CatDat

pullbacks

A category $\mathcal{C}$ has pullbacks if every cospan of morphisms $A \rightarrow S \leftarrow B$ has a pullback $A \times_S B$. This is also known as a fiber product. Equivalently, the slice category $\mathcal{C}/S$ has binary products.

• Dual property: pushouts
• Related properties: wide pullbacks
• nLab Link

Relevant implications

• additive and  pullbacks and  subobject classifier implies   trivial
• binary products and  equalizers implies   pullbacks
• binary products and  pullbacks implies   equalizers
• groupoid implies   filtered limits and  left cancellative and  mono-regular and  pullbacks and  self-dual and  well-powered
• pullbacks and  self-dual implies   pushouts
• pullbacks and  terminal object implies   binary products
• pushouts and  self-dual implies   pullbacks
• wide pullbacks is equivalent to   filtered limits and  pullbacks

Examples

There are 45 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 fields
• category of finite abelian groups
• category of finite orders
• category of finite sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finitely generated abelian groups
• category of free 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 metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of monoids
• category of pointed sets
• category of posets
• category of rings
• category of rngs
• category of schemes
• 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 5 categories without this property.

• category of finite sets and surjections
• category of non-empty sets
• category of sets and relations
• category of smooth manifolds
• walking parallel pair of morphisms

Unknown

There are 1 categories for which the database has no information on whether they satisfy this property.

• delooping of the additive monoid of ordinal numbers