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posted 4mo ago by Derek Elkins‭

#1: Initial revision by Derek Elkins‭ · 2023-02-14T05:32:30Z (4 months ago)
Let $\mathcal I$ be the category with two objects and two parallel arrows between them: $1 \rightrightarrows 2$ and $\mathcal P$ be the cone over $\mathcal I$ with apex labelled $0$. Write $\mathsf f$ and $\mathsf g$ for the two arrows $1 \to 2$. Let $\iota : \mathcal I \hookrightarrow \mathcal P$ be the obvious inclusion. This is clearly a setup for the equalizer. $F : \mathcal P \to \mathcal C$ is an extension of $D : \mathcal I \to \mathcal C$ if $F \circ \iota = D$.

We can formulate the condition on $L$ as being the terminal object in the full subcategory of $[\mathcal P, \mathcal C]$, the category of functors from $\mathcal P$ to $\mathcal C$, induced by the extensions of $D$.

Now we get to some issues with your notion of limit. You suggest the setup above should lead to the equalizer. The universal property of the equalizer, $Eq(f,g) = L(0)$ (with $X_i = D(i)$, $f = D(\mathsf f)$, and $g = D(\mathsf g)$), is given an object $X_0 = F(0)$ and a morphism $h : X_0 \to X_1$ such that $f \circ h = g \circ h$ we get a unique morphism $\bar h: X_0 \to Eq(f,g)$ such that $h = e \circ \bar h$ where $e = L(0 \to 1)$ and $h = F(0 \to 1)$. However, the universal property of your notion is that we have a (unique) trio of morphisms $\tau_i : F(i) \to L(i)$ satisfying $f \circ \tau_1 = \tau_2 \circ f$ and similarly for $g$ and $e \circ \tau_0 = \tau_1 \circ h$. To show that this is not the equalizer, consider $f = id_{\mathbb B}$ and $g = \neg$ where $\mathbb B = \{\bot, \top\}$ treated as Booleans. The equalizer of this (in $\mathbf{Set}$) is the empty set, but then $\tau_0$ must be $id_0$ and $e = h = 0_{\mathbb B}$. The only constraints on $\tau_1$ and $\tau_2$ become $\tau_1 = \tau_2$ and $\neg \circ \tau_1 = \tau_2 \circ \neg$ but both $\tau_1 = \tau_2 = id_{\mathbb B}$ and $\tau_1 = \tau_2 = \neg$ satisfy these equations, so uniqueness fails.

One perspective on the issue is that we've developed a notion of an extension, but we haven't specified a notion of morphism of extensions except arbitrary natural transformations. Arguably, a more compelling notion of morphism for extensions would be natural transformations, $\tau$, *such that* $\tau_\iota = id_D$. With this choice in our above example $\tau_1$ and $\tau_2$ would be identities and we'd get the appropriate universal property. In fact, we'd get *exactly* the category of cones whenever $\mathcal P$ is a cone, so this would definitely generalize the notion of limit. It's also clear from this that simply choosing $\mathcal P$ to a be a cocone won't work for colimits. We would want an *initial* cocone, not a terminal one.

Of course, $\mathcal P$ doesn't have to be a cone, so what does that extra flexibility give?

It's a bit awkward because unlike other notions of (co)limit, the result isn't a particular object. See [weighted (aka indexed) (co)limits](https://ncatlab.org/nlab/show/weighted+limit) for a particularly broad generalization which also moves away from "conical" limits. This is usually what we want, because we want a representing object. For example, the object $X_1 \times X_2$ represents the functor $\mathsf{Hom}({-},X_1)\times\mathsf{Hom}({-},X_2)$.

A closer fit would be [Kan extensions](https://ncatlab.org/nlab/show/Kan+extension). One slightly problematic way of describing right (global) Kan extensions (which are themselves slightly problematic as compared to pointwise Kan extensions) is via the following adjoint situation: $$\mathsf{Nat}(F \circ \iota, D) \cong \mathsf{Nat}(F, \mathsf{Ran}_\iota D)$$
$\mathsf{Ran}_\iota D$ is called the right Kan extension of $D$ along $\iota$. There are no restrictions on any of these functors. In particular, $\iota$ need not be an inclusion. This is closer because $\mathsf{Ran}_\iota D$ is a functor and the $F \circ \iota$ is reminiscent of the extension condition. However, the way you represent categorical products as right Kan extensions is not via an inclusion $\mathcal I \hookrightarrow \mathcal P$, but rather by considering right Kan extensions along the functor ${!} : \mathbf{1} + \mathbf{1} \to \mathbf{1}$. This makes $\mathsf{Ran}_{!}D : \mathbf{1} \to \mathcal C$ which is effectively a object of $\mathcal C$.

Nevertheless, if we *do* consider the right Kan extensions along $\iota$ as defined in the very first paragraph, we get something somewhat similar to what you describe. The biggest issue is that given $\tau : F \circ \iota \to D$, there's no a priori reason for $\bar \tau : F \to \mathsf{Ran}_\iota D$ to satisfy $\bar \tau_\iota = \tau$ or even for $\mathsf{Ran}_\iota(D) \circ \iota = D$. If we did have this property for whatever reason, then when $F \circ \iota = D$ choosing $\tau = id_D$ would give $\bar \tau_\iota = id_D$ leading to the earlier "category of cones" situation.

Ultimately, I don't know what, if anything, your notion corresponds to. For the reasons discussed above, I don't think it is useful as a generalized limit nor does it unify limits and colimits. Categorical intuitions suggest that perhaps we'd want to weaken the $F \circ \iota = D$ condition, and, sure enough, this is exactly what happens with Kan extensions.