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Stream: learning: questions

Topic: Graph of a Profunctor cocontinuous?

Ruby Khondaker (she/her) (Feb 03 2026 at 14:26):

From the nlab page on [[category of elements]], it's known that the functor $\text{el} : \mathbf{Set}^{\mathcal{C}} \to \mathbf{Cat}$ is cocontinuous. Is there any similar result for the category of elements of a profunctor? By which I mean, the construction that takes a functor $P : \mathcal{C}^\text{op} \times \mathcal{D} \to \mathbf{Set}$ and produces a category $\text{el}(P)$ with a projection down to $\mathcal{C} \times \mathcal{D}$.

Ruby Khondaker (she/her) (Feb 03 2026 at 14:29):

If I was more familiar with colimits of categories I might've been able to answer this myself, but those still feel out of reach for me at the moment. Nevertheless, my guess is that you can construct $\text{el}(P)$ by first currying to $\mathcal{D} \to [\mathcal{C}^\text{op}, \mathbf{Set}]$, then taking the category of elements in the second argument to get a functor $\mathcal{D} \to \mathbf{Cat}$, and then taking the grothendieck construction?

Fernando Chu (Feb 03 2026 at 15:19):

Your guess is almost right. You also have to postcompose $\mathcal{D} \to \mathbf{Cat}$ with the opposite involution $op:\mathbf{Cat} \to \mathbf{Cat}$. There was a previous discussion about this construction in this thread, which you might find interesting.

Ruby Khondaker (she/her) (Feb 03 2026 at 15:22):

Ah I see I see. Does that ensure the overall process is cocontinuous?

Fernando Chu (Feb 03 2026 at 15:26):

I don't know. The last step involves the Grothendieck construction on $\mathcal{D} \to \mathbf{Cat}$ (not the usual category of elements) and this is an oplax colimit. So maybe it preserves these?

Ruby Khondaker (she/her) (Feb 03 2026 at 15:27):

I would hope that suffices since “op” also preserves colimits, but that’s a little too slick for me to be sure…

Ruby Khondaker (she/her) (Feb 03 2026 at 19:47):

Ok, I think I may have been able to resolve this.

Suppose $\text{el}$ was indeed cocontinuous. There's a density formula for profunctors:

$P(x, y) \cong \int^{c , d} P(c, d) \times \text{Hom}(x, c) \times \text{Hom}(d, y)$, or in other words $P(-, +) \cong \int^{c, d} P(c, d) \times \text{Hom}(-, c) \times \text{Hom}(d, +)$.

Thus, it suffices to define it on $\text{Hom}(-, c) \times \text{Hom}(d, +) : \mathcal{C}^\text{op} \times \mathcal{D} \to \mathbf{Set}$. The obvious candidate is the product of the slice category over $c$ and the coslice category under $d$, i.e. $\mathcal{C} / c \times d / \mathcal{D}$. This has the requisite projection functor to $\mathcal{C} \times \mathcal{D}$.

So, we'd just need to verify that $\text{el}(P) \cong \int^{c, d} \text{Disc}(P(c, d)) \times \mathcal{C} / c \times d / \mathcal{D}$. I think you can probably do this directly, but you can also check the universal property - that functors $\text{el}(P) \to \mathcal{E}$ correspond to natural transformations $P \to \text{Ob}([\mathcal{C}/- \times +/\mathcal{D}, \mathcal{E}])$, where the right adjoint sends $\mathcal{E}$ to the functor $\mathcal{C}^\text{op} \times \mathcal{D} \to \mathbf{Set}$ defined by $(c, d) \mapsto \text{Ob}([\mathcal{C}/c \times d/\mathcal{D}, \mathcal{E}])$.

This can be proven as follows. Take a functor $F : \text{el}(P) \to \mathcal{E}$. For each $c \in \text{Ob}(\mathcal{C}), d \in \text{Ob}(\mathcal{D}), \alpha \in P(c, d)$, we can define a functor $F_{c, d, \alpha} : \mathcal{C} / c \times d / \mathcal{D} \to \mathcal{E}$ by sending $(f : c' \to c, g : d \to d') \mapsto F(c', d', P(f, g)(\alpha))$ on objects. We can use "heteromorphism" notation to write $P(f, g)(\alpha) = g \circ \alpha \circ f$, which makes functoriality of this map more obvious - it boils down to requiring $g_1 \circ (g_2 \circ \alpha \circ f_2) \circ f_1 = (g_1 \circ g_2) \circ \alpha \circ (f_2 \circ f_1)$.

Collecting these functors together gives a natural transformation $P \Rightarrow \text{Ob}([\mathcal{C}/- \times +/\mathcal{D}, \mathcal{E}])$, I think. So $\text{el}(P)$ has a right adjoint, thus is cocontinuous.

Ruby Khondaker (she/her) (Feb 03 2026 at 19:56):

For context, I've been looking to edit the nlab page for categories of elements for... quite a while. The "covariant" and "contravariant" forms are really special cases of the category of elements of a profunctor, which is a "two-sided" version of the usual category of elements. This also subsumes examples such as comma categories.

Nathanael Arkor (Feb 03 2026 at 20:11):

The nLab page on the [[category of elements]] and [[Grothendieck construction]] have been frustrating me for similar reasons. They would be significantly clarified by presenting the two-sided versions from the beginning.

Ruby Khondaker (she/her) (Feb 03 2026 at 20:12):

Unfortunately I am not quite skilled enough to understand the two-sided Grothendieck construction just yet! But at least in the discrete case (i.e. set-valued functors) I think I can improve the current articles.

Ruby Khondaker (she/her) (Feb 03 2026 at 22:46):

Ok, I've made an edit to the [[category of elements]] nlab page! Hope things seem a little clearer now.

Ruby Khondaker (she/her) (Feb 03 2026 at 22:54):

Bryce Clarke said:

Ruby Khondaker (she/her) said:

Checking nlab, it seems that “category of elements of a profunctor” is one of the notions listed under “graph of a profunctor”? Whereas the category of elements page only takes about Set-valued functors…

I think these nlab pages should be updated. I will add it to my to-do list, unless you feel happy to update them yourself.

I've also finally gotten around to fulfilling (half) of my promise to Bryce :)

Ruby Khondaker (she/her) (Feb 04 2026 at 18:57):

Also now edited the [[graph of a functor]] page! That should finally resolve the questions I started asking in #learning: questions > ✔ Comma Categories vs Categories of Elements.