CatDat

pushouts

A category $\mathcal{C}$ has pushouts if every span of morphisms $A \leftarrow S \rightarrow B$ has a pushout $A \sqcup_S B$. This is also known as a fiber coproduct. Equivalently, the coslice category $S/\mathcal{C}$ has binary coproducts.

• Dual property: pullbacks
• nLab Link

Relevant implications

• binary coproducts and  coequalizers implies   pushouts
• binary coproducts and  pushouts implies   coequalizers
• groupoid implies   epi-regular and  filtered colimits and  pushouts and  right cancellative and  self-dual and  well-copowered
• initial object and  pushouts implies   binary coproducts
• pullbacks and  self-dual implies   pushouts
• pushouts and  self-dual implies   pullbacks
• wide pushouts is equivalent to   filtered colimits and  pushouts

Examples

There are 40 categories with this property.

• category of abelian groups
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative rings
• category of finite abelian groups
• category of finite sets
• category of finite sets and bijections
• category of finite sets and surjections
• category of finitely generated abelian groups
• 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 non-empty sets
• 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
• delooping of a non-trivial finite group
• delooping of an infinite group
• delooping of the additive monoid of natural numbers
• 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

There are 9 categories without this property.

• category of fields
• category of finite orders
• category of finite sets and injections
• category of free abelian groups
• category of metric spaces with non-expansive maps
• category of schemes
• category of sets and relations
• category of smooth manifolds
• walking parallel pair of morphisms

Unknown

There are 2 categories for which the database has no information on whether they satisfy this property.

• category of metric spaces with continuous maps
• delooping of the additive monoid of ordinal numbers