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Here are two results about groups:

(The Third isomorphism theorem) Suppose that I have $A \triangleleft B \triangleleft C$ and $A \triangleleft C$. Then $C/B \cong (C/A)/(B/A)$.

(An exercise I just assigned my students) Suppose that we have $X \triangleleft Z$ and $Y \triangleleft Z$ with $X \cap Y = 1$. Then $(Z/X)/Y \cong (Z/Y)/X$.

Vague question a student just asked me Is there is some general context in which to think of these results and why they look similar?

We can write these as diagrams with exact rows and columns:

$$\begin{matrix} && && 1 && 1 && \\ && && \downarrow && \downarrow && \\ 1 &\to& A &\to& B &\to& B/A &\to& 1 \\ && = && \downarrow && \downarrow && \\ 1 &\to& A &\to& C &\to& C/A &\to& 1 \\ && && \downarrow && \downarrow && \\ && && C/B & \cong & (C/A)/(B/A) && \\ && && \downarrow && \downarrow && \\ && && 1 && 1 && \\ \end{matrix}$$

$$\begin{matrix} && && 1 && 1 && \\ && && \downarrow && \downarrow && \\ && && X &=& X && \\ && && \downarrow && \downarrow && \\ 1 &\to& Y &\to& Z &\to& Z/Y &\to& 1 \\ && = && \downarrow && \downarrow && \\ && Y &\to& Z/X &\to& (Z/X)/Y \cong (Z/Y)/X &\to& 1 \\ && && \downarrow && \downarrow && \\ && && 1 && 1 && \\ \end{matrix}$$

If we were in an abelian category, these would be two forms of the octahedral axiom.

Vague but more technical question Is there something like an semi-abelian category which includes the case of groups, and where we have something like an octahedral axiom?

$\require{AMScd}$Here are two results about groups:

(The third isomorphism theorem) Suppose that I have $A \triangleleft B \triangleleft C$ and $A \triangleleft C$. Then $C/B \cong (C/A)/(B/A)$.

(An exercise I just assigned my students) Suppose that we have $X \triangleleft Z$ and $Y \triangleleft Z$ with $X \cap Y = 1$. Then $(Z/X)/Y \cong (Z/Y)/X$.

Vague question a student just asked me: Is there some general context in which to think of these results and why they look similar?

We can write these as diagrams with exact rows and columns: \begin{gather} \begin{CD} @. @. 1 @. 1 \\ @. @. @VVV @VVV \\ 1 @>>> A @>>> B @>>> B/A @>>> 1 \\ @. @| @VVV @VVV \\ 1 @>>> A @>>> C @>>> C/A @>>> 1 \\ @. @. @VVV @VVV \\ @. @. C/B @>\cong>> (C/A)/(B/A) \\ @. @. @VVV @VVV \\ @. @. 1 @. 1 \end{CD} \\ \begin{CD} @. @. 1 @. 1 \\ @. @. @VVV @VVV \\ @. @. X @= X \\ @. @. @VVV @VVV \\ 1 @>>> Y @>>> Z @>>> Z/Y @>>> 1 \\ @. @| @VVV @VVV \\ 1 @>>> Y @>>> Z/X @>>> (Z/X)/Y \cong (Z/Y)/X @>>> 1 \\ @. @. @VVV @VVV \\ @. @. 1 @. 1 \end{CD} \end{gather} If we were in an abelian category, these would be two forms of the octahedral axiom.

Vague but more technical question: Is there something like a semi-abelian category which includes the case of groups, and where we have something like an octahedral axiom?

Here are two results about groups:

( The Third isomorphism theorem) Suppose that I have $A \triangleleft B \triangleleft C$ and $A \triangleleft C$. Then $C/B \cong (C/A)/(B/A)$.

( An exercise I just assigned my students) Suppose that we have $X \triangleleft Z$ and $Y \triangleleft Z$ with $X \cap Y = 1$. Then $(Z/X)/Y \cong (Z/Y)/X$.

Vague question a student just asked me Is there is some general context in which to think of these results and why they look similar?

We can write these as diagrams with exact rows and columns:

$$\begin{matrix} && && 1 && 1 && \\ && && \downarrow && \downarrow && \\ 1 &\to& A &\to& B &\to& B/A &\to& 1 \\ && = && \downarrow && \downarrow && \\ 1 &\to& A &\to& C &\to& C/A &\to& 1 \\ && && \downarrow && \downarrow && \\ && && C/B & \cong & (C/A)/(B/A) && \\ && && \downarrow && \downarrow && \\ && && 1 && 1 && \\ \end{matrix}$$

$$\begin{matrix} && && 1 && 1 && \\ && && \downarrow && \downarrow && \\ && && X &=& X && \\ && && \downarrow && \downarrow && \\ 1 &\to& Y &\to& Z &\to& Z/Y &\to& 1 \\ && = && \downarrow && \downarrow && \\ && Y &\to& Z/X &\to& (Z/X)/Y \cong (Z/Y)/X &\to& 1 \\ && && \downarrow && \downarrow && \\ && && 1 && 1 && \\ \end{matrix}$$

If we were in an abelian category, these would be two forms of the octahedral axiom.

Vague but more technical question Is there something like an semi-abelian category which includes the case of groups, and where we have something like an octahedral axiom?

Here are two results about groups:

(The Third isomorphism theorem) Suppose that I have $A \triangleleft B \triangleleft C$ and $A \triangleleft C$. Then $C/B \cong (C/A)/(B/A)$.

(An exercise I just assigned my students) Suppose that we have $X \triangleleft Z$ and $Y \triangleleft Z$ with $X \cap Y = 1$. Then $(Z/X)/Y \cong (Z/Y)/X$.

Vague question a student just asked me Is there is some general context in which to think of these results and why they look similar?

We can write these as diagrams with exact rows and columns:

$$\begin{matrix} && && 1 && 1 && \\ && && \downarrow && \downarrow && \\ 1 &\to& A &\to& B &\to& B/A &\to& 1 \\ && = && \downarrow && \downarrow && \\ 1 &\to& A &\to& C &\to& C/A &\to& 1 \\ && && \downarrow && \downarrow && \\ && && C/B & \cong & (C/A)/(B/A) && \\ && && \downarrow && \downarrow && \\ && && 1 && 1 && \\ \end{matrix}$$

$$\begin{matrix} && && 1 && 1 && \\ && && \downarrow && \downarrow && \\ && && X &=& X && \\ && && \downarrow && \downarrow && \\ 1 &\to& Y &\to& Z &\to& Z/Y &\to& 1 \\ && = && \downarrow && \downarrow && \\ && Y &\to& Z/X &\to& (Z/X)/Y \cong (Z/Y)/X &\to& 1 \\ && && \downarrow && \downarrow && \\ && && 1 && 1 && \\ \end{matrix}$$

If we were in an abelian category, these would be two forms of the octahedral axiom.

Vague but more technical question Is there something like an semi-abelian category which includes the case of groups, and where we have something like an octahedral axiom?

Here are two results about groups:

( The Third isomorphism theorem) Suppose that I have $A \triangleleft B \triangleleft C$ and $A \triangleleft C$. Then $C/B \cong (C/A)/(B/A)$.

( An exercise I just assigned my students) Suppose that we have $X \triangleleft Z$ and $Y \triangleleft Z$ with $X \cap Y = \emptyset$. Then $(Z/X)/Y \cong (Z/Y)/X$.

Vague question a student just asked me Is there is some general context in which to think of these results and why they look similar?

We can write these as diagrams with exact rows and columns:

$$\begin{matrix} && && 1 && 1 && \\ && && \downarrow && \downarrow && \\ 1 &\to& A &\to& B &\to& B/A &\to& 1 \\ && = && \downarrow && \downarrow && \\ 1 &\to& A &\to& C &\to& C/A &\to& 1 \\ && && \downarrow && \downarrow && \\ && && C/B & \cong & (C/A)/(B/A) && \\ && && \downarrow && \downarrow && \\ && && 1 && 1 && \\ \end{matrix}$$

$$\begin{matrix} && && 1 && 1 && \\ && && \downarrow && \downarrow && \\ && && X &=& X && \\ && && \downarrow && \downarrow && \\ 1 &\to& Y &\to& Z &\to& Z/Y &\to& 1 \\ && = && \downarrow && \downarrow && \\ && Y &\to& Z/X &\to& (Z/X)/Y \cong (Z/Y)/X &\to& 1 \\ && && \downarrow && \downarrow && \\ && && 1 && 1 && \\ \end{matrix}$$

If we were in an abelian category, these would be two forms of the octahedral axiom.

Vague but more technical question Is there something like an semi-abelian category which includes the case of groups, and where we have something like an octahedral axiom?

Here are two results about groups:

( The Third isomorphism theorem) Suppose that I have $A \triangleleft B \triangleleft C$ and $A \triangleleft C$. Then $C/B \cong (C/A)/(B/A)$.

( An exercise I just assigned my students) Suppose that we have $X \triangleleft Z$ and $Y \triangleleft Z$ with $X \cap Y = 1$. Then $(Z/X)/Y \cong (Z/Y)/X$.

Vague question a student just asked me Is there is some general context in which to think of these results and why they look similar?

We can write these as diagrams with exact rows and columns:

$$\begin{matrix} && && 1 && 1 && \\ && && \downarrow && \downarrow && \\ 1 &\to& A &\to& B &\to& B/A &\to& 1 \\ && = && \downarrow && \downarrow && \\ 1 &\to& A &\to& C &\to& C/A &\to& 1 \\ && && \downarrow && \downarrow && \\ && && C/B & \cong & (C/A)/(B/A) && \\ && && \downarrow && \downarrow && \\ && && 1 && 1 && \\ \end{matrix}$$

$$\begin{matrix} && && 1 && 1 && \\ && && \downarrow && \downarrow && \\ && && X &=& X && \\ && && \downarrow && \downarrow && \\ 1 &\to& Y &\to& Z &\to& Z/Y &\to& 1 \\ && = && \downarrow && \downarrow && \\ && Y &\to& Z/X &\to& (Z/X)/Y \cong (Z/Y)/X &\to& 1 \\ && && \downarrow && \downarrow && \\ && && 1 && 1 && \\ \end{matrix}$$

If we were in an abelian category, these would be two forms of the octahedral axiom.

Vague but more technical question Is there something like an semi-abelian category which includes the case of groups, and where we have something like an octahedral axiom?