CatDat

CIP

A category satisfies CIP ("coproducts inject into products") if it has zero morphisms, products, coproducts, and for every family of objects $(X_i)_{i \in I}$ the canonical morphism $$\textstyle \alpha : \coprod_i X_i \to \prod_{i \in I} X_i$$ defined by $p_j \circ \alpha \circ \iota_i = \delta_{i,j}$ is a monomorphism. This is no standard terminology. This property has been added to clarify relationships between other properties, in particular those concerning the commutation between limits and colimits.

• Dual property: CSP
• Related properties: coproducts, counital, filtered-colimit-stable monomorphisms, products, zero morphisms

Relevant implications

• biproducts andfiltered colimits andfiltered-colimit-stable monomorphisms andproducts implies CIP
• CIP andself-dual implies CSP
• CIP implies coproducts andproducts andzero morphisms
• CSP andself-dual implies CIP

Examples

There are 12 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of Banach spaces with linear contractions
• category of commutative monoids
• category of left modules over a division ring
• category of left modules over a ring
• category of pointed sets
• category of pointed topological spaces
• category of sets and relations
• category of vector spaces
• trivial category
• walking isomorphism

Counterexamples

There are 61 categories without this property.

• category of algebras
• category of combinatorial species
• category of commutative algebras
• category of commutative rings
• category of countable sets
• category of fields
• 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 injections
• category of finite sets and surjections
• category of finitely generated abelian groups
• category of free abelian groups
• category of groups
• category of Hausdorff spaces
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of metric spaces with ∞ allowed
• category of monoids
• category of non-empty sets
• category of pairs of sets
• category of posets
• category of prosets
• category of pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of schemes
• category of sets
• category of sets with finite-to-one maps
• category of sheaves
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of Z-functors
• 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
• 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
• simplex category
• walking commutative square
• walking composable pair
• walking fork
• walking idempotent
• walking morphism
• walking parallel pair
• walking span
• walking splitting

Unknown

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

—