3.4 Completeness and cocompleteness of categories

Categories can be complete, containing all small limits, or cocomplete, containing all small colimits. These properties are crucial for understanding universal constructions and preserving structure between categories.

Set theory exemplifies completeness and cocompleteness through products, equalizers, coproducts, and coequalizers. This pattern extends to other common categories like groups, rings, and topological spaces, forming a foundation for categorical analysis.

Completeness and Cocompleteness in Categories

Definition of complete categories

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• Complete categories possess all small limits
• Small limit results from functor mapping small category to given category
• Cocomplete categories contain all small colimits
• Small colimit stems from functor connecting small category to given category
• Limits act as universal cones exemplified by products, equalizers, pullbacks
• Colimits function as universal cocones demonstrated through coproducts, coequalizers, pushouts
• Small category comprises set of objects rather than proper class

Completeness of set category

• Set category exhibits completeness through existence of products for any set family
• Equalizers exist for all parallel function pairs
• Small limits constructed from products and equalizers
• Set category demonstrates cocompleteness via coproducts (disjoint union) for any set family
• Coequalizers exist for all parallel function pairs
• Small colimits built from coproducts and coequalizers
• Proof strategy involves showing existence of products, equalizers, coproducts, coequalizers
• Utilizes fact that all limits and colimits derivable from these constructions

Completeness across common categories

• Group category complete with products and equalizers cocomplete with free products and coequalizers
• Ring category complete featuring products and equalizers cocomplete with tensor products and coequalizers
• Topological spaces category complete with products and equalizers cocomplete through disjoint unions and coequalizers
• Vector spaces over field both complete and cocomplete with componentwise computation of limits and colimits
• Partially ordered sets as categories exhibit completeness when all subsets have least upper bounds
• Partially ordered sets demonstrate cocompleteness when all subsets possess greatest lower bounds

Significance of categorical completeness

• Structure preservation achieved through limit-preserving functors between complete categories
• Adjoint functor theorem relates to completeness and cocompleteness
• New category construction inherits completeness and cocompleteness from target or base categories (functor categories, slice categories)
• Universal constructions embodied by limits and colimits define crucial categorical concepts
• Algebraic theories yield complete and cocomplete model categories vital in universal algebra and categorical logic
• Elementary toposes in topos theory exhibit completeness and cocompleteness underpinning categorical foundations of mathematics
• Homotopy theory leverages interplay between completeness, cocompleteness, and homotopy in model categories
• Completeness and cocompleteness serve as fundamental properties for modeling mathematical structures in category theory as a foundation