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Stream: learning: questions

Topic: ✔ Comma Categories vs Categories of Elements

Ruby Khondaker (she/her) (Jul 30 2025 at 08:21):

To sum up what I think I learned regarding my original question:

• It's convenient to think of the "category of elements" construction as fundamentally being one on profunctors rather than just presheaves. In this sense, it can be viewed as a generalisation of the arrow category construction.
• Comma categories are then a particularly nice instance of these, arising as the category of elements of $Hom(F-, G-)$ viewed as a profunctor from $D$ to $C$, for $F : C \to E$ and $G : D \to E$.
• The category of elements of a profunctor gives a notion of graph of a profunctor. The other notions of graph come from the compact closed bicategory structure on Prof - explicitly, that profunctors $C \times D \to E$ naturally correspond to profunctors $C \to D^\text{op} \times E$.

Notification Bot (Jul 30 2025 at 08:23):

Ruby Khondaker (she/her) has marked this topic as resolved.

Vincent Moreau (Sep 08 2025 at 23:08):

Bryce Clarke said:

Comma categories are examples of categories of elements of a distributor or profunctor. Given a distributor $P \colon A \nrightarrow B$ viewed as a functor $A^{\mathrm{op}} \times B \to \mathbf{Set}$, its category of elements $\mathbf{El}(P)$ has objects given by triples $(a \in A, b \in B, p \in P(a, b))$ and morphisms $(f, g) \colon (a, b, p) \to (a', b', p')$ given by morphisms $f \colon a \to a'$ in $A$ and $g \colon b \to b'$ in $B$ such that $P(f, 1_{b'})(p') = P(1_{a}, g)(p)$.

In praise of @Bryce Clarke's message which I found very enlightning a big month ago, let me share a small application of this viewpoint that I hope will illustrate its elegance.

I follow the notations of the quoted message. Let $C$ be a monoidal category and $\bot$ be an object of $C$. We consider the associated category of Chu spaces:

• whose objects are triples $(a, b, p)$ where $a$ and $b$ are two objects of $C$ and $p : a \otimes b \to \bot$ is a morphism of $C$,
• whose morphisms $(a, b, p) \to (a', b', p')$ are pairs of morphisms $f : a \to a'$ and $g : b' \to b$ which verify the "adjunction-like" equation

$$p' \circ (f \otimes 1_{b'}) \quad=\quad p \circ (1_{a} \otimes g)$$

This category, of Chu spaces built from $C$ and $\bot$, is exactly the category of elements of an appropriate profunctor! namely $P : C \nrightarrow C^{\mathrm{op}}$ given by the formula

$$P(a, b) \quad:=\quad C(a \otimes b, \bot)$$

Nathanael Arkor (Sep 09 2025 at 08:42):

You should add this observation to the nLab article [[Chu construction]] :)

Matteo Capucci (he/him) (Sep 22 2025 at 14:13):

Vincent Moreau said:

This category, of Chu spaces built from $C$ and $\bot$, is exactly the category of elements of an appropriate profunctor! namely $P : C \nrightarrow C^{\mathrm{op}}$ given by the formula

$$P(a, b) \quad:=\quad C(a \otimes b, \bot)$$

Is this the same as the category of elements of the comma $\otimes / \top \to C \times C$?