2.1 Fibrations and Grothendieck construction

A fibration, introduced by Grothendieck (see Grothendieck (Reference Grothendieck1963), Section 12), is the category-theoretic analog of a fibre bundle. Concretely, a fibration consists of a functor from a category, called the total category, to another category, called the base category, which satisfies a universal property analog to the path-lifting property of the universal covering space of a topological space. An equivalent way of understanding a fibration is to regard it as a collection of categories, called fibres, indexed by the objects of the base category, “glued” together.

This interpretation is made precise by the Grothendieck construction, which establishes an equivalence between (cloven) fibrations and indexed categories. Concretely, the indexed category associated with a (cloven) fibration is the pseudofunctor which sends an object of the base category of the fibration to the fibre over the given object.

In this section, we recall the definitions of a (cloven) fibration, of an indexed category, and we recall the Grothendieck construction as presented in (Borceux (Reference Borceux1994), Theorem 8.3.1).

Let us briefly establish some of the jargon we adopt in the rest of the paper.

To introduce the notion of a fibration, we first need to recall the definition of a cartesian morphism.

Notation 2.4. In the following, a fibration $\Pi \colon {\mathbb{X}}'\to {\mathbb{X}}$ is denoted by $({\mathbb{X}},{\mathbb{X}}',\Pi)$ . Moreover, for a cloven fibration $({\mathbb{X}},{\mathbb{X}}',\Pi)$ , the cartesian lift of a morphism $f\colon A\to A'$ specified by the cleveage, corresponding to an object $E'$ over $A'$ , is denoted by:

\begin{align*} &\varphi ^{(f)}_{E'}\colon f^{\ast} E'\to E' \end{align*}

For the sake of simplicity, when the object $E'$ is clear by the context, we omit the subscript $_{E'}$ . When the symbol adopted for a cloven fibration is decorated with a subscript or with a superscript, the same decoration is applied to the cartesian lifts. For example, for a cloven fibration $({\mathbb{X}}_{\scriptscriptstyle \square },{\mathbb{X}}_{\scriptscriptstyle \square }',\Pi _{\scriptscriptstyle \square })$ , a morphism $f\colon A\to A'$ of ${\mathbb{X}}_{\scriptscriptstyle \square }$ , and an object $E'$ over $A'$ , the corresponding cartesian lift is denoted by $\varphi _{\scriptscriptstyle \square }^{(f)}$ .

Every morphism $\psi \colon E\to E'$ of the total category ${\mathbb{X}}'$ of a fibration $({\mathbb{X}},{\mathbb{X}}',\Pi)$ admits a decomposition into a vertical morphism $\xi \colon E\to \tilde E$ and a cartesian morphism $\varphi \colon \tilde E\to E'$ . To see why, consider the morphism $(f\colon A\to A')\colon \kern -1ex=\Pi (\psi)$ . Since $\Pi$ is a fibration, there is a cartesian lift $\varphi \colon \tilde E\to E'$ of $f$ whose codomain is $E'$ . In particular, since $\varphi$ is a lift of $f$ , $\Pi (\varphi){\circ }{\mathsf{id}}_A=\Pi (\varphi)=f$ . For the cartesian property of $\varphi$ , there is a unique morphism $\xi \colon E\to \tilde E$ of $\mathbb{X}$ such that $\Pi (\xi)={\mathsf{id}}_A$ and $\varphi {\circ }\xi =\psi$ . Since $\xi$ is a lift of ${\mathsf{id}}_A$ , $\xi$ is vertical.

This decomposition is not unique, since there could be more than one cartesian lift of the morphism $f$ . However, when the fibration is equipped with a cleavage, there is a canonical choice of such a decomposition. In particular, for a cloven fibration, each morphism $\psi \colon E\to E'$ of the total category can be decomposed as follows:

However, since the cartesian lift $\varphi ^{(f)}\colon f^{\ast} E'\to E'$ is determined by the cleavage, by $f$ , and by $E'$ , each morphism $\psi \colon E\to E'$ of the total category is fully specified by a pair $(f,\xi ^{(f)})$ formed by a morphism $f\colon \kern -1ex=\Pi (\psi)$ of $\mathbb{X}$ and by a vertical morphism $\xi ^{(f)}\colon E\to f^{\ast} E'$ . This is precisely the intuition underpinning the Grothendieck construction.

The first step is to `split’ a fibration into the collection of its fibres. Such a collection has the structure of an indexed category. Informally, an indexed category on a category $\mathbb{X}$ consists of a collection of categories indexed by the objects of $\mathbb{X}$ together with a collection of functors between these categories, indexed by the morphisms of $\mathbb{X}$ , and for which the indexing of the categories and of the functors are compatible with the composition and the identity morphisms of $\mathbb{X}$ .

Each cloven fibration $({\mathbb{X}},{\mathbb{X}}',\Pi)$ is associated with an indexed category, denoted by $\mathscr{I}(\Pi)\colon {\mathbb{X}}^{\mathsf{op}}\to {\mathsf{CAT}}$ , defined as follows:

• categories. Each object $A$ of $\mathbb{X}$ is associated with the corresponding fibre $\Pi ^{-1}(A)$ ;

• substitution functors. Each morphism $f\colon A\to A'$ of $\mathbb{X}$ is associated with the functor $f^{\ast} \colon \Pi ^{-1}(A')\to \Pi ^{-1}(A)$ which sends each object $E'$ over $A'$ to the object $f^{\ast} E'$ , domain of the cartesian lift $\varphi ^{(f)}\colon f^{\ast} E'\to E'$ of $f$ , and which sends any vertical morphism $\xi \colon E'\to E''$ over $A'$ to the unique vertical morphism $f^{\ast} \xi \colon f^{\ast} E'\to f^{\ast} E''$ over $A$ such that $\varphi ^{(f)}_{E''}{\circ } f^{\ast} \xi =\varphi ^{(f)}_{E'}$ , defined by the universality of the cartesian morphism $\varphi ^{(f)}_{E''}$ ;

• unitor. Given an object $E$ over $A$ , the unitor is the unique vertical natural isomorphism ${\mathfrak{I}}^0\colon E\to {\mathsf{id}}_A^{\ast} E$ such that ${\mathfrak{I}}^0\varphi ^{\left ({\mathsf{id}}_A\right)}={\mathsf{id}}_E$ , induced by the universality of the cartesian morphism $\varphi ^{\left ({\mathsf{id}}_A\right)}$ ;

• compositor. Given an object $E''$ over $A''$ and two morphisms $f\colon A\to A'$ and $g\colon A'\to A''$ of $\mathbb{X}$ , the compositor is the unique vertical natural isomorphism ${\mathfrak{I}}^2\colon f^{\ast} g^{\ast} E''\to (g{\circ } f)^{\ast} E''$ such that $\varphi ^{\left (g{\circ } f\right)}{\circ }{\mathfrak{I}}^2=\varphi ^{\left (g\right)}{\circ }\varphi ^{(f)}$ , induced by the universality of the cartesian morphism $\varphi ^{\left (g{\circ } f\right)}$ .

The process of “separating” a cloven fibration into the indexed category $\mathscr{I}(\Pi)$ of its fibres is analogous to associating a fibre bundle $\pi \colon E\to M$ over a topological space $M$ with the assignment which sends each point $x$ of $M$ to the fibre $\pi ^{-1}(x)$ over $x$ . This process can be reversed via the Grothendieck construction.

Let us start by considering an indexed category $({\mathbb{X}},{\mathfrak{I}})$ , which sends each object $A$ of $\mathbb{X}$ to the category ${\mathbb{X}}^{\left (A\right)}$ and each morphism $f\colon A\to A'$ to the functor $f^{\ast} \colon {\mathbb{X}}^{\left (A'\right)}\to {\mathbb{X}}^{\left (A\right)}$ . Then, the category of elements of $\mathfrak{I}$ is the category $\mathsf{EL}({\mathbb{X}},{\mathfrak{I}})$ defined as follows:

• objects. The objects of $\mathsf{EL}({\mathbb{X}},{\mathfrak{I}})$ are pairs $(A,E)$ formed by an object $A$ of $\mathbb{X}$ together with an object $E$ of ${\mathbb{X}}^{\left (A\right)}$ ;

• morphisms. The morphisms of $\mathsf{EL}({\mathbb{X}},{\mathfrak{I}})$ are pairs $(f,\xi ^{(f)})\colon (A,E)\to (A',E')$ formed by a morphism $f\colon A\to A'$ of $\mathbb{X}$ together with a morphism $\xi ^{(f)}\colon E\to f^{\ast} E'$ of ${\mathbb{X}}^{\left (A\right)}$ ;

• identities. The identity morphism of a pair $(A,E)\in \mathsf{EL}({\mathbb{X}},{\mathfrak{I}})$ is given by the pair $({\mathsf{id}}_A,\xi ^{\left ({\mathsf{id}}_A\right)})$ , where $\xi ^{\left ({\mathsf{id}}_A\right)}\colon E\to {\mathsf{id}}_A^{\ast} E$ is the unitor of $\mathfrak{I}$ ;

• composition. Given two composable morphisms $(f,\xi ^{(f)})\colon (A,E)\to (A',E')$ and $(g,\xi ^{\left (g\right)})\colon (A',E')\to (A'',E'')$ of $\mathsf{EL}({\mathbb{X}},{\mathfrak{I}})$ , their composition is the pair $(g{\circ } f,\xi ^{\left (g{\circ } f\right)})$ where

  \begin{align*} &\xi ^{\left (g{\circ } f\right)}\colon E\xrightarrow {\xi ^{(f)}}f^{\ast} E'\xrightarrow {f^{\ast} \xi ^{\left (g\right)}}f^{\ast} g^{\ast} E''\xrightarrow {{\mathfrak{I}}^2}(g{\circ } f)^{\ast} E'' \end{align*}
  ${\mathfrak{I}}^2$ denoting the compositor of $\mathfrak{I}$ .

The functor ${\mathscr{F}}({\mathfrak{I}})\colon \mathsf{EL}({\mathbb{X}},{\mathfrak{I}})\to {\mathbb{X}}$ which sends each object $(A,E)$ of $\mathsf{EL}({\mathbb{X}},{\mathfrak{I}})$ to the object $A$ of $\mathbb{X}$ and each morphism $(f,\xi ^{(f)})$ of $\mathsf{EL}({\mathbb{X}},{\mathfrak{I}})$ to the morphism $f$ of $\mathbb{X}$ is a cloven fibration.

The assignments $\mathscr{I}$ and $\mathscr{F}$ which send a cloven fibration to an indexed category and vice versa, respectively, extend to an equivalence of $2$ -categories. To see this, let us recall the definition of the $1$ -morphisms and $2$ -morphisms of indexed categories and fibrations.

Definition 2.9. A $2$ -morphism of fibrations $(\theta ,\theta ')\colon (F,F')\to (G,G')$ between two $1$ -morphisms $(F,F'),(G,G')\colon ({\mathbb{X}}_{\scriptscriptstyle \square },{\mathbb{X}}_{\scriptscriptstyle \square }',\Pi _{\scriptscriptstyle \square })\to ({\mathbb{X}}_{\scriptscriptstyle \blacksquare },{\mathbb{X}}_{\scriptscriptstyle \blacksquare }',\Pi _{\scriptscriptstyle \blacksquare })$ of fibrations consists of two natural transformations $\theta \colon F\Rightarrow G$ and $\theta '\colon F'\Rightarrow G'$ such that:

\begin{align*} &\Pi _{\scriptscriptstyle \blacksquare }\theta '=\theta _{\Pi _{\scriptscriptstyle \square }} \end{align*}

Proposition 2.15. The assignment $\mathscr{F}$ which sends an indexed category $({\mathbb{X}},{\mathfrak{I}})$ to the cloven fibration $({\mathbb{X}},\mathsf{EL}({\mathbb{X}},{\mathfrak{I}}),{\mathscr{F}}({\mathfrak{I}}))$ and the assignment $\mathscr{I}$ which sends a cloven fibration $({\mathbb{X}},{\mathbb{X}}',\Pi)$ to the indexed category $({\mathbb{X}},\mathscr{I}(\Pi))$ extend to an equivalence of $2$ -categories:

\begin{align*} &\mathsf{FIB}\simeq \mathsf{INDX} \end{align*}

Proof. Let us start by considering a $1$ -morphism of fibrations $(F,F')\colon ({\mathbb{X}}_{\scriptscriptstyle \square },{\mathbb{X}}_{\scriptscriptstyle \square }',\Pi _{\scriptscriptstyle \square })\to ({\mathbb{X}}_{\scriptscriptstyle \blacksquare },{\mathbb{X}}_{\scriptscriptstyle \blacksquare }',\Pi _{\scriptscriptstyle \blacksquare })$ and let us define the corresponding $1$ -morphism of indexed categories. The base functor is $F\colon {\mathbb{X}}_{\scriptscriptstyle \square }\to {\mathbb{X}}_{\scriptscriptstyle \blacksquare }$ . The collection of functors $F^{\left (A\right)}\colon \Pi _{\scriptscriptstyle \square }^{-1}(A)\to \Pi _{\scriptscriptstyle \blacksquare }^{-1}(FA)$ is given by the restriction of $F'$ to the fibres $\Pi _{\scriptscriptstyle \square }^{-1}(A)$ of $\Pi _{\scriptscriptstyle \square }$ . Indeed, since $F{\circ }\Pi _{\scriptscriptstyle \square }=\Pi _{\scriptscriptstyle \blacksquare }{\circ } F'$ , each $E\in \Pi _{\scriptscriptstyle \square }^{-1}(A)$ is sent to $F'E\in \Pi _{\scriptscriptstyle \blacksquare }^{-1}(FA)$ by $F'$ . The distributor $\kappa ^{(f)}\colon F^{\left (A\right)}{\circ } f^{\ast} _{\scriptscriptstyle \square }\Rightarrow (Ff)^{\ast} _{\scriptscriptstyle \blacksquare }{\circ } F^{\left (A'\right)}$ is defined by the distributor

\begin{align*} &\kappa ^{(f)}\colon F'(f^{\ast} _{\scriptscriptstyle \square } E')\to (Ff)^{\ast} _{\scriptscriptstyle \blacksquare }(F'E') \end{align*}

induced by the universality of the cartesian lift $\varphi ^{\left (Ff\right)}_{\scriptscriptstyle \blacksquare }\colon (Ff)^{\ast} _{\scriptscriptstyle \blacksquare }(F'E')\to F'(f^{\ast} _{\scriptscriptstyle \square } E')$ . To prove that $\kappa ^{(f)}$ is natural observe that, for a vertical morphism $\psi \colon E'\to E''$ , the following diagram commutes:

So, in particular:

\begin{align*} &\varphi ^{\left (Ff\right)}_{{\scriptscriptstyle \blacksquare } E''}{\circ }\kappa ^{(f)}_{E''}{\circ } F'(f_{\scriptscriptstyle \square }^{\ast} \psi)=\varphi ^{\left (Ff\right)}_{{\scriptscriptstyle \blacksquare } E''}{\circ }(Ff)^{\ast} _{\scriptscriptstyle \blacksquare }(F'\psi){\circ }\kappa ^{(f)}_{E'} \end{align*}

From the universality of the cartesian lift $\varphi ^{\left (Ff\right)}_{{\scriptscriptstyle \blacksquare } E''}$ , we obtain:

\begin{align*} &\kappa ^{(f)}_{E''}{\circ } F'(f_{\scriptscriptstyle \square }^{\ast} \psi)=(Ff)^{\ast} _{\scriptscriptstyle \blacksquare }(F'\psi){\circ }\kappa ^{(f)}_{E'} \end{align*}

Similarly, to prove the compatibility between the unitors, compositors and distributors, first notice that the following diagrams commute:

From the universality of the cartesian lifts we conclude:

\begin{align*} &\kappa ^{\left ({\mathsf{id}}_A\right)}{\circ } F^{\left (A\right)}{\mathfrak{I}}^0_{\scriptscriptstyle \square }={{\mathfrak{I}}^0_{\scriptscriptstyle \blacksquare }}_{F^{\left (A\right)}}\\ &{\mathfrak{I}}^2_{{\scriptscriptstyle \blacksquare } F^{\left (A''\right)}}{\circ }(Ff)^{\ast} _{\scriptscriptstyle \blacksquare }\kappa ^{\left (g\right)}{\circ }\kappa ^{(f)}_{g^{\ast} _{\scriptscriptstyle \square }}=\kappa ^{\left (g{\circ } f\right)}{\circ } F^{\left (A\right)}{\mathfrak{I}}^2_{\scriptscriptstyle \square } \end{align*}

Let us now consider a $2$ -morphism $(\theta ,\theta ')\colon (F,F')\Rightarrow (G,G')$ of fibrations between two $1$ -morphisms $(F,F'),(G,G')\colon ({\mathbb{X}}_{\scriptscriptstyle \square },{\mathbb{X}}_{\scriptscriptstyle \square }',\Pi _{\scriptscriptstyle \square })\to ({\mathbb{X}}_{\scriptscriptstyle \blacksquare },{\mathbb{X}}_{\scriptscriptstyle \blacksquare }',\Pi _{\scriptscriptstyle \blacksquare })$ of fibrations and let us define the corresponding $2$ -morphism

\begin{align*} &\mathscr{I}(\theta ,\theta ')\colon \mathscr{I}(F,F')\to \mathscr{I}(G,G') \end{align*}

of indexed categories between the two $1$ -morphisms $(F,F',\kappa),(G,G',\lambda)\colon \mathscr{I}({\mathbb{X}}_{\scriptscriptstyle \square },{\mathbb{X}}_{\scriptscriptstyle \square }',\Pi _{\scriptscriptstyle \square })\to \mathscr{I}({\mathbb{X}}_{\scriptscriptstyle \blacksquare },{\mathbb{X}}_{\scriptscriptstyle \blacksquare }',\Pi _{\scriptscriptstyle \blacksquare })$ . The base natural transformation is $\theta \colon F\Rightarrow G$ . To define the natural transformations $\theta ^{\left (A\right)}\colon F^{\left (A\right)}\Rightarrow \theta ^{\ast} {\circ } G^{\left (A\right)}$ , first notice that, since $\Pi _{\scriptscriptstyle \blacksquare }\theta '=\theta _{\Pi _{\scriptscriptstyle \square }}$ , $\theta '\colon F'E\to G'E$ is over $\theta \colon FA\to GA$ , for $A=\Pi _{\scriptscriptstyle \square } E$ . Therefore, the universality of the cartesian lift $\varphi ^{\left (\theta \right)}\colon \theta ^{\ast} G'E\to G'E$ defines a unique morphism $\theta ^{\left (A\right)}_E\colon F'E\to \theta ^{\ast} G'E$ making the following diagram commutes:

To show that $\theta ^{\left (A\right)}$ is natural, consider a vertical morphism $\psi \colon E\to E'$ :

In particular

\begin{align*} &\varphi ^{\left (\theta \right)}_{E'}{\circ }\theta ^{\ast} G'\psi {\circ }\theta ^{\left (A\right)}_E=\varphi ^{\left (\theta \right)}_{E'}{\circ }\theta ^{\left (A\right)}_{E'}{\circ } F'\psi \end{align*}

and from the universality of $\varphi ^{\left (\theta \right)}_{E'}\colon \theta ^{\ast} G'E'\to G'E'$ we deduce:

\begin{align*} &\theta ^{\ast} G'\psi {\circ }\theta ^{\left (A\right)}_E=\theta ^{\left (A\right)}_{E'}{\circ } F'\psi \end{align*}

To prove the compatibility between the distributors and $\theta '$ , observe that the following diagrams commute:

So, we have

\begin{align*} &\varphi ^{\left (\theta \right)}_{G'}{\circ }\varphi ^{\left (Ff\right)}_{{\scriptscriptstyle \blacksquare }\theta ^{\ast} G'}{\circ }\theta ^{\ast} \lambda ^{(f)}{\circ }\theta ^{\left (A\right)}_{f^{\ast} _{\scriptscriptstyle \square }}=\varphi ^{\left (\theta \right)}_{G'}{\circ }\varphi ^{\left (Ff\right)}_{{\scriptscriptstyle \blacksquare }\theta ^{\ast} G'}{\circ }(Ff)^{\ast} _{\scriptscriptstyle \blacksquare }\theta ^{\left (A'\right)}{\circ }\kappa ^{(f)} \end{align*}

and, from the universality of the cartesian lifts, we obtain

\begin{align*} &\theta ^{\ast} \lambda ^{(f)}{\circ }\theta ^{\left (A\right)}_{f^{\ast} _{\scriptscriptstyle \square }}=(Ff)^{\ast} _{\scriptscriptstyle \blacksquare }\theta ^{\left (A'\right)}{\circ }\kappa ^{(f)} \end{align*}

which is precisely the compatibility between $\theta '$ and the distributors.

Conversely, consider a $1$ -morphism $(F,F',\kappa)\colon ({\mathbb{X}}_{\scriptscriptstyle \square },{\mathfrak{I}}_{\scriptscriptstyle \square })\to ({\mathbb{X}}_{\scriptscriptstyle \blacksquare },{\mathfrak{I}}_{\scriptscriptstyle \blacksquare })$ of indexed categories and let us define the corresponding $1$ -morphism of fibrations. Let us start with the base functor, which is just the base functor $F\colon {\mathbb{X}}_{\scriptscriptstyle \square }\to {\mathbb{X}}_{\scriptscriptstyle \blacksquare }$ . To define the total functor $F'\colon {\mathbb{X}}_{\scriptscriptstyle \square }'\to {\mathbb{X}}_{\scriptscriptstyle \blacksquare }'$ , let us consider an object $(A,E)\in \mathsf{EL}({\mathbb{X}}_{\scriptscriptstyle \square },{\mathfrak{I}}_{\scriptscriptstyle \square })$ and let us define

\begin{align*} &F'(A,E)\colon \kern -1ex=(FA,F^{\left (A\right)}E) \end{align*}

where $F^{\left (A\right)}\colon {\mathbb{X}}_{\scriptscriptstyle \square }^{\left (A\right)}\to {\mathbb{X}}_{\scriptscriptstyle \blacksquare }^{\left (FA\right)}$ , so $(FA,F^{\left (A\right)}E)$ is indeed an object of $\mathsf{EL}({\mathbb{X}}_{\scriptscriptstyle \blacksquare },{\mathfrak{I}}_{\scriptscriptstyle \blacksquare })$ . Now, consider a morphism $(f,\xi ^{(f)})\colon (A,E)\to (A',E')$ of $\mathsf{EL}({\mathbb{X}}_{\scriptscriptstyle \square },{\mathfrak{I}}_{\scriptscriptstyle \square })$ . Let us define $F'(f,\xi ^{(f)})$ as the morphism $(Ff,\xi ^{\left (Ff\right)})$ , where

\begin{align*} &\xi ^{\left (Ff\right)}\colon F^{\left (A\right)}E\xrightarrow {F^{\left (A\right)}\xi ^{(f)}}F^{\left (A\right)}f^{\ast} _{\scriptscriptstyle \square } E'\xrightarrow {\kappa ^{(f)}}(Ff)^{\ast} _{\scriptscriptstyle \blacksquare } F^{\left (A'\right)}E' \end{align*}

where we employed the distributor $\kappa ^{(f)}\colon F^{\left (A\right)}{\circ } f^{\ast} _{\scriptscriptstyle \square }\Rightarrow (Ff)^{\ast} _{\scriptscriptstyle \blacksquare }{\circ } F^{\left (A'\right)}$ . To prove that $F^{\left (A\right)}$ is functorial, we need to employ the compatibilities of the distributors with the unitors and the compositors. Let us start by showing that $F'{\mathsf{id}}_{(A,E)}={\mathsf{id}}_{F'(A,E)}$ :

\begin{align*} &F'\left ({\mathsf{id}}_{(A,E)}\right)\\ &\quad =F'\left ({\mathsf{id}}_A,{\mathfrak{I}}^0_{\scriptscriptstyle \square }\colon E\to {\mathsf{id}}_{A{\scriptscriptstyle \square }}^{\ast} E\right)\\ &\quad=\left (F{\mathsf{id}}_A,F^{\left (A\right)}E\xrightarrow {F^{\left (A\right)}{\mathfrak{I}}^0_{\scriptscriptstyle \square }}F^{\left (A\right)}{\mathsf{id}}_{A{\scriptscriptstyle \square }}^{\ast} E\xrightarrow {\kappa ^{\left ({\mathsf{id}}_A\right)}}(F{\mathsf{id}}_A)^{\ast} _{\scriptscriptstyle \blacksquare } F^{\left (A\right)}E\right) \end{align*}

\begin{align*} &\quad=\left ({\mathsf{id}}_{FA},{\mathfrak{I}}^0_{{\scriptscriptstyle \blacksquare } F^{\left (A\right)}}\colon F^{\left (A\right)}E\to ({\mathsf{id}}_{FA})^{\ast} _{\scriptscriptstyle \blacksquare } F^{\left (A\right)}E\right)\\ &\quad={\mathsf{id}}_{F'(A,E)} \end{align*}

Consider now two morphisms $(f,\xi ^{(f)})\colon (A,E)\to (A',E')$ and $(g,\xi ^{\left (g\right)})\colon (A',E')\to (A'',E'')$ of $\mathsf{EL}({\mathbb{X}}_{\scriptscriptstyle \square },{\mathfrak{I}}_{\scriptscriptstyle \square })$ . First, recall that $F'(f,\xi ^{(f)})=(Ff,\xi ^{\left (Ff\right)})$ and $F'(g,\xi ^{\left (g\right)})=(Fg,\xi ^{\left (Fg\right)})$ , where:

\begin{align*} &\xi ^{\left (Ff\right)}\colon F^{\left (A\right)}E\xrightarrow {F^{\left (A\right)}\xi ^{(f)}}F^{\left (A\right)}f^{\ast} _{\scriptscriptstyle \square } E'\xrightarrow {\kappa ^{(f)}}(Ff)^{\ast} _{\scriptscriptstyle \blacksquare } F^{\left (A'\right)}E'\\ &\xi ^{\left (Fg\right)}\colon F^{\left (A'\right)}E'\xrightarrow {F^{\left (A'\right)}\xi ^{\left (g\right)}}F^{\left (A'\right)}g^{\ast} _{\scriptscriptstyle \square } E''\xrightarrow {\kappa ^{\left (g\right)}}(Fg)^{\ast} _{\scriptscriptstyle \blacksquare } F^{\left (A''\right)}E'' \end{align*}

Moreover:

\begin{align*} &\left (g,\xi ^{\left (g\right)}\right){\circ }\left (f,\xi ^{(f)}\right)=\left (g{\circ } f,E\xrightarrow {\xi ^{(f)}}f^{\ast} _{\scriptscriptstyle \square } E'\xrightarrow {f^{\ast} _{\scriptscriptstyle \square }\xi ^{\left (g\right)}}f^{\ast} _{\scriptscriptstyle \square } g^{\ast} _{\scriptscriptstyle \square } E''\xrightarrow {{\mathfrak{I}}^2_{\scriptscriptstyle \square }}(g{\circ } f)^{\ast} _{\scriptscriptstyle \square } E''\right) \end{align*}

Therefore, $F'\left (\left (g,\xi ^{\left (g\right)}\right){\circ }\left (f,\xi ^{(f)}\right)\right)=\left (F(g{\circ } f),\xi ^{\left (F(g{\circ } f)\right)}\right)$ , where:

\begin{align*} &\xi ^{\left (F(g{\circ } f)\right)}\colon\\ &\quad F^{\left (A\right)}E\xrightarrow {F^{\left (A\right)}\xi ^{(f)}}F^{\left (A\right)}f^{\ast} _{\scriptscriptstyle \square } E'\xrightarrow {F^{\left (A\right)}f^{\ast} _{\scriptscriptstyle \square }\xi ^{\left (g\right)}}F^{\left (A\right)}f^{\ast} _{\scriptscriptstyle \square } g^{\ast} _{\scriptscriptstyle \square } E''\xrightarrow {F^{\left (A\right)}{\mathfrak{I}}^2_{\scriptscriptstyle \square }}F^{\left (A\right)}(g{\circ } f)^{\ast} _{\scriptscriptstyle \square } E''\xrightarrow {\kappa ^{\left (g{\circ } f\right)}}(F(g{\circ } f))^{\ast} _{\scriptscriptstyle \blacksquare } F^{\left (A''\right)}E'' \end{align*}

However, the following diagram commutes:

Therefore:

\begin{eqnarray*} &=&\left (F(g{\circ } f),\xi ^{\left (F(g{\circ } f)\right)}\right)\\ &=&\left (F(g{\circ } f),\kappa ^{\left (g{\circ } f\right)}{\circ } F^{\left (A\right)}{\mathfrak{I}}^2_{\scriptscriptstyle \square }{\circ } F^{\left (A\right)}f^{\ast} _{\scriptscriptstyle \square }\xi ^{\left (g\right)}{\circ } F^{\left (A\right)}\xi ^{(f)}\right)\\ &=&\left (Fg{\circ } Ff,{\mathfrak{I}}^2_{{\scriptscriptstyle \blacksquare } F^{\left (A''\right)}}{\circ }(Ff)^{\ast} _{\scriptscriptstyle \blacksquare }\kappa ^{\left (g\right)}{\circ }(Ff)^{\ast} _{\scriptscriptstyle \blacksquare } F^{\left (A'\right)}\xi ^{\left (g\right)}{\circ }\kappa ^{(f)}{\circ } F^{\left (A\right)}\xi ^{(f)}\right)\\ &=&\left (Fg,\xi ^{\left (Fg\right)}\right){\circ }\left (Ff,\xi ^{\left (Ff\right)}\right)\\ &=&F'\left (g,\xi ^{\left (g\right)}\right){\circ } F'\left (f,\xi ^{(f)}\right) \end{eqnarray*}

Let us now consider a $2$ -morphism $(\theta ,\theta ')\colon (F,F',\kappa)\Rightarrow (G,G',\lambda)$ of indexed categories between two $1$ -morphisms $(F,F',\kappa),(G,G',\lambda)\colon ({\mathbb{X}}_{\scriptscriptstyle \square },{\mathfrak{I}}_{\scriptscriptstyle \square })\to ({\mathbb{X}}_{\scriptscriptstyle \blacksquare },{\mathfrak{I}}_{\scriptscriptstyle \blacksquare })$ of indexed categories. Let us define the corresponding $2$ -morphism of fibrations ${\mathscr{F}}(\theta ,\theta ')\colon {\mathscr{F}}(F,F',\kappa)\Rightarrow {\mathscr{F}}(G,G',\lambda)$ . First, the base natural transformation is given by $\theta \colon F\Rightarrow G$ . The total natural transformation $\theta '\colon F'\to G'$ is the morphism so defined:

\begin{align*} &(\theta '\colon F'(A,E)=(FA,F^{\left (A\right)}E)\to (GA,G^{\left (A\right)}E)=G'(A,E))=(\theta \colon FA\to GA,\theta ^{\left (A\right)}\colon F^{\left (A\right)} E\to \theta ^{\ast} G^{\left (A\right)}E) \end{align*}

Let us prove that $\theta '$ is natural. Consider a morphism $(f,\xi ^{(f)})\colon (A,E)\to (A',E')$ of $\mathsf{EL}({\mathbb{X}}_{\scriptscriptstyle \square },{\mathfrak{I}}_{\scriptscriptstyle \square })$ . Then

where we used the compatibility of $\theta '$ with the distributors. This shows that $\mathscr{I}\colon \mathsf{FIB}\to \mathsf{INDX}$ and ${\mathscr{F}}\colon \mathsf{INDX}\to \mathsf{FIB}$ are $2$ -functors. We leave to the reader to show that they form an equivalence, since this is just an extension to the classical equivalence $\mathsf{FIB}({\mathbb{X}})\simeq \mathsf{INDX}({\mathbb{X}})$ .

2.2 Tangent category theory

Tangent category theory aims to categorically axiomatise some of the fundamental constructions of differential geometry. In particular, a tangent category, first introduced by Rosický (Reference Rosický1984) and further generalised by Cockett and Cruttwell (Reference Cockett and Cruttwell2014), consists of a category equipped with an endofunctor which can be interpreted as the tangent bundle functor. An object in a tangent category can be interpreted as a geometric space equipped with a tangent bundle which encodes a notion of local linearity for this space. Informally, the fibres of the tangent bundle can be interpreted as tangent spaces of the given space, that is the best local linear approximation of the space at a given point.

The notion of linearity in a tangent category is not an intrinsic notion, but rather a contextual notion: the tangent structure establishes what it means for a morphism to be linear. Cockett and Cruttwell (Reference Cockett and Cruttwell2018) introduced the concept of differential bundles in a tangent category, which are the analogs of vector bundles in the category of smooth manifolds, as shown by MacAdam (Reference MacAdam2021). Differential bundles represent precisely those fibre bundle-like objects whose fibres carry a linear structure.

In this section, we would like to recall the main definitions of a tangent category, of a differential bundle in a tangent category and discuss some examples.

We start with the definition of a tangent category. We suggest the reader to consult Cockett and Cruttwell (Reference Cockett and Cruttwell2014). The original definition of Rosickỳ presented in Rosický (Reference Rosický1984) is now known with the denomination of a tangent category with negatives, sometimes also called a Rosickỳ tangent category (see Cruttwell and Lemay (Reference Cruttwell and Lemay2023b) or Ikonicoff et al. (Reference Ikonicoff, Lanfranchi and Lemay2024)). First, let us recall the definition of an additive bundle.

The tangent bundle functor sends an object $A$ to its tangent bundle ${\mathrm{T}} A$ , which is interpreted as the space of tangent vectors of $A$ at each point. Moreover, it sends a morphism $f\colon A\to A'$ , regarded as a smooth map, to the morphism ${\mathrm{T}} f$ which sends each tangent vector $u$ at a given point $x$ to the differential of $f$ at $x$ applied to $u$ , that is, ${\mathrm{d}} f_x(u)$ . The functoriality of $\mathrm{T}$ corresponds to the chain rule of the differential.

The projection, the zero morphism and the sum morphism equip each tangent bundle ${\mathrm{T}} A\to A$ with an additive structure. Informally, this structure axiomatises the additive structure of the tangent spaces of a tangent bundle of a smooth manifold. By interpreting ${\mathrm{T}} A$ as the space of vector tangent vectors over each point of $A$ , the projection sends each tangent vector to its base point, the zero morphism associates each point with the zero tangent vector and the sum morphism allows one to sum together two tangent vectors with the same base point.

When the tangent category has negatives, the additive structure over the fibres becomes a commutative group. This can be interpreted as inverting the orientation of the tangent vectors.

The double tangent bundle ${\mathrm{T}}^2A$ can be regarded as the space of infinitesimal differentiable homotopies of $A$ , that is tangent vectors of tangent vectors. Locally, one can represent such an element as a quadruple $(x,u,v,\omega)$ formed by a point $x$ of $A$ , two tangent vectors $u,v$ of $A$ at $x$ and a tangent vector $\omega$ of ${\mathrm{T}} A$ at $(x,u)$ . Then, the vertical lift sends a tangent vector $u$ of $A$ at a given point $x$ to $(x,0,0,u)$ .

The universality of the vertical lift establishes that the vertical bundle, which is the bundle whose fibres are the kernel of the differential of the projection at each point, is trivial. This can be interpreted as a local linearity condition for the tangent bundle, that is, an object in a tangent category has a local linear behaviour.

Finally, the canonical flip encodes the symmetry of the Hessian matrix, that is the commutativity of partial derivatives for smooth functions.

Example 2.22. The opposite of the category of commutative and unital rigs comes equipped with a tangent structure, which is adjoint to the tangent structure of Example 2.21 , that is the tangent bundle functors form an adjunction. Concretely, the tangent bundle functor of $\mathsf{CRIG}^{\mathsf{op}}$ sends a rig $R$ to the symmetric rig of the module of Kähler differentials ${\mathrm{T}} R\colon \kern -1ex=\mathrm{Sym}_R\Omega ^1_{R/\mathbb{N}}$ of $R$ , that is to the rig generated by all elements $a$ of $R$ together with symbols ${\mathrm{d}} a$ , for each $a$ of $R$ , satisfying the following relations:

\begin{align*} &a\cdot _{{\mathrm{T}} R}b=a\cdot _Rb\\ &{\mathrm{d}}(1)=0\\ &{\mathrm{d}}(ra+sb)=r{\mathrm{d}} a+s{\mathrm{d}} b\\ &{\mathrm{d}}(ab)=b{\mathrm{d}} a+a{\mathrm{d}} b \end{align*}

Regarding the natural transformations are morphisms of $\mathsf{CRIG}$ , the projection $p\colon R\to {\mathrm{T}} R$ sends each $a\in R$ to itself; the zero morphism ${\mathrm{T}} R\to R$ sends each $a$ to itself and each ${\mathrm{d}} a$ to $0$ ; the sum morphism $s\colon {\mathrm{T}} R\to {\mathrm{T}}_2R$ sends each $a$ to itself and each ${\mathrm{d}} a$ to ${\mathrm{d}} a\otimes 1+1\otimes {\mathrm{d}} a$ , where ${\mathrm{T}}_2R={\mathrm{T}} R\otimes _R{\mathrm{T}} R$ ; the vertical lift $l\colon {\mathrm{T}}^2R\to {\mathrm{T}} R$ sends each $a$ to itself, each ${\mathrm{d}} a$ and each ${\mathrm{d}}' a$ to $0$ , and ${\mathrm{d}}'{\mathrm{d}} a$ to ${\mathrm{d}} a$ , where ${\mathrm{d}}'x$ is the Kähler differential of $x\in {\mathrm{T}} R$ ; finally, the canonical flip $c\colon {\mathrm{T}}^2R\to {\mathrm{T}}^2R$ sends each $a$ to itself, ${\mathrm{d}} a$ to ${\mathrm{d}}'a$ , ${\mathrm{d}}'a$ to ${\mathrm{d}} a$ and ${\mathrm{d}}'{\mathrm{d}} a$ to itself.

This tangent structure restricts to the subcategory $\mathsf{CRING}^{\mathsf{op}}$ . Moreover, $\mathsf{CRING}^{\mathsf{op}}$ has negatives whose negation $n\colon {\mathrm{T}} R\to {\mathrm{T}} R$ sends each $a$ to itself and each ${\mathrm{d}} a$ to $-{\mathrm{d}} a$ .

A morphism of tangent categories consists of a functor between the underlying categories, together with a distributive law between the functor and the respective tangent bundle functors. The distributive law can be lax, colax, an isomorphism, or the identity. Consequently, there are four different flavours of morphisms of tangent categories.

3.1 Tangent fibrations and their fibres

Informally, a tangent fibration is a fibration between two tangent categories, compatible with the tangent structures. To introduce the required compatibility condition, consider first a cloven fibration $\Pi \colon {\mathbb{X}}'\to {\mathbb{X}}$ between two categories, each equipped with an endofunctor ${\mathrm{T}}'\colon {\mathbb{X}}'\to {\mathbb{X}}'$ and ${\mathrm{T}}\colon {\mathbb{X}}\to {\mathbb{X}}$ , respectively; suppose also that $\Pi {\circ }{\mathrm{T}}'={\mathrm{T}}{\circ }\Pi$ .

On one hand, consider now a morphism $f\colon A\to A'$ of $\mathbb{X}$ and an object $E'$ over $A'$ . Since $\Pi {\mathrm{T}}'E'={\mathrm{T}}\Pi E'={\mathrm{T}} A$ , there is a cartesian lift $\varphi ^{\left ({\mathrm{T}} f\right)}\colon ({\mathrm{T}} f)^{\ast} ({\mathrm{T}}' E')\to {\mathrm{T}}'E'$ of ${\mathrm{T}} f$ on ${\mathrm{T}}'E'$ . On the other hand, one can also apply the endofunctor ${\mathrm{T}}'$ to the cartesian lift $\varphi ^{(f)}\colon f^{\ast} E'\to E'$ of $f$ on $E'$ and obtain ${\mathrm{T}}'\varphi ^{(f)}\colon {\mathrm{T}}'(f^{\ast} E')\to {\mathrm{T}}'E'$ .

Both $\varphi ^{\left ({\mathrm{T}} f\right)}$ and ${\mathrm{T}}'\varphi ^{(f)}$ are lifts of ${\mathrm{T}} f$ , since $\Pi (\varphi ^{\left ({\mathrm{T}} f\right)})=\Pi ({\mathrm{T}}'\varphi ^{(f)})=f$ . By the universality of the cartesian lift $\varphi ^{\left ({\mathrm{T}} f\right)}$ , there is a unique morphism $\kappa ^{(f)}\colon {\mathrm{T}}'(f^{\ast} E')\dashrightarrow ({\mathrm{T}} f)^{\ast} ({\mathrm{T}}'E')$ making the following diagram commutes:

Requiring the morphism $\kappa ^{(f)}$ to be an isomorphism is to ask the tangent bundle functors $({\mathrm{T}},{\mathrm{T}}')$ to form a $1$ -morphism of fibrations, as stated in Definition 2.7. We follow the convention of referring to the morphisms $\kappa ^{(f)}$ as the distributors of the tangent bundle functors.

The original definition of a tangent fibration was first introduced by Cockett and Cruttwell in ((Reference Cockett and Cruttwell2018), Definition 5.2). Here we specialise this definition for cloven fibrations.

In ((Reference Cockett and Cruttwell2018), Section 5), Cockett and Cruttwell constructed two important examples of tangent fibrations: the tangent fibrations of display bundles and the one of differential bundles. In the next examples, we recall these two constructions. First, we recall a technical concept, introduced by Cockett and Cruttwell in the same paper to ensure the existence of the cartesian lifts. MacAdam reformulated this notion in (MacAdam (Reference MacAdam2021), Definition 1.2.1). Here, we adopt MacAdam’s version.

In the tangent category of smooth manifolds (see Example 2.20), submersions constitute a canonical choice of a tangent display system; in the tangent category of affine schemes (see Example 2.22), one can consider the family of all morphisms as a tangent display system, since the category is finitely complete and the tangent bundle functor is the right adjoint to the (opposite of the) tangent bundle functor of the tangent category of Example 2.21.

Cockett and Cruttwell proved in ((Reference Cockett and Cruttwell2018), Theorem 5.3) that the fibres of a cloven tangent fibration inherit a tangent structure from the total tangent category, strongly preserved by the substitution functors.

Let us briefly recall this construction; consider a tangent fibration $\Pi \colon ({\mathbb{X}}',\mathbb{T}')\to ({\mathbb{X}},\mathbb{T})$ and an object $A$ of $\mathbb{X}$ . The first step is to show that each fibre comes equipped with a tangent structure $\mathbb{T}^{\left (A\right)}$ so defined:

• tangent bundle functor. The tangent bundle functor ${\mathrm{T}}^{\left (A\right)}\colon \Pi ^{-1}(A)\to \Pi ^{-1}(A)$ is the functor obtained by composing ${\mathrm{T}}'\colon \Pi ^{-1}(A)\to \Pi ^{-1}({\mathrm{T}} A)$ with the substitution functor $z^{\ast} \colon \Pi ^{\left (-1\right)}({\mathrm{T}} A)\to \Pi ^{-1}(A)$ , induced by the zero morphism $z\colon A\to {\mathrm{T}} A$ ;

• projection. The projection $p^{\left (A\right)}\colon {\mathrm{T}}^{\left (A\right)}\Rightarrow {\mathsf{id}}_{\Pi ^{-1}A}$ is defined by

  \begin{align*} &p^{\left (A\right)}\colon z^{\ast} {\mathrm{T}}'E\xrightarrow {\varphi ^{\left (z\right)}}{\mathrm{T}}'E\xrightarrow {p'}E \end{align*}
  where $\varphi ^{\left (z\right)}$ is the cartesian lift of the zero morphism of $({\mathbb{X}},\mathbb{T})$ and $p'$ denotes the projection of $({\mathbb{X}}',\mathbb{T}')$ ;

• zero morphism. The zero morphism $z^{\left (A\right)}\colon {\mathsf{id}}_{\Pi ^{-1}A}\Rightarrow {\mathrm{T}}^{\left (A\right)}$ is the unique morphism $z^{\left (A\right)}\colon E\to z^{\ast} {\mathrm{T}}'E$ induced by the universality of the cartesian lift $\varphi ^{\left (z\right)}$ , which makes the following diagram commutes:

  

• n-fold pullback. The $n$ -fold pullback ${\mathrm{T}}^{\left (A\right)}_n$ of the projection $p^{\left (A\right)}$ along itself is the functor obtained by composing ${\mathrm{T}}'_n\colon \Pi ^{-1}(A)\to \Pi ^{-1}({\mathrm{T}}_nA)$ and the substitution functor $z_n^{\ast} \colon \Pi ^{-1}({\mathrm{T}}_nA)\to \Pi ^{-1}(A)$ , where $z_n\colon A\to {\mathrm{T}}_nA$ denotes $\langle z,\ldots ,z\rangle$ . The $k$ -th projection $\pi _k^{\left (A\right)}\colon {\mathrm{T}}^{\left (A\right)}_n\Rightarrow {\mathrm{T}}^{\left (A\right)}$ is the unique morphism induced by the universality of the cartesian lift $\varphi ^{\left (z\right)}$ , which makes the following diagram commutes:

  

• sum morphism. The sum morphism $s^{\left (A\right)}\colon {\mathrm{T}}^{\left (A\right)}_2\Rightarrow {\mathrm{T}}^{\left (A\right)}$ is the unique morphism $s^{\left (A\right)}\colon z^{\ast} _2{\mathrm{T}}'_2E\to z^{\ast} {\mathrm{T}}'E$ induced by the universality of the cartesian lift $\varphi ^{\left (z\right)}$ , which makes the following diagram commutes:

  

• vertical lift. The vertical lift $l^{\left (A\right)}\colon {\mathrm{T}}^{\left (A\right)}\Rightarrow {{\mathrm{T}}^{\left (A\right)}}^2$ is the unique morphism induced by the universality of the cartesian lifts $\varphi ^{\left (z_{\mathrm{T}}\right)}$ and ${\mathrm{T}}'\varphi ^{\left (z\right)}$ , which makes the following diagram commutes:

  

• canonical flip. The canonical flip $c^{\left (A\right)}\colon {{\mathrm{T}}^{\left (A\right)}}^2\Rightarrow {{\mathrm{T}}^{\left (A\right)}}^2$ is the unique morphism induced by the universality of the cartesian lifts $\varphi ^{\left (z_{\mathrm{T}}\right)}$ and ${\mathrm{T}}'\varphi ^{\left (z\right)}$ , which makes the following diagram commutes:

  

When $({\mathbb{X}}',\mathbb{T}')$ has negatives with negation $n'\colon {\mathrm{T}}'\Rightarrow {\mathrm{T}}'$ , we can also define:

• negation. The negation $n^{\left (A\right)}\colon {\mathrm{T}}^{\left (A\right)}\Rightarrow {\mathrm{T}}^{\left (A\right)}$ is the unique morphism induced by the universality of the cartesian lift $\varphi ^{\left (z\right)}$ , which makes the following diagram commutes:

  

The next step is to equip each substitution functor $f^{\ast} \colon \Pi ^{-1}(A')\to \Pi ^{-1}(A)$ induced by a morphism $f\colon A\to A'$ of $\mathbb{X}$ with a suitable distributive law which makes $f^{\ast}$ a strong tangent morphism. Consider the morphism

(3.1) \begin{align} \begin{split} \alpha ^{(f)}\colon &{\mathrm{T}}^{\left (A\right)}f^{\ast} K=z^{\ast} {\mathrm{T}}^{\prime}(f^{\ast} E^{\prime})\xrightarrow {z^{\ast} \kappa ^{(f)}}z^{\ast} ({\mathrm{T}} f)^{\ast} {\mathrm{T}}^{\prime}K\xrightarrow {{\mathfrak{I}}^2}({\mathrm{T}} f{\circ } z)^{\ast} {\mathrm{T}}^{\prime}K \,-\,-\,-\,-\\ &\xrightarrow {{\mathrm{T}} f{\circ } z=z{\circ } f}(z{\circ } f)^{\ast} {\mathrm{T}}^{\prime}K\xrightarrow {{{\mathfrak{I}}^2}^{-1}}f^{\ast} z^{\ast} {\mathrm{T}}^{\prime}K=f^{\ast} {\mathrm{T}}^{\left (A^{\prime}\right)}K \end{split} \end{align}

where we used the compositor ${\mathfrak{I}}^2$ , the distributor $\kappa ^{(f)}$ of the tangent bundle functors and the naturality of $z$ .

Let us rewrite (Cockett and Cruttwell (Reference Cockett and Cruttwell2018), Theorem 5.3) in our jargon.

The next step is to extend the assignment which sends a tangent fibration to the corresponding indexed tangent category to a functor. We first need to introduce morphisms of tangent fibrations and morphisms of indexed tangent categories.

Definition 3.16. Consider two tangent fibrations $\Pi _{\scriptscriptstyle \square }\colon ({\mathbb{X}}_{\scriptscriptstyle \square }',\mathbb{T}_{\scriptscriptstyle \square }')\to ({\mathbb{X}}_{\scriptscriptstyle \square },\mathbb{T}_{\scriptscriptstyle \square })$ and $\Pi _{\scriptscriptstyle \blacksquare }\colon ({\mathbb{X}}'_{\scriptscriptstyle \blacksquare },\mathbb{T}_{\scriptscriptstyle \blacksquare }')\to ({\mathbb{X}}_{\scriptscriptstyle \blacksquare },\mathbb{T}_{\scriptscriptstyle \blacksquare })$ . A lax (colax) morphism of tangent fibrations $(F,\beta ,F',\beta ')\colon \Pi _{\scriptscriptstyle \square }\to \Pi _{\scriptscriptstyle \blacksquare }$ consists of a pair of lax (colax) tangent morphisms

\begin{align*} &(F,\beta)\colon ({\mathbb{X}}_{\scriptscriptstyle \square },\mathbb{T}_{\scriptscriptstyle \square })\to ({\mathbb{X}}_{\scriptscriptstyle \blacksquare },\mathbb{T}_{\scriptscriptstyle \blacksquare })\\ &(F',\beta ')\colon ({\mathbb{X}}_{\scriptscriptstyle \square }',\mathbb{T}_{\scriptscriptstyle \square }')\to ({\mathbb{X}}_{\scriptscriptstyle \blacksquare }',\mathbb{T}_{\scriptscriptstyle \blacksquare }') \end{align*}

for which the following diagram commutes:

in $\mathsf{TNGCAT}$ . Furthermore, $(F,F')\colon \Pi _{\scriptscriptstyle \square }\to \Pi _{\scriptscriptstyle \blacksquare }$ is required to be a morphism of fibrations as in Definition 2.7.

Moreover, a morphism of tangent fibration $(F,\beta ,F',\beta ')$ is:

• • strong if $(F,\beta)$ and $(F',\beta ')$ are strong tangent morphisms;

• • strict if $(F,\beta)$ and $(F',\beta ')$ are strict;

• • strict on the base whenever the base tangent morphism $(F,\beta)$ is a strict tangent morphism.

Consider two tangent fibrations $\Pi _{\scriptscriptstyle \square }\colon ({\mathbb{X}}_{\scriptscriptstyle \square }',\mathbb{T}_{\scriptscriptstyle \square }')\to ({\mathbb{X}}_{\scriptscriptstyle \square },\mathbb{T}_{\scriptscriptstyle \square })$ and $\Pi _{\scriptscriptstyle \blacksquare }\colon ({\mathbb{X}}_{\scriptscriptstyle \blacksquare }',\mathbb{T}_{\scriptscriptstyle \blacksquare }')\to ({\mathbb{X}}_{\scriptscriptstyle \blacksquare },\mathbb{T}_{\scriptscriptstyle \blacksquare })$ and a lax morphism of tangent fibrations $(F,F',\beta ')\colon \Pi _{\scriptscriptstyle \square }\to \Pi _{\scriptscriptstyle \blacksquare }$ and suppose that $(F,F',\beta ')$ is strict on the base. Thanks to Proposition 3.12, $\Pi _{\scriptscriptstyle \square }$ and $\Pi _{\scriptscriptstyle \blacksquare }$ define two indexed tangent categories ${\mathfrak{I}}_{\scriptscriptstyle \square }\colon \kern -1ex=\mathscr{I}(\Pi _{\scriptscriptstyle \square })\colon ({\mathbb{X}}_{\scriptscriptstyle \square },\mathbb{T}_{\scriptscriptstyle \square })\to \mathsf{TNGCAT}$ and ${\mathfrak{I}}_{\scriptscriptstyle \blacksquare }\colon \kern -1ex=\mathscr{I}(\Pi _{\scriptscriptstyle \blacksquare })\colon ({\mathbb{X}}_{\scriptscriptstyle \blacksquare },\mathbb{T}_{\scriptscriptstyle \blacksquare })\to \mathsf{TNGCAT}$ , respectively. The goal is to show that the morphism $(F,F',\beta ')$ defines a lax morphism ${\mathfrak{I}}(F,F',\beta ')\colon {\mathfrak{I}}_{\scriptscriptstyle \square }\to {\mathfrak{I}}_{\scriptscriptstyle \blacksquare }$ of indexed tangent categories which is also strict on the base. Let us start with the base tangent morphism:

• base tangent morphism. The base tangent morphism is $F\colon ({\mathbb{X}}_{\scriptscriptstyle \square },\mathbb{T}_{\scriptscriptstyle \square })\to ({\mathbb{X}}_{\scriptscriptstyle \blacksquare },\mathbb{T}_{\scriptscriptstyle \blacksquare })$ ;

To define the indexed tangent morphism, let us start by noticing that $F'\colon {\mathbb{X}}_{\scriptscriptstyle \square }'\to {\mathbb{X}}_{\scriptscriptstyle \blacksquare }'$ maps an object $E$ of the fibre over $A$ of $\Pi _{\scriptscriptstyle \square }$ to $F'E$ . However, we have $\Pi _{\scriptscriptstyle \blacksquare } F'E=F\Pi _{\scriptscriptstyle \square } E=FA$ , thus $F'E$ lives in the fibre over $FA$ of $\Pi _{\scriptscriptstyle \blacksquare }$ . So, we can restrict $F'$ to the fibres over $A$ and obtain a functor $F^{\left (A\right)}\colon {\mathbb{X}}_{\scriptscriptstyle \square }^{\left (A\right)}\to {\mathbb{X}}_{\scriptscriptstyle \blacksquare }^{\left (FA\right)}$ , where ${\mathbb{X}}_{\scriptscriptstyle \square }^{\left (A\right)}=\Pi _{\scriptscriptstyle \square }^{-1}(A)$ and ${\mathbb{X}}_{\scriptscriptstyle \blacksquare }^{\left (FA\right)}=\Pi _{\scriptscriptstyle \blacksquare }^{-1}(FA)$ . Let us extend each $F^{\left (A\right)}$ to a lax tangent morphism. Consider the following morphism:

\begin{align*} \beta ^{\left (A\right)}\colon F^{\left (A\right)}{\mathrm{T}}_{\scriptscriptstyle \square }^{\left (A\right)}E & =F'z_{\scriptscriptstyle \square }^{\ast} {\mathrm{T}}_{\scriptscriptstyle \square }'E\xrightarrow {\kappa ^{\left (z\right)}}(Fz_{\scriptscriptstyle \blacksquare })^{\ast} F'{\mathrm{T}}_{\scriptscriptstyle \square }'E\xrightarrow {Fz_{\scriptscriptstyle \blacksquare }={z_{\scriptscriptstyle \blacksquare }}_F}({z_{\scriptscriptstyle \blacksquare }}_F)^{\ast} F'{\mathrm{T}}_{\scriptscriptstyle \square }'E\xrightarrow {({z_{\scriptscriptstyle \blacksquare }}_F)^{\ast} \beta '}({z_{\scriptscriptstyle \blacksquare }}_F)^{\ast} {\mathrm{T}}_{\scriptscriptstyle \blacksquare }'F'E\\ & ={\mathrm{T}}_{\scriptscriptstyle \blacksquare }^{\left (FA\right)}F^{\left (A\right)}E \end{align*}

• indexed tangent morphism. The indexed tangent morphism is the collection of lax tangent morphisms $(F^{\left (A\right)},\beta ^{\left (A\right)})\colon ({\mathbb{X}}_{\scriptscriptstyle \square }^{\left (A\right)},\mathbb{T}_{\scriptscriptstyle \square }^{\left (A\right)})\to ({\mathbb{X}}_{\scriptscriptstyle \blacksquare }^{\left (FA\right)},\mathbb{T}_{\scriptscriptstyle \blacksquare }^{\left (FA\right)})$ ;

• distributors. The distributors are precisely the distributors of $(F,F',\beta ')$ , as in Definition 2.7.

Notice that the compatibility between the distributors and the distributive laws is a consequence of $\Pi _{\scriptscriptstyle \blacksquare }\beta '={\mathsf{id}}_{{\mathbb{X}}_{\scriptscriptstyle \blacksquare }}$ and of the universality of the cartesian lifts. Concretely, this compatibility consists of the commutativity of the following diagram

where $\lambda$ denotes the distributors of the $1$ -morphism $(F,F')$ and $\kappa$ denotes the distributors of the tangent bundle functors.

To prove the commutativity of the bottom rectangular diagram, let us post-compose the two paths of morphisms by $\varphi ^{\left (Ff\right)}\varphi _{\scriptscriptstyle \blacksquare }^{\left (z_{\scriptscriptstyle \blacksquare }\right)}$ , as follows:

Since $\Pi _{\scriptscriptstyle \blacksquare }\beta '$ is the identity, from the cartesian property of $\varphi ^{\left (Ff\right)}\varphi _{\scriptscriptstyle \blacksquare }^{\left (z_{\scriptscriptstyle \blacksquare }\right)}$ we conclude that these two paths of morphisms are identical.

Proposition 3.21. The assignment which sends a cloven tangent fibration $\Pi$ to the corresponding indexed tangent category $\mathscr{I}(\Pi)$ extends to a pair of functors. In particular, each lax (colax) morphism of tangent fibrations strict on the base is associated with a lax (colax) morphism of indexed tangent categories, also strict on the base:

\begin{align*} &\mathscr{I}\colon \mathsf{TNGFIB}_{\mathsf{strb}}\to \mathsf{INDXTNG}_{\mathsf{strb}}\\ &\mathscr{I}\colon \mathsf{TNGFIB}_{{\mathsf{co}},\mathsf{strb}}\to \mathsf{INDXTNG}_{{\mathsf{co}},\mathsf{strb}} \end{align*}

These functors restrict to the subcategory of strong and strict morphisms of tangent fibrations which are strict on the base:

\begin{align*} &\mathscr{I}\colon \mathsf{TNGFIB}_{\cong ,\mathsf{strb}}\to \mathsf{INDXTNG}_{\cong ,\mathsf{strb}}\\ &\mathscr{I}\colon \mathsf{TNGFIB}_{=,\mathsf{strb}}\to \mathsf{INDXTNG}_{=,\mathsf{strb}} \end{align*}

3.2 The reduced Grothendieck construction for tangent fibrations

In the previous section, we extended Cockett and Cruttwell’s result to a functorial assignment which sends a tangent fibration to a corresponding indexed tangent category. Crucially, for a tangent fibration $\Pi \colon ({\mathbb{X}}',\mathbb{T}')\to ({\mathbb{X}},\mathbb{T})$ the tangent bundle functor ${\mathrm{T}}'$ of the total tangent category sends an object $E$ of the fibre over $A\in {\mathbb{X}}$ , to ${\mathrm{T}}'E$ , which lives in the fibre over ${\mathrm{T}} A$ . So, in order to define the tangent bundle functor ${\mathrm{T}}^{\left (A\right)}\colon \Pi ^{-1}A\to \Pi ^{-1}A$ we post-composed the total tangent bundle functor ${\mathrm{T}}'$ with the substitution functor $z^{\ast}$ induced by the zero morphism of $({\mathbb{X}},\mathbb{T})$ .

This process inevitably destroys part of the information of the total tangent bundle functor. To convince the reader of this phenomenon, consider a tangent category $({\mathbb{X}},\mathbb{T})$ equipped with a display system as in Example 3.5. This produces a tangent fibration $\Pi \colon \mathscr{D}({\mathbb{X}},\mathbb{T})\to ({\mathbb{X}},\mathbb{T})$ whose total tangent category is the tangent category of tangent display maps $E\to A$ of $\mathbb{X}$ .

As shown in Example 3.14, the corresponding indexed tangent category $\mathscr{I}(\Pi)$ sends each object $A$ of $\mathbb{X}$ to the slice tangent category $({\mathbb{X}},\mathbb{T})/A$ , whose tangent bundle functor $\mathbb{T}^{\left (A\right)}$ sends a tangent display map $q\colon E\to A$ to the vertical bundle ${\mathrm{T}}^{\left (A\right)}q\colon VE\to A$ , defined by the pullback diagram:

In general, $VE$ is only strictly included in ${\mathrm{T}} E$ . For instance, in the tangent category of Example 2.22, consider a ring $R$ and let us consider its tangent bundle $p\colon {\mathrm{T}} R\to R$ , where ${\mathrm{T}} R$ is the ring $R[{\varepsilon }]$ , with ${\varepsilon }^2=0$ , and $p$ sends $\varepsilon$ to $0$ . From the universality of the vertical lift, one finds out that the vertical bundle of $p$ consists of the pullback ${\mathrm{T}}_2R=R[{\varepsilon }_1,{\varepsilon }_2]$ of $p$ along itself, which is the ring of polynomials with coefficients in $R$ in two variables ${\varepsilon }_1$ and ${\varepsilon }_2$ , such that ${\varepsilon }_i{\varepsilon }_j=0$ , for $i,j=1,2$ .

Contrarily, ${\mathrm{T}}({\mathrm{T}} R)={\mathrm{T}}^2R$ is the ring $R[{\varepsilon }][{\varepsilon }']$ , generated by two variables $\varepsilon$ and ${\varepsilon }'$ , such that ${\varepsilon }^2={\varepsilon }'^2=0$ , but ${\varepsilon }'{\varepsilon }\neq 0$ . The map $R[{\varepsilon }_1,{\varepsilon }_2]\to R[{\varepsilon }][{\varepsilon }']$ which sends ${\varepsilon }_1$ to $\varepsilon$ and ${\varepsilon }_2$ to ${\varepsilon }'$ is precisely the strict inclusion of the vertical bundle into the tangent bundle.

In this section, we partially reconstruct the original tangent fibration starting from the associated indexed tangent category, by constructing a functor of type ${\mathscr{F}}\colon \mathsf{INDXTNG}\to \mathsf{TNGFIB}$ . We show that this functor together with the functor of Proposition 3.21 forms an adjunction, which, however, is not an equivalence of categories.

The main idea is to equip the category of elements $\mathsf{EL}({\mathbb{X}},{\mathfrak{I}})$ (see Section 2.1) of the underlying indexed category ${\mathfrak{I}}\colon {\mathbb{X}}^{\mathsf{op}}\to \mathsf{TNGCAT}_{\cong }\to {\mathsf{CAT}}$ with a suitable tangent structure, obtained by “gluing” together the tangent structures of the fibres.

• tangent bundle functor. The tangent bundle functor $\hat {\mathrm{T}}\colon \mathsf{EL}({\mathbb{X}},{\mathfrak{I}})\to \mathsf{EL}({\mathbb{X}},{\mathfrak{I}})$ sends an object $(A,E)$ to $({\mathrm{T}} A,p^{\ast} {\mathrm{T}}^{\left (A\right)}E)$ and a morphism $(f,\xi ^{(f)})\colon (A,E)\to (A',E')$ to the morphism

  \begin{align*} ({\mathrm{T}} f,\xi ^{\left ({\mathrm{T}} f\right)})\colon ({\mathrm{T}} A,p^{\ast} {\mathrm{T}}^{\left (A\right)}E)\to ({\mathrm{T}} A',p^{\ast} {\mathrm{T}}^{\left (A'\right)}E') \end{align*}
  defined by ${\mathrm{T}} f\colon {\mathrm{T}} A\to {\mathrm{T}} A'$ together with the morphism
  \begin{align*} \xi ^{\left ({\mathrm{T}} f\right)}&\colon p^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {p^{\ast} {\mathrm{T}}^{\left (A\right)}\xi ^{(f)}}p^{\ast} {\mathrm{T}}^{\left (A\right)}f^{\ast} E'\xrightarrow {p^{\ast} \alpha ^{(f)}}p^{\ast} f^{\ast} {\mathrm{T}}^{\left (A'\right)}E'\\ & \xrightarrow {{\mathfrak{I}}^2}(f{\circ } p)^{\ast} {\mathrm{T}}^{\left (A'\right)}E'\,-\,-\,-\,- \,-\,-\,-\,-\\ &\xrightarrow {f{\circ } p=p{\circ } {\mathrm{T}} f}(p{\circ } {\mathrm{T}} f)^{\ast} {\mathrm{T}}^{\left (A'\right)}E'\xrightarrow {{{\mathfrak{I}}^2}^{-1}}({\mathrm{T}} f)^{\ast} p^{\ast} {\mathrm{T}}^{\left (A'\right)}E' \end{align*}
  where ${\mathfrak{I}}^2$ denotes the compositor;

• projection. The projection $\hat p\colon \hat {\mathrm{T}}\Rightarrow {\mathsf{id}}_{\mathsf{EL}({\mathbb{X}},{\mathfrak{I}})}$ is the pair

  \begin{align*} &(p,\xi ^{\left (p\right)})\colon ({\mathrm{T}} A,p^{\ast} {\mathrm{T}}^{\left (A\right)}E)\to (A,E) \end{align*}
  formed by $p\colon {\mathrm{T}} A\to A$ and by the morphism:
  \begin{align*} &\xi ^{\left (p\right)}\colon p^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {p^{\ast} p^{\left (A\right)}}p^{\ast} E \end{align*}

• zero morphism. The zero morphism $\hat z\colon {\mathsf{id}}_{\mathsf{EL}({\mathbb{X}},{\mathfrak{I}})}\Rightarrow \hat {\mathrm{T}}$ is the pair

  \begin{align*} &(z,\xi ^{\left (z\right)})\colon (A,E)\to ({\mathrm{T}} A,p^{\ast} {\mathrm{T}}^{\left (A\right)}E) \end{align*}
  formed by $z\colon A\to {\mathrm{T}} A$ and by the morphism:
  \begin{align*} &\xi ^{\left (z\right)}\colon E\xrightarrow {z^{\left (A\right)}}{\mathrm{T}}^{\left (A\right)}E\xrightarrow {{\mathfrak{I}}^0}{\mathsf{id}}_A^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {p{\circ } z={\mathsf{id}}_A}(p{\circ } z)^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {{\mathfrak{I}}^2}z^{\ast} p^{\ast} {\mathrm{T}}^{\left (A\right)}E \end{align*}

• n-fold pullback. $n$ -fold pullback. The $n$ -fold pullback along the projection $p^{\left (A\right)}$ is given by:

  \begin{align*} &\hat {\mathrm{T}}_n(A,E)=({\mathrm{T}}_nA,\pi _1^{\ast} p^{\ast} {\mathrm{T}}^{\left (A\right)}_nE) \end{align*}
  where the $k$ -th projection $\hat {\pi }_k\colon \hat {\mathrm{T}}_n\Rightarrow \hat {\mathrm{T}}$ is given by the pair
  \begin{align*} &(\pi _k,\xi ^{\left (\pi _k\right)})\colon ({\mathrm{T}}_nA,\pi _1^{\ast} p^{\ast} {\mathrm{T}}^{\left (A\right)}_nE)\to ({\mathrm{T}} A,p^{\ast} {\mathrm{T}}^{\left (A\right)}E) \end{align*}
  where $\pi _k\colon {\mathrm{T}}_nA\to {\mathrm{T}} A$ and $\xi ^{\left (\pi _k\right)}$ is the morphism:
  \begin{align*} \xi ^{\left (\pi _k\right)}&\colon \pi _1^{\ast} p^{\ast} {\mathrm{T}}_n^{\left (A\right)}E\xrightarrow {\pi _1^{\ast} p^{\ast} \pi _k^{\left (A\right)}}\pi _1^{\ast} p^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {{{\mathfrak{I}}^2}^{-1}}(p{\circ }\pi _1)^{\ast} {\mathrm{T}}^{\left (A\right)}E\,-\,-\,-\,- \,-\,-\,-\,-\\ &\xrightarrow {p{\circ }\pi _1=p{\circ }\pi _k}(p{\circ }\pi _k)^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {{\mathfrak{I}}^2}\pi _k^{\ast} p^{\ast} {\mathrm{T}}^{\left (A\right)}E \end{align*}

• sum morphism. The sum morphism $\hat s\colon \hat {\mathrm{T}}_2\Rightarrow \hat {\mathrm{T}}$ is the pair

  \begin{align*} &(s,\xi ^{\left (s\right)})\colon ({\mathrm{T}}_2A,\pi _1^{\ast} {\mathrm{T}}^{\left (A\right)}_2E)\to ({\mathrm{T}} A,p^{\ast} {\mathrm{T}}^{\left (A\right)}E) \end{align*}
  formed by $s\colon {\mathrm{T}}_2A\to {\mathrm{T}} A$ and by the morphism:
  \begin{align*} \xi ^{\left (s\right)}\colon &\pi _1^{\ast} p^{\ast} {\mathrm{T}}^{\left (A\right)}_2E\xrightarrow {\pi _1^{\ast} p^{\ast} s^{\left (A\right)}}\pi _1^{\ast} p^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {{{\mathfrak{I}}^2}^{-1}}(p{\circ } \pi _1)^{\ast} {\mathrm{T}}^{\left (A\right)}E\,-\,-\,-\,- \,-\,-\,-\,-\\ &\xrightarrow {p{\circ } s=p{\circ }\pi _1}(p{\circ } s)^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {{\mathfrak{I}}^2}s^{\ast} p^{\ast} {\mathrm{T}}^{\left (A\right)}E \end{align*}

• vertical lift. The vertical lift $\hat l\colon \hat {\mathrm{T}}\Rightarrow {\hat {\mathrm{T}}}^2\colon \kern -1ex=\hat {\mathrm{T}}{\circ }\hat {\mathrm{T}}$ is the pair

  \begin{align*} &(l,\xi ^{\left (l\right)})\colon ({\mathrm{T}} A,p^{\ast} {\mathrm{T}}^{\left (A\right)}E)\to ({\mathrm{T}}^2A,p_{\mathrm{T}}^{\ast} {\mathrm{T}}^{\left ({\mathrm{T}} A\right)}p^{\ast} {\mathrm{T}}^{\left (A\right)}E) \end{align*}
  formed by $l\colon {\mathrm{T}} A\to {\mathrm{T}}^2A$ and by the morphism:
  \begin{align*} \xi ^{\left (l\right)}&\colon p^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {p^{\ast} l^{\left (A\right)}}p^{\ast} {{\mathrm{T}}^{\left (A\right)}}^2E\xrightarrow {p=p{\circ } z{\circ } p=p{\circ } p_{\mathrm{T}}{\circ } l}(p{\circ } p_{\mathrm{T}}{\circ } l)^{\ast} {{\mathrm{T}}^{\left (A\right)}}^2E\,-\,-\,-\,- \,-\,-\,-\,-\\ &\xrightarrow {{\mathfrak{I}}^2}l^{\ast} p_{\mathrm{T}}^{\ast} p^{\ast} {{\mathrm{T}}^{\left (A\right)}}^2E\xrightarrow {l^{\ast} p^{\ast} _{\mathrm{T}}\alpha ^{\left (p\right)}{\mathrm{T}}^{\left (A\right)}}l^{\ast} p^{\ast} _{\mathrm{T}}{\mathrm{T}}^{\left ({\mathrm{T}} A\right)}p^{\ast} {\mathrm{T}}^{\left (A\right)}E \end{align*}

• canonical flip. The canonical flip $\hat c\colon {\hat {\mathrm{T}}}^2\Rightarrow {\hat {\mathrm{T}}}^2$ is the pair

  \begin{align*} &(c,\xi ^{\left (c\right)})\colon ({\mathrm{T}}^2A,p_{\mathrm{T}}^{\ast} {\mathrm{T}}^{\left ({\mathrm{T}} A\right)}p^{\ast} {\mathrm{T}}^{\left (A\right)}E)\to ({\mathrm{T}}^2A,p_{\mathrm{T}}^{\ast} {\mathrm{T}}^{\left ({\mathrm{T}} A\right)}p^{\ast} {\mathrm{T}}^{\left (A\right)}E) \end{align*}
  formed by $c\colon {\mathrm{T}}^2A\to {\mathrm{T}}^2A$ and by the morphism:
  \begin{align*} \xi ^{\left (c\right)}&\colon p^{\ast} _{\mathrm{T}}{\mathrm{T}}^{\left ({\mathrm{T}} A\right)}p^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {p^{\ast} _{\mathrm{T}}{\alpha ^{\left (p\right)}}^{-1}{\mathrm{T}}^{\left (A\right)}}p^{\ast} _{\mathrm{T}} p^{\ast} {{\mathrm{T}}^{\left (A\right)}}^2E\xrightarrow {p^{\ast} _{\mathrm{T}} p^{\ast} c^{\left (A\right)}}p^{\ast} _{\mathrm{T}} p^{\ast} {{\mathrm{T}}^{\left (A\right)}}^2E \,-\,-\,-\,-\\ &\xrightarrow {{{\mathfrak{I}}^2}^{-1}}(p{\circ } p_{\mathrm{T}})^{\ast} {{\mathrm{T}}^{\left (A\right)}}^2E\xrightarrow {p{\circ } p_{\mathrm{T}}=p{\circ }{\mathrm{T}} p{\circ } c=p{\circ } p_{\mathrm{T}}{\circ } c}(p{\circ } p_{\mathrm{T}}{\circ } c)^{\ast} {{\mathrm{T}}^{\left (A\right)}}^2E\,-\,-\,-\,- \,-\,-\,-\,-\\ &\xrightarrow {{\mathfrak{I}}^2}c^{\ast} p_{\mathrm{T}}^{\ast} p^{\ast} {{\mathrm{T}}^{\left (A\right)}}^2E\xrightarrow {c^{\ast} p^{\ast} _{\mathrm{T}}\alpha ^{\left (p\right)}{{\mathrm{T}}^{\left (A\right)}}^2}c^{\ast} p^{\ast} _{\mathrm{T}}{\mathrm{T}}^{\left ({\mathrm{T}} A\right)}p^{\ast} {\mathrm{T}}^{\left (A\right)}E \end{align*}

Moreover, if $({\mathbb{X}},\mathbb{T})$ has negatives with negation $n\colon {\mathrm{T}}\Rightarrow {\mathrm{T}}$ and each $({\mathbb{X}}^{\left (A\right)},\mathbb{T}^{\left (A\right)})$ has negatives with negation $n^{\left (A\right)}\colon {\mathrm{T}}^{\left (A\right)}\Rightarrow {\mathrm{T}}^{\left (A\right)}$ , then also the tangent category of elements has negatives whose negation is defined as follows:

• negation.The negation $\hat n\colon \hat {\mathrm{T}}\Rightarrow \hat {\mathrm{T}}$ is the pair

  \begin{align*} &(n,\xi ^{\left (n\right)})\colon ({\mathrm{T}} A,p^{\ast} {\mathrm{T}}^{\left (A\right)}E)\to ({\mathrm{T}} A,p^{\ast} {\mathrm{T}}^{\left (A\right)}E) \end{align*}
  formed by $n\colon {\mathrm{T}} A\to {\mathrm{T}} A$ and by the morphism:
  \begin{align*} \xi ^{\left (n\right)}&\colon p^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {p^{\ast} n^{\left (A\right)}}p^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {p=p{\circ } n}(p{\circ } n)^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {{\mathfrak{I}}^2}n^{\ast} p^{\ast} {\mathrm{T}}^{\left (A\right)}E \end{align*}

Lemma 3.22. The category $\mathsf{EL}({\mathbb{X}},\mathbb{T},{\mathfrak{I}})$ of elements of an indexed tangent category $({\mathbb{X}},\mathbb{T},{\mathfrak{I}})$ comes equipped with a tangent structure:

\begin{align*} &\hat {\mathbb{T}}\colon =\left (\hat {\mathrm{T}},\hat p,\hat z,\hat s,\hat l,\hat c\right) \end{align*}

Furthermore, if $({\mathbb{X}},\mathbb{T})$ and each tangent category $({\mathbb{X}}^{\left (A\right)},\mathbb{T}^{\left (A\right)})$ admit negatives for each $A$ of $\mathbb{X}$ , so does $\mathsf{EL}({\mathbb{X}},\mathbb{T},{\mathfrak{I}})$ .

Proof. The equational axioms required for $\hat {\mathbb{T}}$ to define a tangent structure are a direct, but tedious, consequence of $({\mathbb{X}},\mathbb{T})$ and each $({\mathbb{X}}^{\left (A\right)},\mathbb{T}^{\left (A\right)})$ being tangent categories. As an example, let us show that the zero morphism $\hat z$ is a section of the projection $\hat p$ , that is:

\begin{align*} &(p,\xi ^{\left (p\right)}){\circ }(z,\xi ^{\left (z\right)})=({\mathsf{id}}_A,\xi ^{\left ({\mathsf{id}}_A\right)})\colon (A,E)\to (A,E) \end{align*}

For starters, notice that the equation holds for the base components of the morphisms:

\begin{align*} &p{\circ } z={\mathsf{id}}_A \end{align*}

Let us focus on the fibre components. First, let us recall that

\begin{align*} &(p,\xi ^{\left (p\right)}){\circ }(z,\xi ^{\left (z\right)})=(p{\circ } z,\xi ^{\left (p{\circ } z\right)}) \end{align*}

where:

\begin{align*} &\xi ^{\left (p{\circ } z\right)}\colon E\xrightarrow {\xi ^{\left (z\right)}}z^{\ast} p^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {z^{\ast} \xi ^{\left (p\right)}}z^{\ast} p^{\ast} E\xrightarrow {{\mathfrak{I}}^2}(p{\circ } z)^{\ast} E={\mathsf{id}}_A^{\ast} E \end{align*}

However, by definition:

\begin{align*} &\xi ^{\left (z\right)}\colon E\xrightarrow {z^{\left (A\right)}}{\mathrm{T}}^{\left (A\right)}E\xrightarrow {{\mathfrak{I}}^0}{\mathsf{id}}_A^{\ast} {\mathrm{T}}^{\left (A\right)}E=(p{\circ } z)^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {{{\mathfrak{I}}^2}^{-1}}z^{\ast} p^{\ast} {\mathrm{T}}^{\left (A\right)}E\\ &\xi ^{\left (p\right)}\colon p^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {p^{\ast} p^{\left (A\right)}}p^{\ast} E \end{align*}

Thus, using that $p^{\left (A\right)}{\circ } z^{\left (A\right)}={\mathsf{id}}_{E}$ , it follows straightforwardly that

\begin{align*} &\xi ^{\left (p{\circ } z\right)}={\mathfrak{I}}^0 \end{align*}

proving the desired equation. One can prove all the other equational axioms in a similar way. The remaining axioms to be proven are the existence of the $n$ -fold pullbacks of the projection along itself, together with the universality of the vertical lift. Let us focus on the existence of the $n$ -fold pullback of the projection along itself.

let us start by considering an object $(A',E')$ of $\mathsf{EL}({\mathbb{X}},\mathbb{T},{\mathfrak{I}})$ and a collection morphisms of $\mathsf{EL}({\mathbb{X}},\mathbb{T},{\mathfrak{I}})$

\begin{align*} &(f_k,\xi ^{(f)}_k)\colon (A',E')\to \hat {\mathrm{T}}(A,E) \end{align*}

such that, for each $i,j=1{,\ldots ,}n$ :

\begin{align*} &\hat p{\circ }(f_i,\xi ^{(f)}_i)=\hat p{\circ }(f_j,\xi ^{(f)}_j) \end{align*}

From the universality of the $n$ -fold pullback of $p$ along itself, we obtain a morphism $\langle f_1{,\ldots ,}f_n\rangle \colon A'\dashrightarrow {\mathrm{T}}_nA$ of $\mathbb{X}$ :

let us now consider the following diagram in ${\mathbb{X}}^{\left (A'\right)}$ :

Since the substitution tangent morphisms are strong, the functor $(p{\circ } f_1)^{\ast}$ preserves the $n$ -fold pullbacks of the projection $p^{\left (A\right)}$ with itself. Therefore, we obtain a unique morphism $E'\dashrightarrow (p{\circ } f_1)^{\ast} {\mathrm{T}}_n^{\left (A\right)}E$ . let us define:

\begin{align*} \xi ^{\left (\langle f_1{,\ldots ,}f_n\rangle \right)}\colon& E'\dashrightarrow (p{\circ } f_1)^{\ast} {\mathrm{T}}_n^{\left (A\right)}E\xrightarrow {f_1=\pi _1{\circ }\langle f_1{,\ldots ,}f_n\rangle }(p{\circ } \pi _1{\circ }\langle f_1{,\ldots ,}f_n\rangle)^{\ast} {\mathrm{T}}_n^{\left (A\right)}E\\ &\xrightarrow {{\mathfrak{I}}^2}\langle f_1{,\ldots ,}f_n\rangle ^{\ast} (p{\circ } \pi _1)^{\ast} {\mathrm{T}}_n^{\left (A\right)}E \end{align*}

Thus, we obtain a morphism:

\begin{align*} &\left \langle (f_1,\xi ^{(f)}_1){,\ldots ,}(f_n,\xi ^{(f)}_n)\right \rangle \colon \kern -1ex=\left (\langle f_1{,\ldots ,}f_n\rangle ,\xi ^{\left (\langle f_1{,\ldots ,}f_n\rangle \right)}\right)\colon (A',E')\to \hat {\mathrm{T}}_n(A,E) \end{align*}

It is easy to prove that this is the unique morphism satisfying the equation

\begin{align*} &\hat {\pi }_k{\circ }\left \langle (f_1,\xi ^{(f)}_1){,\ldots ,}(f_n,\xi ^{(f)}_n)\right \rangle =(f_k,\xi ^{(f)}_k) \end{align*}

for each $k=1{,\ldots ,}n$ . Finally, to prove the universal property of the vertical lift, first notice that since $p^{\ast}$ is a strong tangent morphism, it preserves the universal property of the vertical lift $l^{\left (A\right)}\colon {\mathrm{T}}^{\left (A\right)}\to {{\mathrm{T}}^{\left (A\right)}}^2$ of $({\mathbb{X}}^{\left (A\right)},\mathbb{T}^{\left (A\right)})$ . From this observation and the universal property of the vertical lift of the base tangent category $({\mathbb{X}},{\mathrm{T}})$ we prove the desired universal property.

It is not hard to see that the functor $\mathscr{I}(\Pi)\colon \mathsf{EL}({\mathbb{X}},\mathbb{T};{\mathfrak{I}})\to ({\mathbb{X}},\mathbb{T})$ which sends each object $(A,E)$ of $\mathsf{EL}({\mathbb{X}},\mathbb{T};{\mathfrak{I}})$ to $A$ defines a cloven tangent fibration. In particular, by construction, $\Pi$ strictly preserves the tangent structures. To prove that the tangent bundle functors preserve the cartesian lifts, notice that the cartesian lift of a morphism $f\colon A\to A'$ of $\mathbb{X}$ along $\Pi$ is the morphism

\begin{align*} &\varphi ^{(f)}\colon \kern -1ex=(f,\xi ^{(f)})\colon (A,f^{\ast} E')\to (A',E') \end{align*}

where:

\begin{align*} &\xi ^f\colon f^{\ast} E'\xrightarrow {f^{\ast} {\mathsf{id}}_{E'}}f^{\ast} E' \end{align*}

From this, one can see that $\hat {\mathrm{T}}(\varphi ^{(f)})$ is the morphism $({\mathrm{T}} f,\alpha ^{(f)})$ , where $\alpha ^{(f)}$ is the distributive law of $f^{\ast}$ . Since $f^{\ast}$ is a strong tangent morphism, it follows directly that the distributor, which coincides with $\alpha ^{(f)}$ , is an isomorphism.

The next step is to prove that this assignment extends to a functor ${\mathscr{F}}\colon \mathsf{INDXTNG}_{\mathsf{strb}}\to \mathsf{TNGFIB}_{\mathsf{strb}}$ which forms an adjunction with $\mathscr{I}$ of Proposition 3.21.

Let us start by considering two indexed tangent categories ${\mathfrak{I}}_{\scriptscriptstyle \square }\colon ({\mathbb{X}}_{\scriptscriptstyle \square },\mathbb{T}_{\scriptscriptstyle \square })\to \mathsf{TNGCAT}$ and ${\mathfrak{I}}_{\scriptscriptstyle \blacksquare }\colon ({\mathbb{X}}_{\scriptscriptstyle \blacksquare },\mathbb{T}_{\scriptscriptstyle \blacksquare })\to \mathsf{TNGCAT}$ and a lax (colax) morphism of indexed tangent categories $(F,F',\beta ',\kappa)\colon {\mathfrak{I}}_{\scriptscriptstyle \square }\to {\mathfrak{I}}_{\scriptscriptstyle \blacksquare }$ which is strict on the base. The goal is to define a lax (colax) morphism of tangent fibrations strict on the base, between the corresponding tangent fibrations $\Pi ({\mathfrak{I}}_{\scriptscriptstyle \square })$ and $\Pi ({\mathfrak{I}}_{\scriptscriptstyle \blacksquare })$ :

• base tangent morphism. The base tangent morphism is given by $F\colon ({\mathbb{X}}_{\scriptscriptstyle \square },\mathbb{T}_{\scriptscriptstyle \square })\to ({\mathbb{X}}_{\scriptscriptstyle \blacksquare },\mathbb{T}_{\scriptscriptstyle \blacksquare })$ ;

• total tangent morphism. The lax (colax) tangent morphism between the total categories is given by the functor $\mathsf{EL}(F,F')\colon \mathsf{EL}({\mathbb{X}}_{\scriptscriptstyle \square },{\mathfrak{I}}_{\scriptscriptstyle \square })\to \mathsf{EL}({\mathbb{X}}_{\scriptscriptstyle \blacksquare },{\mathfrak{I}}_b)$ which sends a pair $(A,E$ to $(FA,F'E)$ and a morphism $(f,\xi ^{(f)})\colon (A,E)\to (A',E')$ to the morphism $(Ff,\xi ^{\left (Ff\right)})\colon (FA,F'E)\to (FA',F'E')$ where $\xi ^{\left (Ff\right)}$ is the morphism:

  \begin{align*} &\xi ^{\left (Ff\right)}\colon F'E\xrightarrow {F'\xi ^{(f)}}F'f^{\ast} E\xrightarrow {\kappa ^{(f)}}(Ff)^{\ast} F'K \end{align*}
  The lax distributive law $\beta '\colon \mathsf{EL}(F,F'){\circ }\hat {\mathrm{T}}_{\scriptscriptstyle \square }\Rightarrow \hat {\mathrm{T}}_{\scriptscriptstyle \blacksquare }{\circ }\mathsf{EL}(F,F')$ is the morphism $(\beta ,\xi ^{\left (\beta \right)})\colon (F{\mathrm{T}}_{\scriptscriptstyle \square } A,F'p_{\scriptscriptstyle \square }^{\ast} {\mathrm{T}}_{\scriptscriptstyle \square }^{\left (A\right)}E)\to ({\mathrm{T}}_{\scriptscriptstyle \blacksquare } FA,(p_{\scriptscriptstyle \blacksquare })_F^{\ast} {\mathrm{T}}_{\scriptscriptstyle \blacksquare }^{\left (FA\right)}F'E)$ given by $F{\mathrm{T}}_{\scriptscriptstyle \square } A={\mathrm{T}}_{\scriptscriptstyle \blacksquare } FA$ and by the morphism
  \begin{align*} &\xi ^{\left (\beta \right)}\colon F'p_{\scriptscriptstyle \square }^{\ast} {\mathrm{T}}_{\scriptscriptstyle \square }^{\left (A\right)}E\xrightarrow {\kappa ^{\left (p_{\scriptscriptstyle \square }\right)}{\mathrm{T}}_{\scriptscriptstyle \square }^{\left (A\right)}}(F p_{\scriptscriptstyle \square })^{\ast} F'{\mathrm{T}}_{\scriptscriptstyle \square }^{\left (A\right)}E\xrightarrow {(Fp_{\scriptscriptstyle \square })^{\ast} \beta ^{\left (A\right)}}(Fp_{\scriptscriptstyle \square })^{\ast} {\mathrm{T}}_{\scriptscriptstyle \blacksquare }^{\left (FA\right)}F'E=(p_{\scriptscriptstyle \blacksquare })_F^{\ast} {\mathrm{T}}_{\scriptscriptstyle \blacksquare }^{\left (FA\right)}F'E \end{align*}
  where we used that $Fp_{\scriptscriptstyle \square }=(p_{\scriptscriptstyle \blacksquare })_F$ . When the morphism of tangent fibrations is colax, the corresponding colax distributive law of $\mathsf{EL}(F,F')$ is the morphism $(\beta ,\xi ^{\left (\beta \right)})\colon ({\mathrm{T}}_{\scriptscriptstyle \blacksquare } FA,(p_{\scriptscriptstyle \blacksquare })_F^{\ast} {\mathrm{T}}_{\scriptscriptstyle \blacksquare }^{\left (FA\right)}F'E){\nrightarrow }(F{\mathrm{T}}_{\scriptscriptstyle \square } A,F'p_{\scriptscriptstyle \square }^{\ast} {\mathrm{T}}_{\scriptscriptstyle \square }^{\left (A\right)}E)$ , given by ${\mathrm{T}}_{\scriptscriptstyle \blacksquare } FA=F{\mathrm{T}}_{\scriptscriptstyle \square } A$ and by the morphism:
  \begin{align*} &\xi ^{\left (\beta \right)}\colon (p_{\scriptscriptstyle \blacksquare })_F^{\ast} {\mathrm{T}}_{\scriptscriptstyle \blacksquare }^{\left (FA\right)}F'E=(Fp_{\scriptscriptstyle \square })^{\ast} {\mathrm{T}}_{\scriptscriptstyle \blacksquare }^{\left (FA\right)}F'E\xrightarrow {(Fp_{\scriptscriptstyle \square })^{\ast} \beta ^{\left (A\right)}}(F p_{\scriptscriptstyle \square })^{\ast} F'{\mathrm{T}}_{\scriptscriptstyle \square }^{\left (A\right)}E\xrightarrow {{\kappa ^{\left (p_{\scriptscriptstyle \square }\right)}}^{-1}{\mathrm{T}}_{\scriptscriptstyle \square }^{\left (A\right)}}F'p_{\scriptscriptstyle \square }^{\ast} {\mathrm{T}}_{\scriptscriptstyle \square }^{\left (A\right)}E \end{align*}

Proposition 3.24. The assignment that sends an indexed tangent category $\mathfrak{I}$ to the corresponding cloven tangent fibration $\Pi ({\mathfrak{I}})$ extends to a pair of functors. In particular, each lax (colax) morphism of indexed tangent categories strict on the base is associated with a lax (colax) morphism of tangent fibrations, also strict on the base:

\begin{align*} &{\mathscr{F}}\colon \mathsf{INDXTNG}_{\mathsf{strb}}\to \mathsf{TNGFIB}_{\mathsf{strb}}\\ &{\mathscr{F}}\colon \mathsf{INDXTNG}_{{\mathsf{co}},\mathsf{strb}}\to \mathsf{TNGFIB}_{{\mathsf{co}},\mathsf{strb}} \end{align*}

These functors restrict to the subcategory of strong and strict morphisms of indexed tangent categories which are strict on the base:

\begin{align*} &{\mathscr{F}}\colon \mathsf{INDXTNG}_{\cong ,\mathsf{strb}}\to \mathsf{TNGFIB}_{{\mathsf{co}},\mathsf{strb}}\\ &{\mathscr{F}}\colon \mathsf{INDXTNG}_{=,\mathsf{strb}}\to \mathsf{TNGFIB}_{{\mathsf{co}},\mathsf{strb}} \end{align*}

We can finally state the main theorem of this section.

Proof. First, recall that by the Grothendieck construction, given a tangent fibration $\Pi \colon ({\mathbb{X}}',\mathbb{T}')\to ({\mathbb{X}},\mathbb{T})$ there is an isomorphism of fibrations between the underlying fibration of $\Pi$ and the one of ${\mathscr{F}}(\mathscr{I}(\Pi))$ . Similarly, for an indexed tangent category $({\mathbb{X}},\mathbb{T};{\mathfrak{I}})$ , the underlying indexed category is isomorphic to the one underlying $\mathscr{I}({\mathscr{F}}({\mathfrak{I}}))$ . So, we only need to compare the tangent structure of $\mathsf{EL}({\mathbb{X}},\mathbb{T};\mathscr{I}(\Pi))$ with $\mathbb{T}'$ , and the indexed tangent structure of $\mathscr{I}({\mathscr{F}}({\mathfrak{I}}))$ with the one of $\mathfrak{I}$ .

Let us start with an indexed tangent category $({\mathbb{X}},\mathbb{T};{\mathfrak{I}})$ . It is not hard to see that for an object $A\in {\mathbb{X}}$ the tangent category associated with $A$ via $\mathscr{I}({\mathscr{F}}({\mathfrak{I}}))$ is defined as follows. The category is precisely ${\mathbb{X}}^{\left (A\right)}={\mathfrak{I}}(A)$ , since ${\mathscr{F}}({\mathfrak{I}})^{-1}A={\mathbb{X}}^{\left (A\right)}$ . The tangent bundle functor sends an object $E$ of ${\mathbb{X}}^{\left (A\right)}$ to $z^{\ast} p^{\ast} {\mathrm{T}}^{\left (A\right)}E$ , where ${\mathrm{T}}^{\left (A\right)}$ is the tangent bundle functor of $({\mathbb{X}}^{\left (A\right)},\mathbb{T}^{\left (A\right)})={\mathfrak{I}}(A)$ . However, notice that $z^{\ast} p^{\ast} {\mathrm{T}}^{\left (A\right)}E$ is isomorphic to ${\mathrm{T}}^{\left (A\right)}E$ via the natural isomorphism:

\begin{align*} &z^{\ast} p^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {{{\mathfrak{I}}^2}^{-1}}(p{\circ } z)^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {p{\circ } z={\mathsf{id}}_{\mathbb{X}}}{\mathsf{id}}^{\ast} {\mathrm{T}}^{\left (A\right)}E\xrightarrow {{{\mathfrak{I}}^0}^{-1}}{\mathrm{T}}^{\left (A\right)}E \end{align*}

Similarly, a morphism $\psi \colon E\to E'$ corresponds to a morphism $({\mathsf{id}}_A,\psi)\colon (A,E)\to (A,E')$ in the tangent category of elements. Thus, the tangent bundle functor of $\mathscr{I}({\mathscr{F}}({\mathfrak{I}}))(A)$ sends $\psi$ to $z^{\ast} p^{\ast} \psi$ . Therefore, the tangent categories ${\mathfrak{I}}(A)$ and $\mathscr{I}({\mathscr{F}}({\mathfrak{I}}))(A)$ are isomorphic via a natural isomorphism

\begin{align*} &\mathscr{I}{\circ }{\mathscr{F}}\Rightarrow {\mathsf{id}}_{\mathsf{INDXTNG}_{\mathsf{strb}}} \end{align*}

which constitutes the counit of the adjunction.

The next step is to define the unit. Let us consider a tangent fibration $\Pi \colon ({\mathbb{X}}',\mathbb{T}')\to ({\mathbb{X}},\mathbb{T})$ and let us compare $({\mathbb{X}}',\mathbb{T}')$ with $\mathsf{EL}({\mathbb{X}},\mathbb{T};\mathscr{I}(\Pi))$ . The objects of the latter are pairs $(A,E)$ formed by $A\in {\mathbb{X}}$ and $E\in \Pi ^{-1}A$ , morphisms are pairs $(f,\xi ^{(f)})\colon (A,E)\to (A',E')$ formed by a morphism $f\colon A\to A'$ of $\mathbb{X}$ together with a morphism $\xi ^{(f)}\colon E\to f^{\ast} E'$ of $\Pi ^{-1}A$ , that is, $\Pi (\xi ^{(f)})={\mathsf{id}}_A$ . The tangent bundle functor $\hat {\mathrm{T}}$ sends a pair $(A,E)$ to $({\mathrm{T}} A,p^{\ast} z^{\ast} {\mathrm{T}}'E)$ and a morphism $(f,\xi ^{(f)})\colon (A,E)\to (A',E')$ to $({\mathrm{T}} f,\xi ^{\left ({\mathrm{T}} f\right)})$ where:

\begin{align*} &\xi ^{\left ({\mathrm{T}} f\right)}\colon p^{\ast} z^{\ast} {\mathrm{T}}'E\xrightarrow {p^{\ast} z^{\ast} {\mathrm{T}}'\xi ^{(f)}}p^{\ast} z^{\ast} {\mathrm{T}}'f^{\ast} E'\xrightarrow {p^{\ast} z^{\ast} \kappa ^{(f)}}p^{\ast} z^{\ast} ({\mathrm{T}} f)^{\ast} {\mathrm{T}}'E'\cong p^{\ast} f^{\ast} z^{\ast} {\mathrm{T}}'E'\cong ({\mathrm{T}} f)^{\ast} p^{\ast} z^{\ast} {\mathrm{T}}'E' \end{align*}

However, since $p$ is not invertible, because $z{\circ } p$ is not the identity, the tangent bundle functor $\hat {\mathrm{T}}$ is not isomorphic to ${\mathrm{T}}'$ . We can still define a natural transformation ${\mathrm{T}}'\Rightarrow \hat {\mathrm{T}}$ , induced by the universality of the cartesian lift $\varphi ^{\left (p\right)}$ and $\varphi ^{\left (z\right)}$ , as the unique morphism which makes the following diagram commutes:

This natural transformation together with the isomorphism of categories between the category of elements and ${\mathbb{X}}'$ defines the unit of the adjunction.

Since the base tangent category is preserved by the functors of the adjunctions of Theorem3.25, these adjunction restrict to tangent fibrations and indexed tangent categories on a fixed base tangent category.

3.3 Tangent fibrations as pseudoalgebras

Street (Reference Street1974) presented (cloven) fibrations as pseudoalgebras of a $2$ -monad and he employed this characterisation to extend the notion of a fibration in the context of $2$ -category theory. Since tangent categories form a $2$ -category, it is natural to wonder whether or not Street’s definition of fibrations applied to the context of tangent categories provides the same notion of a tangent fibration of Definition 3.1.

This section is dedicated to exploring this question. Let us start by recalling Street’s construction. Consider the $2$ -category $\mathsf{CAT}$ of categories and let $\mathbb{X}$ be a fixed category. On the slice $2$ -category ${\mathsf{CAT}}/{\mathbb{X}}$ , one can define a $2$ -functor $M$ which sends a functor $\Pi \colon {\mathbb{X}}'\to {\mathbb{X}}$ to $M(\Pi)\colon {\mathbb{X}}\downarrow \Pi \to {\mathbb{X}}$ , where ${\mathbb{X}}\downarrow \Pi$ denotes the comma category of $\Pi$ and the functor $M(\Pi)$ sends each $(f\colon A\to \Pi (E))\in {\mathbb{X}}\downarrow \Pi$ to $A\in {\mathbb{X}}$ .

$M\colon {\mathsf{CAT}}/{\mathbb{X}}\to {\mathsf{CAT}}/{\mathbb{X}}$ comes equipped with a $2$ -monad structure and cloven fibrations are precisely pseudoalgebras of $M$ . Let us recall the notion of a pseudoalgebra for a $2$ -monad.

Concretely, a pseudoalgebra of the $2$ -monad $M\colon {\mathsf{CAT}}/{\mathbb{X}}\to {\mathsf{CAT}}/{\mathbb{X}}$ is a functor $\Pi \colon {\mathbb{X}}'\to {\mathbb{X}}$ together with a $2$ -functor $\mathsf{trans}\colon {\mathbb{X}}\downarrow \Pi \to {\mathbb{X}}'$ which sends each $f\colon A\to \Pi (E)$ to an object $f^{\ast} E$ of ${\mathbb{X}}'$ . Moreover, the $2$ -functor $\mathsf{trans}$ sends the square diagram

to a morphism $f^{\ast} E\to E$ of ${\mathbb{X}}'$ , which corresponds to the cartesian lift of $f\colon A\to \Pi (E)$ on $E$ . Finally, the $2$ -morphisms $\zeta$ and $\theta$ provide the unitor and the compositor.

For an arbitrary $2$ -category $\mathbb{K}$ , the $2$ -monad $M$ is replaced with the $2$ -monad which sends each object $\Pi \colon {\mathbb{X}}'\to {\mathbb{X}}$ of the slice $2$ -category ${\mathbb{K}}/{\mathbb{X}}$ to $M(\Pi)\colon {\mathbb{X}}\downarrow \Pi \to {\mathbb{X}}$ , where ${\mathbb{X}}\downarrow \Pi$ denotes the comma object of $\Pi$ , provided the existence of such comma objects.

To see whether or not pseudoalgebras of the $2$ -monad $M$ in the $2$ -category of tangent categories correspond to the notion of a tangent fibration of Definition 3.1, first we need to establish if such a $2$ -category has comma objects. In particular, we need to see whether or not the comma category ${\mathbb{X}}\downarrow \Pi$ carries a tangent structure when $(\Pi ,\alpha)$ is a tangent morphism. In general, this is not the case, however, by considering a subclass of tangent morphisms, we obtain the following characterisation.

Lemma 3.29. Let $(\Pi ,\alpha)\colon ({\mathbb{X}}',\mathbb{T}')\to ({\mathbb{X}},\mathbb{T})$ be a strong tangent morphism. Then, the comma category ${\mathbb{X}}\downarrow \Pi$ carries a tangent structure whose tangent bundle functor sends each $f\colon A\to \Pi (E)$ to:

\begin{align*} &{\mathrm{T}} A\xrightarrow {{\mathrm{T}} f}{\mathrm{T}}\Pi (E)\xrightarrow {\alpha ^{-1}}\Pi {\mathrm{T}}'(E) \end{align*}

Moreover, this tangent category is the comma object of the $2$ -category $\mathsf{TNGCAT}_{\cong }$ of tangent categories and strong tangent morphisms which preserve the $n$ -fold pullbacks of the projection and the universality of the vertical lift.

Proof. Let us briefly sketch the proof. To show that the comma category ${\mathbb{X}}\downarrow \Pi$ comes equipped with a tangent structure, one uses the compatibilities of the strong distributive law $\alpha$ with the tangent structures and that the functor $\Pi$ preserves the $n$ -fold pullbacks of the projection along itself and the universality of the vertical lift.

To prove that such a construction defines comma objects, let us consider another tangent category $({\mathbb{X}}'',\mathbb{T}'')$ together with two strong tangent morphisms $(G,\beta)\colon ({\mathbb{X}}'',\mathbb{T}'')\to ({\mathbb{X}},\mathbb{T})$ and $(H,\gamma)\colon ({\mathbb{X}}'',\mathbb{T}'')\to ({\mathbb{X}}',\mathbb{T}')$ and a tangent natural transformation:

Let us consider the functor

\begin{align*} K\colon &{\mathbb{X}}''\to {\mathbb{X}}\downarrow \Pi \\ &A\mapsto (\varphi _A\colon GA\to \Pi (HA))\\ &(f\colon A\to B)\mapsto ((\Pi (Hf),Gf)\colon \varphi _A\to \varphi _B) \end{align*}

together with the distributive law $\delta \colon K{\circ }{\mathrm{T}}''\Rightarrow \bar {\mathrm{T}}{\circ } K$ , so defined:

One can prove that this strong tangent morphism preserves the $n$ -fold pullbacks of the projection along itself, the universality of the vertical lift, and that it is the unique strong tangent morphism such that

where $\pi _1$ and $\pi _2$ denote the projections of the comma object ${\mathbb{X}}\downarrow \Pi$ , which are strict tangent morphisms, and such that:

\begin{align*} &\varphi =\psi _{(K,\delta)} \end{align*}

where:

is the tangent natural transformation which picks out the object of the comma category.

Thanks to Lemma 3.29, we can define the $2$ -monad $M\colon \mathsf{TNGCAT}_{\cong }/({\mathbb{X}},\mathbb{T})\to \mathsf{TNGCAT}_{\cong }/({\mathbb{X}},\mathbb{T})$ which sends a strong tangent morphism $(\Pi ,\alpha)\colon ({\mathbb{X}}',\mathbb{T}')\to ({\mathbb{X}},\mathbb{T})$ to $M(\Pi ,\alpha)\colon ({\mathbb{X}},\mathbb{T})\downarrow (\Pi ,\alpha)\to ({\mathbb{X}},\mathbb{T})$ .

There is only one important difference between tangent fibrations and pseudoalgebras of $M$ : in the latter, the underlying tangent morphism $(\Pi ,\alpha)\colon ({\mathbb{X}}',\mathbb{T})'\to ({\mathbb{X}},\mathbb{T})$ is strong but not necessarily strict. One could have tried to consider the $2$ -category $\mathsf{TNGCAT}_{=}$ of tangent categories and strict tangent morphism instead of $\mathsf{TNGCAT}_{\cong }$ . However, in this context, the corresponding pseudoalgebras are instead fibrations between tangent categories whose tangent bundle functors strictly (not strongly) preserve the cartesian lifts.