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category isomorphism (Definition)

Let $ \mathcal{C}$ and $ \mathcal{D}$ be categories. An isomorphism $ T:\mathcal{C}\to\mathcal{D}$ is a (covariant) functor which has a two-sided inverse. In other words, there is a (covariant) functor $ S:\mathcal{D}\to\mathcal{C}$ such that $ T\circ S=I_{\mathcal{D}}$ and $ S\circ T=I_{\mathcal{C}}$, where $ I_{\mathcal{D}}$ and $ I_{\mathcal{C}}$ are the identity functors of $ \mathcal{D}$ and $ \mathcal{C}$ respectively. Two categories $ \mathcal{C}$ and $ \mathcal{D}$ are isomorphic if there exists a functor $ T:\mathcal{C}\to\mathcal{D}$ that is an isomorphism.

Remarks

  1. An isomorphism (functor) from $ \mathcal{C}$ to $ \mathcal{D}$ is just an isomorphism (in the sense of morphism) in the functor category $ \mathcal{D}^{\mathcal{C}}$.
  2. Two isomorphic categories are equivalent. The converse is not true. For example, the category of all finite sets is equivalent to its subcategory of all finite ordinals. But clearly these two categories are not isomorphic. Isomorphism has a “size” restriction, whereas natural equivalence does not.



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Cross-references: natural equivalence, ordinals, finite, subcategory, finite sets, converse, functor category, morphism, identity functors, inverse, functor, categories
There are 3 references to this entry.

This is version 6 of category isomorphism, born on 2004-05-11, modified 2007-06-16.
Object id is 5852, canonical name is CategoryIsomorphism.
Accessed 1408 times total.

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