In an article I am writing, I am led to the following generalization of the notion of functor. Let $C$ and $D$ and be two categories. A generalized functor $F : C \to D$ is given by:

  • a function $f : C_0 \to D_0$
  • for all $c \in C_0$, a set $F_c$
  • for all $c_1,c_2 \in C_0$, a function $F_{c_1,c_2}:C(c_1,c_2) \times F_{c_2} \to F_{c_1} \times D(f(c_1),f(c_2))$.

All this must be compatible with the product and units, that is:

  • for all $c \in C$ and all $x \in F_c$, $F_{c,c} (1_c, x) = (x,1_{f(c)})$
  • for all $c_1,c_2,c_3 \in C$, the following composites are equal:

$$C(c_1,c_2) \times C(c_2,c_3) \times F_{c_3} \to C(c_1,c_3) \times F_{c_3} \to F_{c_1} \times D(f(c_1),f(c_3)) $$ and $$C(c_1,c_2) \times C(c_2,c_3) \times F_{c_3} \to C(c_1,c_2) \times F_{c_2} \times D(f(c_2),f(c_3)) \to F_{c_1}\times D(f(c_1),f(c_2))\times D(f(c_2),f(c_3)) \to F_{c_1}\times D(f(c_1),f(c_3))$$

In particular, a functor is a generalized functor, where every $F_c$ is the singleton set.

Are you aware of a similar structure arising somewhere?

One way to see this structure is the following. One can see objects $a,b$ of a category as points, and then see $C(a,b)$ as an arrow form $a$ to $b$. If you say that the composition of two arrows is given by the Cartesian product, then you can see the composition in $C$ as a $2$-arrow from $C(a,b) \times C(b,c) \to C(a,c)$.

composition as a $2$-cell

Now how do you represent a functor in this setting? You can't represent the function $C(a,b) \to D(f(a),f(b))$ by a $2$-arrow, because the 'arrows' $C(a,b)$ and $D(f(a),f(b))$ are not parallel (the first one goes from $a$ to $b$, while the second goes from $f(a)$ to $f(b)$).

functor as an application

One way to deal with that is to add two arrows: one called $F_{a}$ from $a$ to $f(a)$ and the other called $F_{b}$ from $b$ to $f(b)$. Now we have a square that we can fill with a $2$-arrow $F_{a,b}$.

functor as a 2-cell

If we keep in mind the correspondence 'arrows are sets' and '$2$-arrows are functions' from the previous paragraph, we obtain the definition of a generalized functor.

Two more things to note:

  • One can define generalized natural transformations between generalised functors. The usual naturality arrows $\eta_a : f(a) \to g(a)$ are then replaced by functions $G_a \to F_a \times D(f(a),g(a))$:

natural transformation as a 2-cell

  • This can be extended to $n$-functors by replacing the sets $F_a$ by $(n-1)$-categories and the functions $F_{a,b}$ by $(n-1)$-functors.
  • 4
    $\begingroup$ Can you say intuitively what this structure does? $\endgroup$ – Jonathan Beardsley Mar 12 '15 at 13:20
  • $\begingroup$ @JonBeardsley I edited my question. I hope it is more clear. $\endgroup$ – Maxime Lucas Mar 12 '15 at 14:02
  • $\begingroup$ Do you have an example of such a thing that you care about? $\endgroup$ – Qiaochu Yuan Mar 13 '15 at 9:56
  • $\begingroup$ It's really hard to read if you write this in terms of hom-sets. How about drawing some diagrams to say what's going on? $\endgroup$ – Paul Taylor Mar 14 '15 at 10:36
  • $\begingroup$ @PaulTaylor I would love to, but I cannot find how to make diagrams work on MathOverflow. How do I do that? $\endgroup$ – Maxime Lucas Mar 17 '15 at 10:59

Here is one way to look at it: if V is a monoidal category and $\mathbf{B} V$ is the corresponding one-object bicategory, then a V-category in the usual sense is the same thing as a lax functor $\mathrm{c}(C_0) \to \mathbf{B} V$, where $C_0$ is a set and $\mathrm{c}(\cdot)$ is the fully faithful functor that takes a set to the codiscrete category on it (i.e. the one with $C_0$ as object set and exactly one morphism between each pair of objects). If C and D are V-categories then a V-functor from C to D is given by a function $f \colon C_0 \to D_0$ together with an identity-component oplax natural transformation (an icon) $\hat C \to \hat D \circ \mathrm{c}(f)$, where $\hat C, \hat D$ are the lax functors corresponding to C and D. What you have seems to be a general oplax transformation, without the condition that its components be identities. For that reason it's probably a not-unreasonable definition to write down, although I for one have never seen an example in the wild.

At the end of your question you are getting somewhere close to the notion of a category enriched in a bicategory, see here.


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