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power object (Definition)

Preliminary remarks.

Let $ A$ be a set. The power set $ P(A)$ is the set of all subsets of $ A$; as such, this idea belongs to the realm of set theory. However, properly formulated, the power set notion can be extended to categories more general than $ \mathit{Set}$.

Every subset $ S\subset A$ determines, and is in turn determined by the characteristic map $ \chi_S:A \to \{0, 1\}$ defined by

\begin{displaymath}\chi_S(a)= \begin{cases} 1 & a\in S,\ 0 & a\notin S; \end{cases}\quad a\in A. \end{displaymath}
In other words, elements of the power set $ P(A)$ correspond in a one-to-one fashion to maps from $ A$ to $ \{0,1\}$. Suppose now that $ A$ is an object in a category $ {\mathcal{C}}$; this will be specified as $ A \in Ob {\mathcal{C}}$, with $ Ob {\mathcal{C}}$ being the class of objects of category $ {\mathcal{C}}$.

It is tempting to generalize the power set construction by considering general homomorphisms into some special object $ \Omega \in Ob {\mathcal{C}}$ that can somehow serve as a category-theoretical analogue of the two element set. This, however, will give us $ P(A)$ as a Hom-set, whereas we want $ P(A)$ to be another object of $ {\mathcal{C}}$. What to do?

A guiding insight[1] is to regard morphisms with codomain $ A$ as “generalized elements” (gelements) of $ A$. Let's return, for the moment, to the category of sets. Henceforth we use $ \mathrm{Map}(X,Y)$ to denote the set of maps from $ X$ to $ Y$, and use the symbol $ 2$ to denote the two element set $ \{0,1\}$. Recall that the set $ \mathrm{Map}(A,\mathrm{Map}(X,Y))$ is naturally isomorphic to the set $ \mathrm{Map}(A\times X,Y)$. In other words, the gelements of $ \mathrm{Map}(X,Y)$ are those special gelements of $ Y$ whose domains are products $ A\times X$. It follows that a gelement of $ P(A)$, which is to say a map from $ X$ to $ \mathrm{Map}(A,2)$, is naturally the same thing as a map from $ X\times A$ to $ 2$. The latter corresponds to a subset of $ X\times A$. We thereby arrive at the key insight:

Subsets of $ X\times A$ are naturally isomorphic to maps from $ X$ to $ P(A)$.
The rest, as they say, is details.

Definition of power object.

Let $ {\mathcal{C}}$ be a category with finite limits (products and pullbacks exist). For a morphism $ m$ of the category, let $ [m]$ denote the equivalence class consisting of all morphisms $ mf$ with $ f$ an isomorphism. Recall that a subobject of an object $ X\in Ob {\mathcal{C}}$ is an equivalence class $ [m]$ where $ m$ is a monomorphism.

Let $ \mathrm{Sub}(-)$ denote the contravariant functor from $ {\mathcal{C}}$ to $ \mathit{Set}$ that sends an object $ X\in Ob {\mathcal{C}}$ to $ \mathrm{Sub}(X)$, the set of all subobjects of $ X$. For a morphism $ f:X\to Y$ we define $ \mathrm{Sub}(f): \mathrm{Sub}(Y)\to \mathrm{Sub}(X)$ to be the set map that acts by sending a monomorphism $ m:S\to Y$ along $ f$ to a monomorphism $ \hat{m}:S\times_f X\to X$ obtained as the pullback of $ m$ along $ f$ (see the figure below). Note: monomorphisms are pullback stable.

Figure: functoriality of $ \mathrm{Sub}$
\includegraphics[width=4cm]{powerobject-fig1}

Next, let us fix an object $ A \in Ob {\mathcal{C}}$ and define what it means to be a power object of $ A$. Let $ F_A= \mathrm{Sub}(-\times A)$ be the functor from $ {\mathcal{C}}$ to $ \mathit{Set}$ that sends $ X\in {\mathcal{C}}$ to $ \mathrm{Sub}(X\times A)$. To establish the functoriality of $ F_A$ we note that it is the composition of the functor $ -\times A: {\mathcal{C}}\to {\mathcal{C}}$ and the functor $ \mathrm{Sub}:{\mathcal{C}}\to \mathit{Set}$. We say that an object $ B\in {\mathcal{C}}$ is a power object of $ A$ if $ B$ represents the functor $ F_A$. If such a $ B$ exists, then by the Yoneda lemma, it is unique up to isomorphism. In other words, $ B$ is a power object of $ A$ if there every monomorphism $ m:S\to X\times A$ corresponds in a natural fashion to a morphism $ \hat{m}\in \mathrm{Hom}_{\mathcal{C}}(X,B)$, with equivalent monomorphisms corresponding to the same $ \hat{m}$.

Note that categories $ {\mathcal{C}}_L$ that have finite limits and are endowed with the power objects defined above satisfy the two axioms of topoi and are therefore called topoi (or toposes).

Bibliography

1
Michael Barr and Charles Wells, ``Toposes, Triples, and Theories'', http://www.cwru.edu/artsci/math/wells/pub/ttt.html



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See Also: power set, subobject, subobject classifier, axioms of elementary topoi, topos

Keywords:  PowerSet, Subobject, SubobjectClassifier, topoi, topos
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Cross-references: topoi, axioms of topoi, equivalent monomorphisms, Yoneda Lemma, represents, composition, fix, monomorphisms are pullback stable, contravariant functor, monomorphism, subobject, isomorphism, equivalence class, pullbacks, limits, finite, products, domains, isomorphic, category of sets, moment, codomain, morphisms, homomorphisms, class, object, maps, one-to-one, categories, set theory, subsets, power set
There are 4 references to this entry.

This is version 25 of power object, born on 2006-02-17, modified 2008-08-02.
Object id is 7630, canonical name is PowerObject.
Accessed 1173 times total.

Classification:
AMS MSC: 18A20 (Category theory; homological algebra :: General theory of categories and functors :: Epimorphisms, monomorphisms, special classes of morphisms, null morphisms)
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