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is the category with groups as objects and group homomorphisms as morphisms.
Similarly there there is the full subcategory of finite groups.
More generally, if is any category with finite products, there is a category of group objects in . This category if equivalent to the category of product-preserving functors from the Lawvere theory for groups to . For instance for SmthMfd this yields the category of Lie groups; while for Set it reduces again to the default case.
If one associates to a group its delooping one-object groupoid , it is sometimes of interest to regard the collection of groups instead as a 2-category, namely as the full sub--category of Grpd on one-object groupoids. In this case the -morphisms between homomorphisms of groups come from “intertwiners”: inner automorphisms of the target group – hence this 2-category is not equivalent to the 1-category of groups.
On the other hand, if we regard as a full sub--category of , the -category of pointed groupoids, then this is locally discrete and equivalent to the ordinary 1-category . This is because the only pointed intertwiner between two homomorphisms is the identity.
Precisely analogous statements hold for the category Alg of algebras.
(In this section, all statements about are valid more generally for where is a topos with a natural numbers object.)
The category is one of the prototypical examples of a semiabelian category, and so enjoys some nice properties. For example, it is regular and even exact, and protomodular so that one can expect a certain battery of diagram chasing lemmas to hold in it.
The category of groups is also balanced. This follows from a somewhat remarkable theorem:
Every monomorphism in is an equalizer.
The proofs most commonly seen in the literature are elementary but nonconstructive; a typical example may be found here at regular monomorphism. Here we give a constructive proof.
Let be monic, and let be the canonical surjective function . Let be the free abelian group on with the canonical injection, and let denote the set of functions , with the pointwise abelian group structure inherited from . This carries a -module structure defined by
For any , the function defined by defines a derivation, i.e., a map satisfying the equation . Consider now the wreath product, i.e., the semidirect product , where the multiplication is defined by . By the derivation equation, we have a homomorphism defined by , and there is a second homomorphism defined by . We claim that is the equalizer of the pair . For,
(All we needed was some injection into an abelian group; embedding into the free abelian group is a pretty canonical choice.)
This proof can be adapted to show that monomorphisms in the category of finite groups (group objects in ) are also equalizers. All that needs to be modified is the choice of , which we could take to be a free -vector space generated by .
The proof requires that the unit of the free abelian group monad (or the free vector space monad) is monic, constructively. This is not entirely obvious, and is not a well-known fact, but details can be found in this MO thread.
The category of groups is balanced: every epic mono is an isomorphism.
This follows because an epic equalizer is an equalizer of two maps that (by epi-ness) must be the same, hence the equalizer is an isomorphism.
Every epimorphism in the category of groups is a coequalizer.
Since every morphism factors as a regular epi followed by a mono , having epi implies is a epic mono. Epic monos being isomorphisms, is then forced to be regular epic as well.
Despite the fact that every morphism in factors as an epi followed by a regular mono, it is not true that is regular. Indeed, (regular) monos are in not stable under pushouts. This follows essentially from the plenitude of simple objects in : if is not simple but embeds in a simple group , then there is a nontrivial quotient , and in the pushout diagram
the object will be a proper quotient of and therefore , so that the pushout of the mono which is fails to be mono.
An axiomatization of the category of groups is given in
Last revised on May 6, 2023 at 12:19:52. See the history of this page for a list of all contributions to it.