Groups and Categories

Groups in a Category

Group Object

Let $\mathsf{C}$ be a category with finite products. A group object in $\mathsf{C}$ consists of objects and arrows as:

satisfying the following conditions:

• $m$ is associative, that is the following commutes:
  where $\cong$ is the canonical associativity isomorphism for products.
• $u$ is a unit for $m$, that is both triangles in the following commute:
• $i$ is an inverse with respect to $m$, that is both sides of this commute:

Homomorphism

A morphism $h:G\to H$ of groups in $\mathsf{C}$ is called a homomorphism if

• $h$ preserves $m$
• $h$ preserves $u$
• $h$ preserves $i$

e.g. The idea of a group in a category captures the familiar notion of a group with additional structure:

• A group in the usual sense is a group object in the category $\mathsf{Set}$.
• A topological group is a group in $\mathsf{Top}$, the category of topological spaces.
• A group object in $\mathsf{Group}$ is an abelian group.
  

Category of Groups

Prop If $N◃G$ is a normal subgroup of $G$, then the following forms a coequalizer:
Proof

Groups as Categories

Prop A group is a category with one object, in which every arrow is an isomorphism.

Def Congruence
A congruence on a category $\mathbf{C}$ is an equivalence relation $\sim$ on morphisms such that:

• $f\sim g$ implies $\text{dom}(f)=\text{dom}(g)$ and $\text{cod}(f)=\text{cod}(g)$.
• $f\sim g$ implies $b\circ f\circ a\sim b\circ g\circ a$ for all morphisms $a:A\to X$ and $b:Y\to B$, where $\text{dom}(f)=X=\text{dom}(g)$ and $\text{cod}(f)=Y=\text{cod}(g)$.

Prop The intersection of a family of congruences is a congruence.

Def Congruence Category
Let $\sim$ be a congruence on the category $\mathbf{C}$, and define the congruence category ${\mathbf{C}}^{\sim }$ by

• $\text{obj}({\mathbf{C}}^{\sim })=\text{obj}\mathbf{C}$
• $\text{mor}({\mathbf{C}}^{\sim })=\{⟨f,g⟩\mid f\sim g\}$
• ${\tilde{1}}_C=⟨1_C,1_C⟩$
• $⟨f^{\prime },g^{\prime }⟩\circ ⟨f,g⟩=⟨f^{\prime }f,g^{\prime }g⟩$

Def Quotient Category
We define the quotient category $\mathbf{C}/\sim$ as follows:

• $\text{obj}(C/\sim )=\text{obj}\mathbf{C}$.
• $\text{mor}(\mathbf{C}/\sim )=(\text{mor}\mathbf{C})/\sim$.
• The morphisms have the form $[f]$ where $f\in \text{mor}\mathbf{C}$.
• $1_{[C]}=[1_C]$, and $[g]\circ [f]=[g\circ f]$.

Prop There is an evident quotient functor $\pi :\mathbf{C}\to \mathbf{C}/\sim$. It then makes the following a coequalizer of categories:

Proof

Def Kernel of Functor
Any functor $F:\mathbf{C}\to \mathbf{D}$ determines a congruence ${\sim }_F$ on $\mathbf{C}$ by setting$f\sim g⟺\text{dom}(f)=\text{dom}(g),\text{cod}(f)=\text{cod}(g),F(f)=F(g)$And we define $\ker (F)={\mathbf{C}}^{{\sim }_F}$ as the kernel of $F$.

Thrm For all functor $F:\mathbf{C}\to \mathbf{D}$, given any congruence $\sim$ on $\mathbf{C}$, one has: $f\sim g⟹f{\sim }_{F}g$if and only if there is a factorization $\tilde{F}:\mathbf{C}/\sim \to \mathbf{D}$ such that the following commutes:

Proof

Corollary Every functor $F:\mathbf{C}\to \mathbf{D}$ factors as $F=\tilde{F}\circ \pi$:

where $\pi$ is bijective on objects and surjective on Hom-sets, and $\tilde{F}$ is injective
on Hom-sets.

Finitely Presented Categories

e.g. The category with two uniquely isomorphic objects is not free on any graph, since it’s finite, but has loops. But it is finitely presented with graph

and relations $gf=1_A$ and $fg=1_B$.