connected limits
A category has connected limits if it has limits of diagrams indexed by connected categories.
- Dual property: connected colimits
- Related properties: connected, complete, filtered limits
- nLab Link
Relevant implications
essentially discrete implies locally essentially small and connected limits
trivialcomplete implies finitely complete and filtered limits and wide pullbacks and connected limits
trivialconnected limits is equivalent to wide pullbacks and equalizers
self-dual and connected limits implies connected colimits
trivial by self-dualityself-dual and connected colimits implies connected limits
trivial by self-duality
Examples
- category of abelian groups
- category of Banach spaces with linear contractions
- category of commutative rings
- category of fields
- category of finite sets and injections
- category of groups
- category of left R-modules
- category of locally ringed spaces
- category of M-sets
- category of measurable spaces
- category of metric spaces with ∞ allowed
- category of monoids
- category of pointed sets
- category of posets
- category of rings
- category of rngs
- category of sets
- category of simplicial sets
- category of small categories
- category of topological spaces
- category of vector spaces
- category of Z-functors
- discrete category on two objects
- empty category
- partial order [0,1]
- partial order of extended natural numbers
- partial order of natural numbers
- partial order of ordinal numbers
- preorder of integers w.r.t. divisiblity
- trivial category
- walking isomorphism
- walking morphism
Counterexamples
- category of combinatorial species
- category of finite abelian groups
- category of finite orders
- category of finite sets
- category of finite sets and bijections
- category of finite sets and surjections
- category of finitely generated abelian groups
- category of free abelian groups
- category of metric spaces with continuous maps
- category of metric spaces with non-expansive maps
- category of non-empty sets
- category of schemes
- category of sets and relations
- category of smooth manifolds
- delooping of a non-trivial finite group
- delooping of an infinite group
- delooping of the additive monoid of natural numbers
- walking parallel pair of morphisms
Unknown
For these categories the database has no info if they satisfy this property or not.