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complete category
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(Definition)
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A category $\mathcal{C}$ is said to be a complete category if every small diagram has a limit, that is, a limiting cone exists over every small diagram (diagram such that collections of objects and morphisms are sets).
Of course, in a complete category, a product exists for any given set of objects. Also, a set of morphisms with common domain and codomain has an equalizer. Conversely, we have
in a category $\mathcal{C}$ , if the product exists for an arbitrary set of objects, and the equalizer exists for any pair of morphisms with common domain and codomain, then $\mathcal{C}$ is complete.
Examples
A category $\mathcal{C}$ is said to be finitely complete if every finite diagram (sets of objects and morphisms are finite) has a limit.
A similar sufficient condition for a category $\mathcal{C}$ to be finitely complete is for $\mathcal{C}$ to possess a terminal object and that a pullback exists for every pair of morphisms with common codomain.
Examples
- Any complete category is clearly finitely complete.
- The subcategories of the above examples consisting of all objects with finite cardinality are finitely complete (but not complete).
Remark. The dual notion of a complete category is that of a cocomplete category, and the dual of a finitely complete category is called a finitely cocomplete category.
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"complete category" is owned by CWoo.
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Cross-references: cardinality, subcategories, pullback, terminal object, sufficient, similar, finite, topological space, unital ring, R-module, vector space, group, complete, conversely, equalizer, codomain, domain, product, morphisms, objects, collections, limiting cone, limit, diagram, category
There are 8 references to this entry.
This is version 6 of complete category, born on 2006-09-17, modified 2009-01-17.
Object id is 8373, canonical name is CompleteCategory.
Accessed 4259 times total.
Classification:
| AMS MSC: | 18A35 (Category theory; homological algebra :: General theory of categories and functors :: Categories admitting limits , functors preserving limits, completions) |
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Pending Errata and Addenda
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