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multifunctor

(Definition)

Let $\mathcal{C}$ be the product of categories $\mathcal{C}_1,\ldots,\mathcal{C}_n$ and $\mathcal{D}$ be any category. A multifunctor $F:\mathcal{C}\to \mathcal{D}$ satisfies the following

for every $A\in\operatorname{Ob}(\mathcal{C})$ , $F(A)\in \operatorname{Ob}(\mathcal{D})$ ,
for every $\alpha\in \operatorname{Mor}(\mathcal{C})$ , $F(\alpha)\in \operatorname{Mor}(\mathcal{D}): F(A)\to F(B)$ , for some objects $A,B\in \operatorname{Ob}(\mathcal{C})$ ,
There is a function $\phi: \lbrace 1,\ldots, n\rbrace\to \lbrace \operatorname{id},\operatorname{op}\rbrace$ where $\operatorname{id}$ is the identity functor and $\operatorname{op}$ is the opposite functor, with
$\displaystyle \overline{C}:= \phi(1)(\mathcal{C}_1) \times \cdots \times \phi(n)(\mathcal{C}_n)$ and $\displaystyle \qquad \overline{\alpha_i}:=\phi(i)(\alpha_i)$
for any morphisms $\alpha_i$ in $\mathcal{C}_i$ , such that $\overline{F}: \overline{\mathcal{C}}\to \mathcal{D}$ given by
$\displaystyle \overline{F}(A):=F(A)$ and $\displaystyle \qquad \overline{F}(\alpha_1,\ldots,\alpha_n):=F(\overline{\alpha_1},\ldots, \overline{\alpha_n})$
is a covariant functor.

Condition 3 says that, while

may not be a functor, by appropriately changing some of the categories $\mathcal{C}_i$ to their opposites, the newly defined $\overline{F}$ becomes a functor. When there is no danger, we may identify $\overline{F}$ with

.

Remarks.

The function $\phi$ in condition 3 above can be changed so that $\overline{F}$ is a contravariant functor instead.
Since $\overline{F}$ is a functor, by restricting $\overline{F}$ to any coordinate gives us a functor as well. Formally, given object $A=(A_1,\ldots,A_n) \in \overline{\mathcal{C}}$ , if we define $F_A:\phi(i)(\mathcal{C}_i)\to D$ by setting
$\displaystyle F_A(B_i)= \overline{F}(\hat{A_i}(B_i))$ and $\displaystyle \qquad F_A(\beta_i)= \overline{F}(\hat{A_i}(\beta_i)),$
for each , where
1. $\hat{A_i}(B_i)$ is the object in $\overline{\mathcal{C}}$ whose -th coordinate is and agrees with everywhere else, and
2. $\hat{A_i}(\beta_i)$ is the morphism in $\overline{\mathcal{C}}$ whose -th coordinate is $\beta_i$ , and the identity morphism (on ) everywhere else,
then is a covariant functor.
Furthermore, since $\overline{F}$ is covariant, this means for any $\alpha=(\alpha_1,\ldots, \alpha_n): A\to B$ , we have the decomposition
$\displaystyle \overline{F}(\alpha)= \overline{F}(\hat{\alpha_1})\circ \cdots \circ \overline{F}(\hat{\alpha_n})$
where $\hat{\alpha_i}:=\hat{A_i}(\alpha_i)$ as defined in property 2 above.
In addition, we see that $\overline{F}(\hat{\alpha_i})\circ \overline{F}(\hat{\alpha_j}) = \overline{F}(\hat{\alpha_j}) \circ \overline{F}(\hat{\alpha_i})$ for $i\ne j$ .
In fact, properties 2, 3, and 4 are enough to insure that $\overline{F}$ is a covariant functor, for

$\displaystyle \overline{F}(\alpha\circ \beta)$ $\displaystyle =$ $\displaystyle \overline{F}((\alpha_1,\ldots,\alpha_n)\circ (\alpha_n,\ldots, \beta_n))$

$\displaystyle =$ $\displaystyle \overline{F}(\alpha_1\circ \beta_1,\ldots, \alpha_n\circ \beta_n)$

$\displaystyle =$ $\displaystyle \overline{F}(\widehat{\alpha_1\circ \beta_1})\circ \cdots \circ \overline{F}(\widehat{\alpha_n\circ \beta_n})$

$\displaystyle =$ $\displaystyle F_A(\alpha_1\circ \beta_1)\circ \cdots \circ F_A(\alpha_n\circ \beta_n)$

$\displaystyle =$ $\displaystyle (F_A(\alpha_1)\circ F_A(\beta_1))\circ \cdots \circ (F_A(\alpha_n)\circ F_A(\beta_n))$

$\displaystyle =$ $\displaystyle (\overline{F}(\hat{\alpha_1})\circ \overline{F}(\hat{\beta_1}))\circ \cdots \circ (\overline{F}(\hat{\alpha_n})\circ \overline{F}(\hat{\beta_n}))$

$\displaystyle =$ $\displaystyle (\overline{F}(\hat{\beta_1})\circ \overline{F}(\hat{\alpha_1}))\circ \cdots \circ (\overline{F}(\hat{\beta_n})\circ \overline{F}(\hat{\alpha_n}))$

$\displaystyle =$ $\displaystyle \overline{F}(\widehat{\beta_1\circ \alpha_1})\circ \cdots \circ \overline{F}(\widehat{\beta_n\circ \alpha_n})$

$\displaystyle =$ $\displaystyle \overline{F}(\beta_1\circ \alpha_1,\ldots, \beta_n\circ \alpha_n)$

$\displaystyle =$ $\displaystyle \overline{F}(\beta\circ \alpha)$

This means we can replace the statement that $\overline{F}$ is a covariant functor in condition 3 of the definition by the three properties above.
is called a bifunctor or trifunctor whenever or .

Hom functors. The most famous bifunctor is the $\hom$ functor from $\mathcal{C}\times \mathcal{C}\to \textbf{Set}$ . Given objects in $\mathcal{C}$ , $\hom(A,B)$ is the set of all morphisms from to . In addition, given morphisms $\alpha:A\to B$ and $\beta:X\to Y$ , $\hom(\alpha,\beta)$ is the morphism from $\hom(B,X)$ $\hom(A,Y)$ taking $f:B\to X$ to $g:=\beta \circ f \circ \alpha:A\to Y$ :

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ B\ar[d]_f & A\ar[l]_{\alpha} \ar[d]^g\ X\ar[r]_{\beta} & Y} } \end{xy}$

Let us verify that $\hom$ is indeed a “binary” multifunctor. Given any object

, we see that $\hom(A,-)$ is covariant functor, for

$\displaystyle \hom(1_A,\beta\circ \alpha)(f)$	$\displaystyle =$	$\displaystyle (\beta\circ\alpha)\circ f\circ 1_A = \beta\circ (\alpha\circ f \circ 1_A)$
	$\displaystyle =$	$\displaystyle \beta\circ \hom(1_A,\alpha)(f) = \beta\circ \hom(1_A,\alpha)(f)\circ 1_A$
	$\displaystyle =$	$\displaystyle \hom(1_A,\beta)(\hom(1_A,\alpha)(f)) = \hom(1_A,\beta)\circ \hom(1_A,\alpha)(f).$

By the same reasoning, we see that, on the other hand, $\hom(-,B)$ is contravariant for any object

. So we want to show that $\overline{\hom}:\mathcal{C}^{\operatorname{op}}\times\mathcal{C}\to \textbf{Set}$ is a covariant functor. Having just verified property 2 (see remarks above), we are left with properties 3 and 4. As the equation $g=\hom(\alpha,\beta)(f)$ turns into $g=\overline{\hom}(\alpha^*,\beta)(f)$ , the diagram above turns into the commutative diagram below

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ B\ar[d]_f \ar[r]^{\alpha^*} & A \ar[d]^g\ X\ar[r]_{\beta} & Y} } \end{xy}$

where $\alpha^*:B\to A$ is the opposite arrow of $\alpha$ . Now, properties 3 and 4 can be easily verified by the following commutative diagrams:

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ B\ar[d]_f \ar[r]^{{1_B}^*} & B \... ...ta} & Y \ar[r]_{1_Y} & Y & && X\ar[r]_{1_X} & X \ar[r]_{\beta} & Y } } \end{xy}$

Therefore, $\overline{\hom}(\alpha^*,\beta)=\overline{\hom}(\alpha^*,1_Y) \circ \overline{... ..._B}^*,\beta)=\overline{\hom}({1_A}^*,\beta) \circ \overline{\hom}(\alpha^*,1_X)$ , and $\overline{\hom}$ is a covariant functor, or that $\hom$ is a bifunctor.

Bibliography

1: A. J. Berrick, M. E. Keating, Categories and Modules, with K-theory in View, Cambridge University Press (2000).

"multifunctor" is owned by CWoo.

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See Also: functor

Also defines:

bifunctor, trifunctor, hom functor

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Cross-references: opposite arrow, commutative diagram, diagram, equation, addition, property, decomposition, identity, coordinate, opposites, functor, morphisms, opposite functor, identity functor, function, objects, category, product of categories
There are 8 references to this entry.

This is version 13 of multifunctor, born on 2007-01-25, modified 2007-09-26.
Object id is 8819, canonical name is Multifunctor.
Accessed 2010 times total.

Classification:

18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)

Pending Errata and Addenda

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