Grp.yaml

id: Grp
name: category of groups
notation: $\Grp$
objects: groups
morphisms: group homomorphisms
description: This is the prototype of a finitary algebraic category.
nlab_link: https://ncatlab.org/nlab/show/Grp

tags:
  - algebra

related_categories:
  - FinGrp
  - Grp_c
  - Ab
  - Mon
  - SemiGrp

satisfied_properties:
  - property_id: locally small
    reason: There is a forgetful functor $\Grp \to \Set$ and $\Set$ is locally small.

  - property_id: pointed
    reason: The trivial group is a zero object.
    check_redundancy: false

  - property_id: finitary algebraic
    reason: Take the algebraic theory of a group.

  - property_id: mono-regular
    reason: See Prop. 4.2 at the <a href="https://ncatlab.org/nlab/show/regular+monomorphism#Examples" target="_blank">nLab</a>.

  - property_id: Malcev
    reason: See Example 2.2.4 in <a href="https://ncatlab.org/nlab/show/Malcev,+protomodular,+homological+and+semi-abelian+categories" target="_blank">Malcev, protomodular, homological and semi-abelian categories</a>.

  - property_id: conormal
    reason: Since epimorphisms are surjective (see below), this is the first isomorphism theorem for groups.

  - property_id: effective cocongruences
    reason: A proof can be found <a href="/pdf/cocongruences_of_groups.pdf">here</a>.

unsatisfied_properties:
  - property_id: normal
    reason: Every non-normal subgroup provides a counterexample.

  - property_id: cogenerator
    reason: 'We apply <a href="/lemma/missing_cogenerator">this lemma</a> to the collection of simple groups: Any non-trivial homomorphism from a simple group to a group must be injective, and for every infinite cardinal $\kappa$ there is a simple group of size $\geq \kappa$ (for example, the alternating group on $\kappa$ elements).'

  - property_id: skeletal
    reason: This is trivial.

  - property_id: coregular
    reason: This is because injective group homomorphisms are not stable under pushouts, see e.g. <a href="https://math.stackexchange.com/questions/601463/" target="_blank">MSE/601463</a> or <a href="https://math.stackexchange.com/questions/5088032" target="_blank">MSE/5088032</a>.

  - property_id: counital
    reason: The canonical morphism $F_2 = \IZ \sqcup \IZ \to \IZ \times \IZ$ is not a monomorphism since $F_2$ is not abelian.

  - property_id: CIP
    # TODO: remove code duplication with "counital" proof
    reason: The canonical morphism $F_2 = \IZ \sqcup \IZ \to \IZ \times \IZ$ is not a monomorphism since $F_2$ is not abelian.

  - property_id: regular quotient object classifier
    reason: 'Assume that $\Grp$ has a (regular) quotient object classifier, i.e. a group $P$ such that every surjective homomorphism $G \to H$ is the cokernel of a unique homomorphism $\varphi : P \to G$. Equivalently, every normal subgroup $N \subseteq G$ is $\langle \langle \varphi(P) \rangle \rangle$ for a unique homomorphism $\varphi : P \to G$, where $\langle \langle - \rangle \rangle$ denotes the normal closure. If $c_g : G \to G$ denotes the conjugation with $g \in G$, then the images of $\varphi$ and $c_g \circ \varphi$ have the same normal closures, so the homomorphisms must be equal. In other words, $\varphi$ factors through the center $Z(G)$. But then every normal subgroup of $G$, in particular $G$ itself, would be contained in $Z(G)$, which is wrong for every non-abelian group $G$.'

  - property_id: cocartesian cofiltered limits
    reason: >-
      For cofiltered diagrams of groups $(H_i)$ and a group $G$ the canonical homomorphism
      $$\textstyle \alpha : G \sqcup \lim_i H_i \to \lim_i (G \sqcup H_i)$$
      is injective, but often fails to be surjective because the components of an element in the image have bounded <i>free product length</i> (the number of factors appearing in the reduced form). Specifically, consider the free groups $G = \langle y \rangle$ and $H_n = \langle x_1,\dotsc,x_n \rangle$ for $n \in \IN$ with the truncation maps $H_{n+1} \to H_n$, $x_{n+1} \mapsto 1$. Define
      $$p_n := x_1 \, y \, x_2 \, y \, \cdots \, x_{n-1} \, y \, x_n \, y^{-(n-1)} \in G \sqcup H_n.$$
      If we substitute $x_{n+1}=1$ in $p_{n+1}$, we get $p_n$. Thus, we have $p = (p_n) \in \lim_n (G \sqcup H_n)$. This element does not lie in the image of $\alpha$ since the free product length of $p_n$ (which is well-defined) is $2n$, which is unbounded.

  - property_id: CSP
    reason: The canonical homomorphism $\coprod_{n \geq 0} \IZ \to \prod_{n \geq 0} \IZ$ is not surjective because its domain is countable and its codomain is uncountable. Hence it is no epimorphism.

  - property_id: cofiltered-limit-stable epimorphisms
    reason: We already know that <a href="/category/Ab">$\Ab$</a> does not have this property. Now apply the contrapositive of the dual of <a href="/lemma/filtered-monos">this lemma</a> to the forgetful functor $\Ab \to \Grp$ which indeed preserves epimorphisms.

special_objects:
  initial object:
    description: trivial group
  terminal object:
    description: trivial group
  coproducts:
    description: free products
  products:
    description: direct products with pointwise operations

special_morphisms:
  isomorphisms:
    description: bijective homomorphisms
    reason: This characterization holds in every algebraic category.
  monomorphisms:
    description: injective homomorphisms
    reason: 'This holds in every finitary algebraic category: the forgetful functor to $\Set$ is faithful, hence reflects monomorphisms, and it is continuous (even representable), hence preserves monomorphisms.'
  epimorphisms:
    description: surjective homomorphisms
    reason: 'For the non-trivial direction, if $f : G \to H$ is an epimorphism, we may factor it as $G \to f(G) \to H$, and $f(G) \to H$ is still an epimorphism, but also an inclusion and hence a monomorphism. Since we already know that the category is balanced, $f(G) \to H$ must be an isomorphism.'
  regular monomorphisms:
    description: same as monomorphisms
    reason: This is because the category is mono-regular.
  regular epimorphisms:
    description: surjective homomorphisms
    reason: This holds in every finitary algebraic category.