Unification and generalization of limit and colimit

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 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. 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.

posted almost 2 years ago

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Derek Elkins‭

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