CatDat

multi-cocomplete

A multi-colimit of a diagram $D\colon \mathcal{S} \to \mathcal{C}$ is a set $I$ of cocones under $D$ such that every cocone under $D$ uniquely factors through a unique cocone belonging to $I$. This property refers to the existence of multi-colimits of small diagrams. Note that any diagram with no cocone admits a multi-colimit, which is the empty set of cocones.

• Dual property: multi-complete
• Related properties: cocomplete, multi-initial object
• nLab Link

Relevant implications

• cocomplete implies multi-cocomplete
• finite products andmulti-cocomplete implies cocomplete
• locally finitely multi-presentable is equivalent to finitely accessible andmulti-cocomplete
• locally multi-presentable is equivalent to accessible andmulti-cocomplete
• multi-algebraic is equivalent to generalized variety andmulti-cocomplete
• multi-cocomplete andself-dual implies multi-complete
• multi-cocomplete implies multi-initial object
• multi-complete andself-dual implies multi-cocomplete

Examples

There are 47 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of fields
• category of finite sets and injections
• category of groups
• category of Hausdorff spaces
• category of left modules over a division ring
• category of left modules over a ring
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with ∞ allowed
• category of monoids
• category of pairs of sets
• category of pointed sets
• category of pointed topological spaces
• category of posets
• category of prosets
• category of rings
• category of rngs
• category of sets
• category of sheaves
• 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
• dual of the category of sets
• empty category
• poset [0,1]
• poset of extended natural numbers
• poset of natural numbers
• poset of ordinal numbers
• proset of integers w.r.t. divisibility
• trivial category
• walking commutative square
• walking composable pair
• walking fork
• walking isomorphism
• walking morphism
• walking span

Counterexamples

There are 23 categories without this property.

• category of combinatorial species
• category of finite abelian groups
• category of finite groups
• category of finite ordered sets
• 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 pseudo-metric spaces with non-expansive maps
• category of schemes
• category of sets and relations
• category of smooth manifolds
• delooping of a non-trivial finite group
• delooping of an infinite countable group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• simplex category
• walking idempotent
• walking parallel pair

Unknown

There are 0 categories for which the database has no information on whether they satisfy this property.

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