Tangent Category

A tangent structure on a Category $\mathcal{C}$ consists of

• Tangent Functor: A Functor $T:\mathcal{C}\to\mathcal{C}$

• Tangent Bundle: Natural Transformations $p:T\to mathrm{Id}{ifx..lse\par i}$, $\eta:mathrm{Id}{ifx..lse\par i}\to T$ and $\mu:T_{2}\to T$ such that

  • Finite pullback powers $p_{n}:T_{n}(M)\to M$ of $p$ exist and are preserved by $T$.
  • $(p_{M},\eta_{M},\mu_{M})$ forms an Additive Bundle for each $M:\mathcal{C}$.

  We will write $f+g$ to denote the map $\mu\circ(f\times_{M}g)$ for $f,g:X\to T(M)$ with $p\circ f=p\circ g$.

• Vertical Lift: There is a Natural Transformation $\ell:T\to T^{2}$ such that for each $M:\mathcal{C}$,

  is a Morphism of Additive Bundles $(p,\eta_{M},\mu_{M})\to(Tp,T(\eta_{M}),T(\mu_{M}))$.

• Canonical Flip: There is a Natural Transformation $c:T^{2}\to T^{2}$ such that for each $M:\mathcal{C}$,

  is a Morphism of Additive Bundles $(Tp,T(\eta_{M}),T(\mu_{M}))\to(pT,\eta_{T(M)},\mu_{T(M)})$

• Coherence of Vertical Lift and Flip We have:

  • $c\circ c=mathrm{id}{ifx..lse\par i}$
  • $\ell\circ c=\ell$

  Moreover, the following diagrams must commute:

• Universality of the Vertical Lift

  For all $M:\mathcal{C}$, the following square is a pullback square:

Examples

• The category of Commutative Semirings forms a tangent category. This example is taken from An Embedding Theorem for Tangent Categories

  • Tangent functor: $T(R)=R[x]/x^{2}$, which sends $R$ to the Semiring of Dual Numbers. Iterating $T^{n}(R)$ yields the ring $R[x_{1},\ldots,x_{n}]/x_{1}^{2},\ldots x_{n}^{2}$ where we remove all higher-order terms.

  • Tangent bundle:

    	$\displaystyle p({r}+{a}\varepsilon)$	$\displaystyle=r$
    	$\displaystyle\eta(r)$	$\displaystyle={r}+{0}\varepsilon$
    	$\displaystyle\mu({r}+{a}\varepsilon,{r}+{b}\varepsilon)$	$\displaystyle={r}+{(a+b)}\varepsilon$

    The Kernel Pair of $p$ is the ring $R[x,y]/x^{2},y^{2},xy$, so finitary pullback powers exist. Moreover, they are preserved by $T$, as $(R[x,y]/x^{2},y^{2},xy)[z]/z^{2}$ is isomorphic to the pullback

    The bundle indexing constraints are obviously validated, and the additive bundle laws are clearly satisfied.

  • Vertical Lift:

    $\ell({r}+{a}\varepsilon)=r+0\varepsilon_{0}+0\varepsilon_{1}+a\varepsilon_{0} \varepsilon_{1}$