nLab multifunctor

Idea

By a multifunctor one may mean:

1. a “functor of several variables”, i.e., the categorification of the notion of a multifunction)

2. a morphism of multicategories.

Functor of several variables

There are at least two ways to generalize the notion of a functor to the case where its domain may an n-tuple of categories:

1. jointly functorial maps

2. separately functorial maps

which we discuss in turn.

In all of the following,

• $D$ denotes a category which serves as the codomain;

• $C_1, \cdots, C_n$ denotes an $n$-tuple of categories serving as the multi-domain;

• ${\vert -\vert}$ denotes passage to the class of objects of a (strict) category.

Jointly functorial maps

A jointly functorial map from $C_1, \cdots, C_n$ to $D$ consists of:

1. a multifunction on objects

   $$F \,\colon\, \big( {|C_1|}, \ldots, {\vert C_n\vert} \big) \longrightarrow {|D|}$$

2. For all n-tuples of morphisms

   $$f_i \,\in\, C_i(a_i,b_i), \;\;\;\; 1 \leq i \leq n$$

   a morphism of the form

   $$F(f_1,\ldots, f_n) \,\colon\, F(a_1,\ldots, a_n) \to F(b_1,\ldots, b_n)$$

   in $D$.

such that:

1. identity morphisms are preserved, in that

   $$F(id_{a_1}, \ldots, id_{a_n}) \;=\; id_{F(a_1, \ldots, a_n)}$$

2. composition is respected, in thay

   $$F(f_1 \circ g_1,\ldots, f_n \circ g_n) \;=\; F(f_1,\ldots, f_n) \circ F(g_1,\ldots, g_n) \,.$$

Such a jointly functorial map is the same as an ordinary functor out of the product category of the $n$-tuple of domain categories:

$$C_1 \times \cdots \times C_n \longrightarrow D \,.$$

In the case $n = 1$ this is an ordinary functor, while for $n = 2$ this is a “bifunctor”. And if one understands multifunctions of zero arguments as functions out of the empty product of domain categories, which is the terminal category, then for $n = 0$ this is just a choice of object of $D$.

Separately functorial maps

On the other hand, rather than requiring an “action” on morphisms from each domain category simultaneously, one may want to require an action of each domain category separately, which we could call separately functorial. I.e., a separately functorial map $\big(C_1,\ldots, C_n\big) \to D$ consists of:

1. A multifunction of objects

   $$F \colon \big( {|C_1|},\ldots, {\vert C_n\vert} \big) \longrightarrow D$$

2. Such that for each domain category $C_i$, and objects $a_1,\ldots\widehat{a_i},\ldots, a_n$, the map $F(a_1,\ldots,\widehat {a_i},\ldots, a_n) \colon C_i \to D$ extends to a functor from $C_i$ to $D$.

This definition is instead equivalent to an ordinary functor out of the funny tensor product of the domain categories

$$C_1 \Box \cdots \Box C_n \longrightarrow D \,.$$

For $n = 0$ and $n = 1$ this definition coincides with that of jointly functorial maps above, bu for $n \geq 2$ arguments it is different.

For more see also at funny tensor product – Separate functoriality

Relation Between Joint and Separate Functoriality

Every jointly functorial map defines a separately functorial map by defining the action as

$$F(a_1,\ldots,\widehat {a_i},\ldots, a_n)(f_i) = F(id,\ldots,f_i,\ldots,id)$$

On the other hand, there is not way to extend an arbitrary separately functorial map in this way to be jointly functorial. We can attempt to define

$$F(f_1,\ldots) = F(\widehat{a_1},\ldots)(f_1) \circ F(a_1,\widehat{a_2},\ldots)(f_2) \circ \cdots$$

But note that for $n\geq 2$ this involves an arbitrary choice of which order to compose these actions, and this will only be a jointly functorial action if all such choices are equal. This can be stated as saying for any pair $i\leq j\leq n$ that

$$F(a_1,\ldots,\widehat{a_i},\ldots a_n)(f_i) \circ F(a_1,\ldots,\widehat{a_j},\ldots,a_n)(f_j) = F(a_1,\ldots,\widehat{a_j},\ldots a_n)(f_j) \circ F(a_1,\ldots,\widehat{a_i},\ldots,a_n)(f_i)$$

Related concepts

• two-variable adjunction

References

On “functors of several variables”:

• eom: Multi-functor