CatDat

wide pullbacks

A category $\mathcal{C}$ has wide pullbacks if for every object $S$ the slice category $\mathcal{C}/S$ has arbitrary products.

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

Relevant implications

• complete implies   finitely complete and  filtered limits and  wide pullbacks and  connected limits

  trivial
• connected limits is equivalent to   wide pullbacks and  equalizers

  The direction $\Rightarrow$ is trivial. The direction $\Leftarrow$ can be found at the nLab.
• wide pullbacks and  terminal object implies   complete

  See the nLab.
• wide pullbacks is equivalent to   pullbacks and  filtered limits

  To prove $\Leftarrow$, a wide pullback can be constructed as a filtered limit of finite pullbacks, and finite pullbacks can be reduced to binary products (the empty-indexed pullback always exists). Conversely, assume that wide pullbacks exist in $\mathcal{C}$. For every object $A$ then the slice category $\mathcal{C} / A$ has wide pullbacks and a terminal object, hence is complete. Since a filtered limit can be finally reduced to such a slice, we are done.
• self-dual and  wide pullbacks implies   wide pushouts

  trivial by self-duality
• self-dual and  wide pushouts implies   wide pullbacks

  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 bijections
• 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
• delooping of a non-trivial finite group
• delooping of an infinite group
• 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 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 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.

• delooping of the additive monoid of ordinal numbers