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[parent] power of an object (Definition)

Let $ \mathcal{C}$ be a category and $ A$ an object in $ \mathcal{C}$. Suppose $ n$ is a non-negative integer. The $ n$-th power of $ A$ is defined as the direct product of $ A$ with itself $ n$ times. In other words, the $ n$-th power of $ A$ is an object $ P$ in $ \mathcal{C}$, together with $ n$ parallel morphisms $ \pi_1, \ldots, \pi_n \in \hom(P,A)$, such that if there are $ n$ parallel morphisms $ p_1, \ldots, p_n \in \hom(B,A)$, then there is a unique morphism $ h:B\to P$ such that $ \pi_i\circ h=p_i$, where $ i=1,\ldots, n$. The commutative diagram below illustrates the situation:

% latex2html id marker 305 $\displaystyle \xymatrix@C=0.5cm{& B \ar@{.>}[d]^{h}... ...[ddr]^{\pi_n}=''2'' & \\ & & \\ A & \cdots & A \ar@{}''1'';''2''\vert-{\cdots}}$

The $ n$-th power of $ A$ is denoted by $ A^n$.

Below are some properties of the power of an object in a category:

For example, in the category of sets, the $ n$-th power of a set $ A$ is the set of $ n$-tuples where each entry is an element of $ A$.

Remark. The copower of an object is defined dually. All of the properties above can be dualized. For example, the 0-th copower of an object is an initial object. The $ n$-th copower of an object $ A$ in Set is the disjoint union of $ n$-copies of $ A$.



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Also defines:  power, copower

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Cross-references: disjoint union, initial object, category of sets, product, terminal object, split epimorphism, projection morphisms, properties, commutative diagram, morphism, parallel morphisms, direct product, integer, object, category
There are 48 references to this entry.

This is version 2 of power of an object, born on 2008-10-31, modified 2008-10-31.
Object id is 11223, canonical name is PowerOfAnObject.
Accessed 254 times total.

Classification:
AMS MSC: 18A30 (Category theory; homological algebra :: General theory of categories and functors :: Limits and colimits )
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