CatDat

equalizers

An equalizer of a pair of morphisms $f,g : A \to B$ is an object $E$ with a morphism $e : E \to A$ such that $f \circ e = g \circ e$ and which is universal with respect to this property. This property refers to the existence of equalizers.

• Dual property: coequalizers
• nLab Link

Relevant implications

• thin implies   equalizers and  left cancellative

  Any two parallel morphisms are equal, so their equalizer is the identity, and every morphism is a monomorphism as well.
• complete is equivalent to   products and  equalizers

  Mac Lane, V.2, Cor. 2
• finitely complete is equivalent to   finite products and  equalizers

  Mac Lane, V.2, Cor. 1
• 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 $Y \to Y \times Y$.
• connected limits is equivalent to   wide pullbacks and  equalizers

  The direction $\Rightarrow$ is trivial. The direction $\Leftarrow$ can be found at the nLab.
• equalizers implies   Cauchy complete

  If $e : X \to X$ is an idempotent, then the equalizer of $e, \mathrm{id}_X : X \rightrightarrows X$ provides a splitting of $e$.
• equalizers and  countable products implies   sequential limits

  Mac Lane, V.2, Prop. 3. The proof can easily be adapted to this case. Namely, the limit of $\cdots \to X_2 \to X_1 \to X_0$ is the equalizer of two suitable endomorphisms of $\prod_{n \geq 0} X_n$.
• abelian is equivalent to   additive and  equalizers and  coequalizers and  mono-regular and  epi-regular

  by definition
• groupoid and  equalizers implies   thin

  The equalizer of any parallel pair $f,g$ must be an isomorphism, so $f=g$.
• right cancellative and  equalizers implies   thin

  [dualized] If $f,g$ are two parallel morphisms, then their coequalizer is a regular epimorphism, but also a monomorphism by assumption, so it must be an isomorphism. But this means that $f = g$.
• self-dual and  equalizers implies   coequalizers

  trivial by self-duality
• self-dual and  coequalizers implies   equalizers

  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 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 the additive monoid of ordinal 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 bijections
• category of finite sets and surjections
• category of non-empty sets
• category of sets and relations
• category of smooth manifolds
• delooping of a non-trivial finite group
• delooping of an infinite group
• 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.

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