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

• binary products and  equalizers implies   pullbacks

  The pullback of $f : X \to S$ and $g : Y \to S$ is the equalizer of $p_1 \circ f, \, p_2 \circ g : X \times Y \rightrightarrows S$.
• binary products and  pullbacks implies   equalizers

  The equalizer of $f,g : X \rightrightarrows Y$ is the pullback of $(f,g) : X \to Y \times Y$ with the diagonal $Y \to Y \times Y$.
• pullbacks and  terminal object implies   binary products

  If $1$ is a terminal object, then $X \times_1 Y = X \times Y$.
• 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.
• additive and  pullbacks and  subobject classifier implies   trivial

  see MSE/4086192
• groupoid implies   self-dual and  mono-regular and  pullbacks and  filtered limits and  left cancellative and  well-powered

  easy
• self-dual and  pullbacks implies   pushouts

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

  trivial by self-duality

Examples

• 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

• 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

For these categories the database has no info if they satisfy this property or not.

• delooping of the additive monoid of ordinal numbers