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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Correction to 'multifunctor'
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Bifunctor definition by lars_h
Correction id: 13115
Filed on: 2007-09-25 09:17:11
Status: Accepted on 2007-09-26 01:52:47
Type: Erratum
Correction text:
There seems to be a mismatch between this definition of
a bifunctor (as a binary multifunctor) and other definitions,
such as
http://eom.springer.de/B/b016190.htm
What I miss is axiom (2) on how a bifunctor is compatible
with composition, which AFAICT does not follow from the
conditions in the Multifunctor PM article.
An alternative way to state this condition is that composition
of $F(A,f)$ and $F(h,B)$ (morphisms in different arguments)
should commute (with $A$ and $B$ adjusted as necessary to
make the condition syntactically correct); a commutative
diagram to that effect can be found in
http://en.wikipedia.org/wiki/Hom_functor
which continues to say "The commutativity of the above
diagram implies that Hom(â,â) is a bifunctor".
I suspect (but cannot verify as I don't have access to any
authorative definition of multifunctor) that multifunctors
should satisfy such a condition as well. In that case, it
should probably be the case that a multifunctor $F$ as in
the article *should* be a (covariant) functor -- because the
missing condition is precisely the axiom on how a functor
is compatible with composition of morphisms -- and that
any contravariance should rather be introduced by taking
the opposite category as factor when forming the product
category $\mathcal{C}$.
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No comment from object owner CWoo.
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