Finite, small, and large limits and colimits; complete and cocomplete categories

Depends on

• Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties
• Assuming Choice, cardinality of a small category and κ-small diagrams
• Small, locally small, and large categories

Used by

• A category is complete exactly when it has all small products and equalizers, and cocomplete exactly when it has all small coproducts and coequalizers Corollary
• A category satisfying the explicit SAFT intersection hypotheses is cocomplete Corollary
• A limit weighted by a Set-valued weight on a small index category exists in a complete target, and the weighted colimit in a cocomplete one Corollary
• A reflective subcategory of a complete category is complete Corollary
• Assuming Choice, every small complete category and every small cocomplete category is a preorder Corollary
• Ends exist over a small index category in a complete target, and coends in a cocomplete one Corollary
• If A is small, then [A,C] is complete or cocomplete whenever C is respectively complete or cocomplete Corollary
• Set, Cat, and every complete category are cartesian monoidal Corollary
• A complete locally small category with no small coseparating set Counterexample
• Filtered categories and filtered colimits Definition
• Left exact and right exact functors Definition
• The axioms AB3 and AB3* Definition
• A left Kan extension along the inclusion of the rationals in the reals Example
• The adjoint functor theorem for ordered sets Example
• Under the definable-class diagram convention, the empty set is the product of the large family of all sets Example
• FALSE: A continuous functor on a complete category necessarily has a left adjoint False statement
• FALSE: every category has all small limits False statement
• FALSE: every functor on CᵒᵖtimesC has an end False statement
• A cone over an identity diagram is weakly initial, and the identity diagram has a limit exactly when the category has an initial object Lemma
• Over a cocomplete base, a monadic category is cocomplete exactly when it has coequalizers Proposition
• A complete locally small category with a jointly weakly initial set has an initial object, without class-indexed choice Theorem
• A complete locally small category with a small coseparating set and intersections of all subobject collections has an initial object Theorem
• AB5 is equivalent to exactness of filtered colimits Theorem
• Every small limit can be constructed as an equalizer between products over the objects and arrows of the index category Theorem
• General adjoint functor theorem, objectwise initial-object form Theorem
• Pointwise Kan extensions exist under smallness and completeness hypotheses Theorem
• Set has all small colimits, realized as a quotient of a set-indexed disjoint union Theorem
• Set has all small limits, realized as compatible tuples in a set-indexed product Theorem
• Special adjoint functor theorem, objectwise form with explicit intersection smallness or preservation data Theorem
• Under dependent choice, a finitary monad on a complete cocomplete locally small category has complete and cocomplete algebras Theorem

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• E. Riehl, Category Theory in Context, Definitions 3.2.1 and 3.2.3 (standard reference, not scraped)