PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Very high Entry average rating: Very high
complete category (Definition)

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 a product exists for an arbitrary set of objects, and an equalizer exists for an arbitrary set 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.



"complete category" is owned by CWoo.
(view preamble)

View style:

See Also: exponential object, Cartesian closed category

Other names:  finitely complete, finitely cocomplete, cocomplete
Also defines:  finitely complete category, cocomplete category, finitely cocomplete category
Log in to rate this entry.
(view current ratings)

Cross-references: cardinality, subcategories, pullback, terminal object, sufficient, similar, finite, topological space, unital ring, vector space, group, complete, equalizer, codomain, domain, product, morphisms, objects, collections, limiting cone, limit, category
There are 5 references to this entry.

This is version 2 of complete category, born on 2006-09-17, modified 2006-09-17.
Object id is 8373, canonical name is CompleteCategory.
Accessed 2520 times total.

Classification:
AMS MSC18A35 (Category theory; homological algebra :: General theory of categories and functors :: Categories admitting limits , functors preserving limits, completions)

Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)