : \mathcal{I} \to \mathcal{C}$. Characterising $\{ W, F \}$ explicitly for each individual weight $W : \mathcal{I} \to \textbf{Set}$ leads to the "official" definition of weighted limits, which can then be applied to categories $\mathcal{C}$ not assumed to be complete. Writing each object in $[\mathcal{I}, \textbf{Set}]^\textrm{op}$ as a limit of a diagram built from the representables translates into a uniform way of writing weighted limits of any diagram $\mathcal{I} \to \mathcal{C}$ as conical limits of (other) diagrams in $\mathcal{C}$.
If you really insist on an analogy with real analysis, then perhaps I should warn that thinking of weighted limits as generalised limits is not helpful. It may be better to think of weighted limits as being analogous to functionals defined by integration against a measure ... but even this is a bit of a stretch, in my opinion.