Additive Bundle

An additive bundle over an object $A:\mathcal{C}$ in a Category $\mathcal{C}$ is a Commutative Monoid Object in the Slice Category $\mathcal{C}/A$.

Explicitly, this consists of the following data:

• A map $p:X\to A$ such that finite pullback powers of $p$ exist with projections $\pi_{i}:X_{n}\to X$ and bundle map $p_{n}:X_{n}\to A$. Note that we also have

• Maps $\mu:X_{2}\to X$ and $\eta:A\to X$ with indexing constraints

Such that

• $\mu$ is Commutative

• $\mu$ is Left Unital with respect to $\eta$

• $\mu$ is Associative

Additive bundles generalize Beck Modules, which act as a general notion of a coefficient object for Cohomology.

Moreover, the map $\eta:A\to X$ makes $X\to A$ an object in the Fibration of Points.

Examples

• An additive bundle in the Category of Semirings over $R$ is equivalent to the data of an $(R,R)$-Bimodule.

  Let $(p:R\to A,\mu:R\times_{A}R\to R,\eta:A\to R)$ be an additive bundle in the Category of Semirings. We shall view $p$ as a Displayed Semiring, and write $x\rightarrowtriangle r$ to denote that $p(x)=r$.

  Now, let's unfold all the conditions. $\mu$ is a semiring homomorphism $X_{2}\to X$, so for all $w,x\rightarrowtriangle r$ and $y,z\rightarrowtriangle s$, we have the following interchange laws.

  • $\mu(w+y,x+z)=\mu(w,x)+\mu(y,z)\$

  • $\mu(wy,xz)=\mu(w,x)\mu(y,z)$

  • $\mu(0,0)=0$
  • $\mu(1,1)=1$

  Moreover, $\mu$ is left and right unital with respect to $\eta$, so for all $x\rightarrowtriangle r$, we have:

  • $\mu(\eta(r),x)=x=\mu(x,\eta(r))$

  Finally, $\eta$ is a semiring homomorphism as well, so we have

  • $\eta(0)=0$

  Now, consider the Kernel $\ker p$; this canonically carries the structure of an $(A,A)$-bimodule. We shall now show that the internal commutative monoid structure of the bundle restricts to the additive structure of $\ker p$.

  The argument follows the usual Eckmann-Hilton script:

  	$\displaystyle\mu(x,y)$	$\displaystyle=\mu(x+0,0+y)$
  		$\displaystyle=\mu(x,0)+\mu(0,y)$
  		$\displaystyle=\mu(x,\eta(0))+\mu(\eta(0),y)$
  		$\displaystyle=x+y$

  Moreover, the action $\eta:A\to R$ equips $R$ with the structure of an $(A,A)$ bimodule, with

  • $r\cdot x=\eta(r)x$
  • $x\cdot r=x\eta(r)$

  Another Eckmann-Hilton swindle lets us deduce that for $x\rightarrowtriangle r,y\rightarrowtriangle s$, we have $\mu(s\cdot x,y\cdot r)=yx$

  	$\displaystyle\mu(s\cdot x,y\cdot r)$	$\displaystyle=\mu(\eta(s)x,y\eta(r))$
  		$\displaystyle=\mu(\eta(s),y)\mu(x,\eta(r))$
  		$\displaystyle=yx$

  In particular, if $x,y\rightarrowtriangle 0$, we have

  	$\displaystyle xy$	$\displaystyle=\mu(0\cdot y,x\cdot 0)$
  		$\displaystyle=\mu(0,0)$
  		$\displaystyle=0$

  This means that $X$ is Isomorphic to the Square-Zero Extension $R\oplus\ker p$, whose semiring structure is given by

  • $(a_{0},x_{0})+(a_{1},x_{1})=(a_{0}+a_{1},m_{0}+m_{1})$
  • $(a_{0},x_{0})\cdot(a_{1},x_{1})=(a_{0}a_{1},a_{0}\cdot m_{1}+m_{0}\cdot a_{1})$

  And additive bundle structure by

  • $p(a_{0},x_{0})=a_{0}$

  • $\eta(a)=(a,0)$

  • $\mu((a,x_{0}),(a,x_{1}))=(a,x_{0}+x_{1})$

In a Displayed Category

More generally, we can define an additive bundle over $A:\mathcal{B}$ in a Displayed Category $\mathcal{E}\rightarrowtriangle\mathcal{B}$ as an object $X\rightarrowtriangle B$ such that

• Joint Cartesian Lifts $X_{n}$ of the family $\lambda(i:[n]).\;X$ along $\lambda(i:[n]).\;mathrm{id}{ifx..lse\par i}$ exist. Notably, this means that the Fibre Category $\mathcal{E}_{A}$ has a Terminal Object $1$ that arises from a joint cartesian lift of the empty family.
• There are vertical maps $+:X_{n}\to X$, $0:1\to X$ that satisfy the obvious unitality, commutativity, and associativity squares.

References

• Differential Structure, Tangent Structure, and SDG