Monday, August 3, 2009

Duality Principle

There is a powerful tool we have at our disposal and we never even knew about it: we can reverse the direction of the arrows. At first this is not too powerful, or so it should seem. (When I first read about it, I grunted and said "So what?" Its power becomes apparent later on...) First some citations:

  • Saunders Mac Lane, Categories for the Working Mathematician Graduate Texts in Mathematics (vol 5) Springer-Verlag, Second Edition (1998);
  • Jiri Ad├ímek, Horst Herrlich, George E. Strecker Abstract and Concrete Categories: The Joy of Cats freely available online (2004)
  • Serge Lang Algebra Springer-Verlag, Third Edition (2000)
  • Duality entry in nLab wiki

We begin by introducing the notion of a dual category:

Definition 1. The "dual Category to $A$" (denoted $A^{op}$) consists of
  • a collection of objects $Ob(A)$, and
  • for each pair of objects $x,y\in OB(A)$ the set $\hom_{A^{op}}(x,y)=\hom_{A}(y,x)$.
equipped with the structure of a category such that it satisfies the properties of a category.

All the structure and properties of a dual category are "inherited" from a category in the obvious way (i.e. "by reversing the arrows").

We indicate dual objects in general by use of the prefix "co-", e.g. codomain of a co-morphism. When we reverse the direction of a morphism $f:x\to{y}$, we get $f^{op}:y\to{x}$ where $f^{op}$ is dual to $f$ and $y$ is the codomain.

When we have some property $\mathcal{P}$, we can find its dual $\mathcal{P}^{op}$. In general, the dual of a dual of a property is "the same" as the original $(\mathcal{P}^{op})^{op}\cong\mathcal{P}$.

We will now concern ourselves with the algorithm to finding the dual of a property about objects by example. Consider the property about objects $x\in{A}$:

  • $\mathcal{P}_{A}(x)$: for any $A$-object $a$ there exists exactly one $A$-morphism $f:a\to{x}$
Step 1: in $\mathcal{P}_{A}(x)$ replace all occurrences of $A$ by $A^{op}$, giving us
  • $\mathcal{P}_{A^{op}}(x)$: for any $A^{op}$-object $a$ there exists exactly one $A^{op}$-morphism $f:a\to{x}$
Step 2: we "translate it into the logically equivalent statement". This is just translatting the $A^{op}$ stuff into $A$ stuff, more explicitly:
  • $\mathcal{P}^{op}_{A}(x)$: for any $A$-object $a$ there exists exactly one $A$-morphism $f:x\to{a}$.
That concludes our algorithm for properties concerning objects in a category $A$.

There is a similar algorithm for properties of morphisms of $A$. Consider the following property of morphisms $f:x\to{y}$ in $A$:

  • $\mathcal{Q}_{A}(f)$: there exists an $A$-morphism $g:y\to{x}$ with $g\circ{f}:x\to{x}$ be the identity $g\circ{f}=1_{x}$.
Step 1: replace all occurrences of $A$ with $A^{op}$ giving us the following:
  • $\mathcal{Q}_{A^{op}}(f)$: there exists an $A^{op}$-morphism $g:y\to{x}$ with $g\circ{f}:x\to{x}$ be the identity $g\circ{f}=1_{x}$.
Step 2: we "translate into the logically equivalent", i.e. replace $A^{op}$ with the equivalent statements involving $A$:
  • $\mathcal{Q}^{op}_{A}(f)$: there exists an $A$-morphism $g:x\to{y}$ with $f\circ{g}:y\to{y}$ be the identity $f\circ{g}=1_{y}$.

These algorithms allow us to introduce one of the most foundational concepts in category theory:

Duality Principle for Categories. Whenever a property $\mathcal{P}$ holds for all categories, then the property $\mathcal{P}^{op}$ holds for all categories.

No comments:

Post a Comment