2.1 Terminology and notation

Mostly, we use the same terminology and notation as the HoTT-Book (The Univalent Foundations Program, 2013). However, there are a few exceptions. To denote terms of type $\prod _{x \colon\! A}B(x)$ , we use the notation for $\lambda$ -expressions from pure mathematics, i.e. $x\mapsto f(x)$ . There are no implicit propositional truncations. If the propositional truncation of a statement is used, it is indicated by the word “merely.” Phrases like “for all” and “there is” are to be interpreted as $\prod$ - and $\sum$ -types. For example, the sentence

\begin{equation*} \mathrm{For\ all}\ x \colon\! A,\ \mathrm{we\ have}\ t \colon\! B(x). \end{equation*}

is to be read as the statement describing the term $(x \colon A)\mapsto t$ of type $ \prod _{x \colon A} B(x)$ . We sometimes write $f_a$ for the application of a dependent function $f \colon \prod _{x \colon A} B(x)$ to $a \colon A$ , instead of $f(a)$ .

Furthermore, similar to (Shulman, Reference Shulman2018), when dealing with identity types, we avoid topology and geometry-related words. For example, we write “equality” instead of “path” and “2-cell” instead of “homotopy,” to avoid confusion with the notions of paths and homotopies for the classical geometric objects we like to study by including them in our theory as 0-types. We use $p\,\bullet\,q$ to denote the concatenation of equalities $p$ and $q$ . We say that $x$ is unique with some properties if the type of all $x$ with these properties is contractible.

2.2 Preliminaries from homotopy type theory

We use a fragment of the Type Theory from (The Univalent Foundations Program, 2013). Function extensionality is always assumed to hold. Furthermore, we assume a propositional truncation modality “ $\|\_\|$ ” and univalent universes.

In the next section, we will give axioms for a modality “ $\Im$ ,” which will be assumed throughout the article. Some knowledge of the basic concepts in (The Univalent Foundations Program, 2013) is assumed. In addition, we will use more facts about pullbacks than presented in (The Univalent Foundations Program, 2013), which we will list in this section.

It is very useful to switch between pullback squares and equivalences over a morphism. We start with the latter concept.

For every morphism over $f:A\to B$ as above, we can construct a squareFootnote ³

where the top map is given as $(a,p_a)\mapsto (f(a),\varphi _a(p_a))$ . This square will turn out to be a pullback in the sense we are going to describe now, if and only if $\varphi$ is an equivalence over $f$ .

For a cospan given by the maps $f:A\to C$ and $g:B\to C$ , we can construct a pullback square:

Then, for any other completion of the cospan to a square

where $\eta :\prod _{x:X}g(x)=f(x)$ is a 2-cell letting it commute, an induced map to the pullback is given by $x\mapsto (\varphi _A(x),\varphi _B(x),\eta _x)$ .

To reverse the construction of a square for a morphism over “ $f$ ” above, we can start with a general square:

Let $P:A\to \mathcal U$ and $Q:B\to \mathcal U$ be the fiber types of the vertical maps, i.e.:

\begin{align*} &P(a:A):\equiv \sum _{x:X}p_A(x)=a \\ &Q(b:B):\equiv \sum _{y:Y}p_B(y)=b \end{align*}

Then, for all $a:A$ , a morphism $\varphi _a:P(a)\to Q(a)$ is given as:

\begin{equation*} \varphi _a((a,(x,p))):\equiv (f(a),(g(x),\eta _x\,\bullet \, f(p))).\end{equation*}

So $\varphi$ is a morphism from $P$ to $Q$ over $f$ . The following statement is quite useful and will be used frequently in this article:

Now, the following corollary can be derived by using the fact that equivalences are stable under pullback:

Corollary 2.4. Let $f:A\to B$ be an equivalence, $P:A\to \mathcal U$ , $Q:B\to \mathcal U$ dependent types, and $\varphi :\prod _{x:A}P(x)\to Q(f(x))$ an equivalence over $f$ . Then the induced map:

\begin{equation*} \left (\sum _{x:A}P(x)\right ) \to \left (\sum _{x:B} Q(x)\right )\end{equation*}

is an equivalence.

2.3 Modalities

From this section on, we will always assume a modality $\Im$ . We use the definition of a uniquely eliminating modality from (Rijke, Shulman and Spitters, Reference Rijke, Shulman and Spitters2020), which is equivalent to the definition given in (The Univalent Foundations Program, 2013, Section 7.7). More on modalities and their relation to concepts in category theory can be found in (Rijke, Shulman and Spitters, Reference Rijke, Shulman and Spitters2020). We deviate from the usual symbol for modalities, which would be “ $\bigcirc$ ” to remind us that while we will technically work with a general modality, we have some particular kind of modality in mind. Furthermore, the work in this article could be reused in a type theory that provides more modal operators from differential cohesion, for example, in the work in progress (Myers, Reference Jaz Myers2022b), which also uses homotopy type theory as a basis and where $\Im$ is called crystalline modality.

The modality $\Im$ is also used in category theory based differential cohesionFootnote ⁴ and is called infinitesimal shape.Footnote ⁵

Axiom 2.5. From this point on, we assume existence of a map $\Im : \mathcal U\to \mathcal U$ and maps $\iota _A: A \to \Im A$ for all types $A$ , subject to this condition: For any $B:\Im A\to \mathcal U$ , the map

\begin{equation*} \_\circ \iota _A:\left (\prod _{a \colon\! \Im A}\Im B(a)\right ) \to \left (\prod _{a \colon\! A}\Im B(\iota _A(a))\right ) \end{equation*}

is an equivalence.

We call the inverse of the equivalence $\Im$ -elimination. Elimination in type theory is a principle that lets us define maps starting in an inductive type like the natural numbers. For example, eliminating from the natural numbers $\mathbb{N}$ to a dependent proposition $P:\mathbb{N}\to \mathcal U$ means essentially to prove the proposition for each possible way to construct a natural number, which is either to take it to be the constant $0$ or the successor $s(n)$ of another natural number $n$ .

The analogy to $\Im$ -elimination is that to eliminate from $\Im A$ into the dependent modal type $\Im B(\_)$ , we only need to provide a value for the case that $x:\Im A$ is of the form $\iota _A(y)$ . This is exactly what the inverse of the map in axiom 2.5 allows us to do. A different way to put this is that $\Im A$ has the same elimination principle as an inductive type with constructor $\iota _A:A\to \Im A$ would have, except that it can only be used to construct functions with modal codomain.

Note that it is possible to conclude a variant of $\Im$ -elimination from axiom 2.5, where $\Im$ is not applied to the type family $B$ , but the type family is required to have values in $\Im$ -modal types.Footnote ⁶

Note that the equivalence in axiom 2.5 specializes to the universal property of a reflection if the family $B$ is constant:

i.e. for all types $B$ and all $f:A\to \Im B$ , we get a unique $\psi$ letting the triangle commute up to a 2-cell. Unique means here that there is a contractible type of maps with 2-cells letting the triangle commute. That type is also a fiber of the equivalence “ $\_\circ \iota _A$ ,” so we do know that it is contractible.

We will make use of this in showing that $\Im$ is idempotent in the following sense:

Proof. By the universal property we just discussed, we get a candidate for an inverse to $\iota _{\Im A}$ , which we call $\varphi$ :

By construction, $\varphi$ is already a left inverse of $\iota _{\Im A}$ . We consider the diagram

and conclude that $\varphi$ is also a right inverse by uniqueness.

Like reflections determine a subcategory, $\Im$ determines a subuniverse of the universe $\mathcal U$ of all types.Footnote ⁷

From what we proved above, all types $\Im A$ will be modal.

As we explained in the introduction, we will not be very interested in spaces of the form “ $\Im X$ ,” but more in the “quotient map” $\iota _X:X\to \Im X$ , which we will view as identifying infinitesimally close points.

Like a functor, $\Im$ extends to maps and we get a naturality square for $\iota$ :

It is also straightforward to prove that the application of $\Im$ to maps commutes with composition of maps up to equality and preserves identities up to equality. In general, we expect that any coherence between these equalities needed in practice can be constructed.

Remark 2.9. For any 2-cell $\eta : f \Rightarrow g$ , we have a 2-cell between the images:

\begin{equation*} \Im \eta : \Im f \Rightarrow \Im g\text {.}\end{equation*}

$\Im$ -Modal types have various closedness properties, which we review in the following lemma.

One immediate consequence is $\Im 1 \simeq 1$ – this is the only provably $\Im$ -modal type. We cannot expect to prove more types to be $\Im$ -modal, since there is always the modality that maps all types to $1$ , so $1$ could be the only $\Im$ -modal type.

The following is a slight variation of (Rijke, Shulman, and Spitters, Reference Rijke, Shulman and Spitters2020) [Lemma 1.24], and plays a central role in the abstract (Wellen, Reference Wellen2018), which was the beginning of (Cherubini and Rijke, Reference Cherubini and Rijke2021):

Proposition 2.11. Let $A$ be a type and $B \colon \Im A \to \mathcal U$ a dependent type. Then the induced map is an equivalence:

\begin{equation*} \Im \left (\sum _{x \colon\! A} B(\iota _A(x))\right ) \simeq \left (\sum _{x \colon\! \Im A} \Im (B(x)) \right ). \end{equation*}

A more category theoretic implication of this proposition is that for the map

\begin{equation*} \pi _1: \left (\sum _{x \colon\! A} B(\iota _A(x))\right ) \to A \end{equation*}

taking fibers commutes with application of $\Im$ . Here, $\pi _1$ is an example of a formally étale map, which we will introduce in the next section. More abstractly, this relates to the principle in algebraic topology that homotopy fibers coincide with ordinary fibers of certain fibrations. This point is highlighted and used in (Myers, Reference Jaz Myers2022a).

3.1 Formal disks

We will start to build geometric notions on top of the modality $\Im$ and its unit $\iota$ . In the intended applications, the modality $\Im$ provides us with a notion of infinitesimal proximity. To see if two points $x,y$ in some type $A$ are infinitesimally close to each other, we map them to $\Im A$ and ask if the images are equal.

Definition 3.1. Let $x,y \colon A$ . Then we have a type that could be read “ $x$ is infinitesimally close to $y$ ” and is given as:

\begin{equation*} x\sim y:\equiv (\iota _A(x)=\iota _A(y)). \end{equation*}

Of course, this is in general not a proposition, but it is useful to think about $\iota _A(x)=\iota _A(y)$ in this way. The name “infinitesimally close” is a poor choice for a general modality,Footnote ⁸ so the reader should keep in mind from now on that the terminology is adapted to a modality in the intended applications.Footnote ⁹

It turns out that all morphisms of types already respect this notion of infinitesimal closedness, i.e., if two points are infinitesimally close to each other, their images are close as well.

Remark 3.2. If $x,y \colon A$ are infinitesimally close, then for any map $f \colon A\to B$ , the images $f(x)$ and $f(y)$ are infinitesimally close. More precisely, we have an induced function:

\begin{equation*} \tilde {f}:(x\sim y)\to (f(x)\sim f(y))\end{equation*}

Proof. We construct a map between the two types $\iota _A(x)=\iota _A(y)$ and $\iota _B(f(x))=\iota _B(f(y))$ . By 2.8, we can apply $\Im$ to maps and get a map $\Im f \colon \Im A \to \Im B$ . So we can apply $\Im f$ to an equality $\gamma \colon \iota _A(x)=\iota _A(y)$ to get an equality:

\begin{equation*}\Im f (\gamma ) \colon \Im f(\iota _A(x))=\Im f(\iota _A(y))\end{equation*}

Again by 2.8, we know that we have a naturality square:

and hence equalities $\eta _f(x) \colon \Im f(\iota _A(x))= \iota _B(f(x))$ and $\eta _f(y) \colon \Im f(\iota _A(y))= \iota _B(f(y))$ . This yields an equality of the desired type:

\begin{equation*} \eta _f(x)^{-1}\,\bullet \, \Im f(\gamma ) {\,\bullet \,} \eta _f(y) \end{equation*}

A formal disk at a point is the “collection” of all other points infinitesimally close to it:

The characterization (iii) is a verbatim translation of its topos-theoretic analog (Reference SchreiberSchreiber) [Definition 5.3.50] to homotopy type theory. Therefore, in the model from (Reference SchreiberSchreiber), composing a function on a manifold $M$ with $\iota _M(x)$ would yield an $\infty$ -order jet of that function. Jets are higher order analogs of tangent vectors and the derivates of a function at a point $x$ can still be observed on a formal disk around $x$ . To say that formal disks are just infinitesimal extensions of the point is supported by the following observation.

Proof. Using Proposition 2.11 and Proposition 2.10 (d), we compute:

\begin{align*} \mathbb{D}_x&\equiv \sum _{y:X}\iota _X(x)=\iota _X(y) \\ &=\sum _{y:X}\Im (\iota _X(x)=(\iota _X(y))) \\ &=\Im \left (\sum _{z:\Im X}\iota _X(x)=z\right ) \\ &=1\rlap{.} \end{align*}

As morphisms of manifolds induce maps on tangent spaces, maps of types induce morphisms on formal disks, containing information on the derivates of a morphism of all orders:

Remark 3.5. If $f: A\to B$ is a map, there is a dependent function:

\begin{equation*} df: \prod _{x \colon\! A} \mathbb {D}_x\to \mathbb {D}_{f(x)}\end{equation*}

We denote the evaluation at $a \colon A$ with

\begin{equation*} df_a: \mathbb {D}_a \to \mathbb {D}_{f(a)} \end{equation*}

and call it the (generalized) differential of $f$ at $a$ .

Proof. To define $df$ , we take the sum over the map from 3.2:

\begin{equation*} df_a{ \colon \!\!\!\equiv }\, (x, \epsilon ) \mapsto (f(x), \eta _f^{-1}(x){\,\bullet \,}\Im f (\epsilon ) {\,\bullet \,}\eta _f(x))\end{equation*}

– where $\eta _f(x)$ is the equality from the naturality of $\iota$ .

Some of the familiar rules for differentiation can be derived from this generality. We will need only the chain rule:

Lemma 3.6. Let $f:A\to B$ and $g:B\to C$ be maps. Then the following holds for all $x:A$

\begin{equation*} d(g\circ f)_x=(dg)_{f(x)}\circ df_x.\end{equation*}

Proof. Note that, in general, the differential $df_x$ is equal to the map induced by the universal property of $\mathbb{D}_{f(x)}$ as a pullback. We can use this to get the desired “functoriality”:

– the induced map $d(g\circ f)_x$ and the composition $(dg)_{f(x)}\circ df_x$ solve the same factorization problem, so they are equal.

In differential geometry, the tangent bundle is an important basic construction consisting of all the tangent spaces in a manifold, capturing first-order infinitesimal information. In this abstract all-order setting, we can mimic the construction by combining all the formal disks of a space in a bundle, capturing infinitesimal information of all orders at once.

Note that despite the seemingly symmetric second definition, we want $T_\infty A$ to be a bundle having formal disks as its fibers, so it is important to distinguish between the two projections and their meaning. If we look at $T_\infty A$ as a bundle, meaning a morphism $p \colon T_\infty A\to A$ , we always take $p$ to be the first projection in both cases. This convention agrees with the first definition – taking the sum yields a bundle with fibers of the first projection equivalent to the formal disks.

For any $f \colon A \to B$ , we defined the induced map $df$ on formal disks. This extends to formal disk bundles.

Definition 3.8. For a map $f \colon A\to B$ , there is an induced map on the formal disk bundles, given as:

\begin{equation*} T_\infty f{ \colon \!\!\!\equiv }\, (a, \epsilon ) \mapsto (f(a), df_a(\epsilon ))\end{equation*}

In differential geometry, the tangent bundle may or may not be trivial. This is some interesting information about a space. If we have a smooth group structure on a manifold $G$ , i.e. a Lie-group, we may consistently translate the tangent space at the unit to any other point. This may be used to construct an isomorphism of the tangent bundle with the projection from the product of $G$ with the tangent space at the unit.

It turns out that this generalizes to formal disk bundles, and the group structure may be replaced by the weaker notion of a homogeneous type.

The notion of homogeneous type was developed by the author to satisfy two needs. The first is to match the intuition of a pointed space, that is equipped with a continuous family of translations that map the base point to any given point. The second need is to have just the right amount of data in all the proofs and constructions concerning homogeneous types. It has not been investigated in what circumstances this definition of homogeneous spaces coincides with the various notions of homogeneous spaces in geometry – apart from the obvious examples given below.

In the following, we will build a family of equivalences from one formal disk of a homogeneous type to any other formal disk of the space. We start by observing how equivalences and equalities act on formal disks.

We are now ready to state and prove the triviality theorem.

In geometry, it is usually possible to add tangent vectors. Our formal disks can at least inherit the group-like properties of a homogeneous type:

Proof. We look at the sequence

where $u_e$ is the inclusion of the formal disk, given as the first projection. We will proceed by constructing a homogeneous structure on $\Im A$ , note some properties of $\iota _A$ , which could be part of a definition of morphism of homogeneous types, and finally give some “kernel”-like construction of the structure on $\mathbb{D}_e$ .

For $x:A$ , there is a translation $t_x:A\simeq A$ , since $\Im$ preserves equivalences, this yields a $\Im t_x: \Im A\simeq \Im A$ . By $\Im$ -elimination, this extends to a family of translations

\begin{equation*} t^{\prime}:\prod _{y:\Im A}\Im A\simeq \Im A \text {, with $t^{\prime}_{\iota _A(x)}=\Im t_x$.} \end{equation*}

Let $e^{\prime}:\equiv \iota _A(e)$ , then $\Im A$ is homogeneous if we can produce a

\begin{equation*} p^{\prime}:\prod _{y:\Im A}t^{\prime}_y(e^{\prime})=y. \end{equation*}

By $\Im$ -eliminating on $y$ , we reduce the problem to

\begin{equation*} \prod _{x:A}t^{\prime}_{\iota _A(x)}(\iota _A(e))=\iota _A(x) \end{equation*}

By definition, the left hand side is $\Im (t_x)(\iota _A(e))$ and by naturality of $\iota$ , we have an equality $\Im (t_x)(\iota _A(e))=\iota _A(t_x(e))$ . So by applying $\iota _A$ to the equality $p_x:t_x(e)=x$ , we get a solution.

We start to construct the homogeneous structure on $\mathbb{D}_e$ by letting $e^{\prime\prime}:\equiv (e,{\mathrm{refl}})$ be the unit. For the translations, we look at the dependent type $(x:A)\mapsto \iota _A(e)=\iota _A(x)$ and establish the following chain of equivalences for $y:A$ with $\iota _A(e)=\iota _A(y)$ :

\begin{align*} &\ \iota _A(e)=\iota _A(x) \\ \simeq &\ t^{\prime}_{\iota _A(y)}\iota _A(e)=t^{\prime}_{\iota _A(y)}\iota _A(x) \\ \simeq &\ t^{\prime}_{\iota _A(y)}\iota _A(e)=\iota _A(t_y(x)) \\ \simeq &\ \iota _A(y)=\iota _A(t_y(x)) \\ \simeq &\ \iota _A(e)=\iota _A(t_y(x)) \end{align*}

The resulting equivalence is an equivalence over $t_y$ . So by 2.4, this induces an equivalence on the sum, which is $\mathbb{D}_e$ .

This construction yields a family of equivalences $t^{\prime\prime}:\prod _{y:\mathbb{D}_e}\mathbb{D}_e\simeq \mathbb{D}_e$ . To finish the prove of the theorem, we need to construct a family of equalities $\prod _{x:\mathbb{D}_e}t^{\prime\prime}_x(e^{\prime\prime})=x$ . This is another computation using the same methods we have seen so far and we refer to the formalization Footnote ¹⁰ instead of giving the details here.

3.2 Formally étale maps

The approach to formally étale maps presented here has been developed further in the ongoing synthetic algebraic geometry project.Footnote ¹¹

In algebraic geometry, formally étale maps are supposed to be analogous to local diffeomorphisms in differential geometry. Below, we will give a not-so-well-known definition that matches the notion of algebraic geometry for finitely presented morphisms of schemesFootnote ¹² and coincides with the local diffeomorphisms between manifolds in the case of differential geometry.Footnote ¹³

Definition 3.14. A map $f \colon A\to B$ is formally étale if its naturality square is a pullback:

To see why this definition expresses that a map is an isomorphism on a infinitesimal scale, we can look at the following situation:

– whenever we have a point $b:B$ that we can lift to $A$ , there will be a unique lift of the whole formal disk around $b$ to $A$ by the universal property of the pullback $A$ and the naturality of $\Im$ .

This definition of formally étale maps was used extensively in (Reference SchreiberSchreiber) and (Khavkine and Schreiber, Reference Khavkine and Schreiber2017).Footnote ¹⁴ The same definition under different names was also used in category theory to study the relation between reflective subcategories and factorization systems (Cassidy, Hébert and Kelly, Reference Cassidy, Hébert and Kelly1985). Here, the maps with a cartesian naturality square for the reflector are the right maps of a factorization system, whereas the left maps are those mapped to isomorphisms by the reflector. The factorization system can also be defined as a modality and studied internally (Cherubini and Rijke, Reference Cherubini and Rijke2021).

We will continue with some basic observations:

Together with the following, we have all the properties of formally étale maps needed in this article:

One might wonder if the converse of this statement holds. With a mild condition on $A$ , which is related to the concept of formal smoothness in algebraic geometry, this is the case:Footnote ¹⁵

Proof. As in the lemma, we use the equivalence of pullback squares and fibered equivalences. So to show that the $\iota$ -naturality square for $f$ is a pullback, we have to show that for all $x \colon \Im A$ , the induced map on fibers

\begin{equation*} \psi _x:\iota _A^{-1}(x) \to \iota _B^{-1}((\Im f)(x)) \end{equation*}

is an equivalence.

By surjectivity of $\iota _A$ , there is merely a $\tilde{x}$ and $p:\iota _A(\tilde{x})=x$ . Since we show a proposition, we can use $\tilde{x}$ and the equivalence $e_1:\mathbb{D}_{\tilde{x}}\simeq \iota _A^{-1}(x)$ . By naturality, we also have $e_2:\mathbb{D}_{f(\tilde{x})}\simeq \iota _B^{-1}((\Im f)(x))$ .

It remains to show that $\psi _x=e_2\circ df_{ \tilde{x}}\circ e_1^{-1}$ . Induction on $p$ simplifies $e_1$ and $e_2$ to the identity and we just have to note that $df_{ \tilde{x}}$ was defined as a induced map on the fibers $\mathbb{D}_{ \tilde{x}}$ and $\mathbb{D}_{f(\tilde{x})}$ .

The following is also proven in a different way in (Cherubini and Rijke, Reference Cherubini and Rijke2021) as corollary 5.2 (b).

Theorem 3.18. Let $f:A\to B$ be formally étale and

a pullback square. Then $f^{\prime}$ is formally étale.

Proof. Let us denote the bottom map with $\psi :B^{\prime}\to B$ . We start by describing $A^{\prime}$ as a pullback:

\begin{align*} A^{\prime} &\simeq \left (\sum{f^{\prime}}^{-1}\right ) \simeq \left (\sum f^{-1}\circ \psi \right ) \simeq \left (\sum (\Im f)^{-1}\circ \iota _B\circ \psi \right ) \\ & \simeq \left (\sum (\Im f)^{-1}\circ \Im \psi \circ \iota _{B^{\prime}}\right ) \end{align*}

Now we can apply 2.11 to compute $\Im A^{\prime}$ :

\begin{equation*} \Im A^{\prime}\simeq \Im \left (\sum (\Im f)^{-1}\circ \Im \psi \circ \iota _{B^{\prime}}\right ) \simeq \left (\sum (\Im f)^{-1}\circ \Im \psi \right ) \end{equation*}

Note that the right-hand side is the pullback of $\Im A$ along $\Im \psi$ . This means that applying $\Im$ to the pullback square given in the statement of the theorem is again a pullback, and by pullback pasting, the naturality square of $f^{\prime}$ is a pullback.

There is much more to be said about formally étale maps that is very useful, but not used in this article. One example that is interesting from a geometric perspective is that formally étale maps are the right class of a factorization system, whose left class are the $\Im$ -equivalences. A consequence is that all maps can be factored into an $\Im$ -equivalence followed by a formally étale map:

Remark 3.20. Let $f:A\to B$ be a map. The map $f$ factors over

\begin{equation*} C_f:\equiv \sum _{x \colon\! \Im A, y \colon\! B}(\Im f)(x)=\iota _B(y) \end{equation*}

by $l_f:\equiv (u:A)\mapsto (\iota _A(u),f(u),\eta _f)$ and $r_f:\equiv ((x,y,p)\mapsto y)$ to $B$ . Furthermore, $\Im (l_f)$ is an equivalence and $r_f$ is formally étale.

We sketch a proof – a full analysis of the factorization system can be found in (Cherubini and Rijke, Reference Cherubini and Rijke2021, Section 7).

We will put formally étale maps to use in Section 4.2. The next section makes no reference to $\Im$ .

4.1 Fiber bundles

As mentioned in the introduction, the spaces we have in mind might have both a differential geometric structure and higher identity types. This section is about maps that correspond to fiber bundles, which are by definition locally trivial with respect to the higher identity or homotopical structure, but are expected to be locally trivial also with respect to a geometric structure that might be present in an application. By local triviality, we just mean that there is a surjectionFootnote ¹⁶ into the base of the fiber bundle $p:E\to B$ such that pulling back along the surjection yields a projection from a product with a fixed given type $F$ . It will turn out to be logically equivalent to ask all fibers of $p$ to be merely equivalent to $F$ .

In a basic intended applications, $E$ and $B$ might just be manifolds given by 0-types, and the reader might wonder if this notion of fiber bundle is too unrestrictive. However, it turns out that asking this internally turns into a surprisingly strong statement externally. In (Cherubini et al., Reference Cherubini, Coquand and Hutzler2023), it was discovered for a model based on a Grothendieck topos relevant to algebraic geometry that internal surjections have local sections with respect to the Grothendieck topology. It is reasonable to assume that similar principles work in differential geometry, and applying this to the surjective projection

\begin{equation*} \left (\sum _{x:B}\|p^{-1}(x)=F\|\right ) \to B \end{equation*}

would show that the fiber bundles defined in this section are actually locally trivial in the sense of classical definitions of fiber bundles. A $\infty$ -topos-theoretic version of this approach to fiber bundles may be found in (Nikolaus, Schreiber and Stevenson, Reference Nikolaus, Schreiber and Stevenson2015).

In this section, we will give four definitions of these fiber bundlesFootnote ¹⁷ and prove that they are equivalent. It will be useful in Section 4.3 to switch between the different definitions.

For the following statements about fiber bundles, we will make a lot of unavoidable use of a univalent universe $\mathcal U$ and propositional truncation. We will frequently use that all maps of types $p:E\to B$ appear in a pullback square

where $\widetilde{\mathcal U}$ is called the universal family and obtained by summing over the dependent type $(A \colon\! \mathcal U)\mapsto A$ . The bottom map $p^{-1}$ determines $p$ up to canonical equivalence over $B$ and is called the classifying map of $p$ . If $E$ is a sum over a dependent type $q \colon B\to \mathcal U$ , and $p$ the projection to $B$ , then $q$ is the classifying map.

This way of using a univalent universe corresponds to looking at it as a moduli space or classifying space of types. We could replace the $\mathcal U$ with some other moduli space to get specialized notions of fiber bundles with additional structure on the fibers.

Before we start, we will look at some preliminaries about surjective and injective maps. A surjective map is a map with merely inhabited fibers, or in other words, a $\|\_\|$ -connected map. An injective map has $\|\_\|$ -truncated fibers.Footnote ¹⁸

Lemma 4.4. For any map $f: A\to B$ there is a unique triangle:

where $e$ is surjective, $m$ injective and ${\text{image}}(f)$ is given by

\begin{equation*}{\text {image}}(f){ \colon \!\!\!\equiv } \sum _{b:B} \left \| \sum _{a \colon A} f(a)=b \right \|\text {.} \end{equation*}

A proof of the general case for $\|\_\|_n$ may be found in (The Univalent Foundations Program, 2013, chapter 7.6).

In Topology, an $F$ -fiber bundle is a map $p \colon E\to B$ that is locally trivial with all its fibers are isomorphic to $F$ . Local triviality means that $B$ may be covered by open sets $U_i$ , such that on each $U_i$ the restricted map $p_{|p^{-1}(U_i)}$ is isomorphic to the projection $F\times U_i\to U_i$ . We may rephrase this in a more economical way: From our cover, we construct a surjective map $w \colon \coprod _{i\in I}U_i\to B$ . Then the local triviality translate to the pullback of $p$ along $w$ being isomorphic to the product projection $F\times \coprod _{i\in I}U_i\to \coprod _{i\in I}U_i$ .

For fiber bundles in geometry, we would require more from a general surjective map, or cover, $w \colon W\to B$ than that pulling back along it turns $p$ into a product projection. However, for the notion we discuss in this section, this turns out to be already enough.

Definition 4.5. Let $p \colon E\to B$ be a map of types. For another map $w \colon W \to B$ we say $w$ is a trivializing cover for $p$ if $w$ is a surjective map and there is a pullback square:

The map $p$ is called an $F$ -fiber bundle if there merely is such a trivializing $p$ .

We give an equivalent dependent version of this definition,Footnote ¹⁹ which will be a lot easier to work with:

Definition 4.6. Let $E \colon B\to \mathcal U$ be a dependent type. We say that a surjection $w: W \to B$ is a trivializing cover for $E$ if

\begin{equation*} \prod _{x \colon W} E(w(x)) \simeq F\text {.} \end{equation*}

The dependent type $E$ is called an $F$ -fiber bundle if there merely is such a trivializing cover.

We can switch between the two definitions in the usual way: Given an $F$ -fiber bundle $p \colon E\to B$ in the first sense, the dependent type of its fibers $p^{-1} \colon B \to \mathcal U$ will be an $F$ -fiber bundle in the second sense, by direct application of 2.3. To go back, we take the projection from the sum of an $F$ -fiber bundle $E \colon B\to \mathcal U$ .

Note that in both cases, the propositional truncation of the trivializing datum is necessary to turn the definition into a proposition. In the following, we will see that we could have defined $F$ -fiber bundles more easily with their classifying maps to a type called ${\mathrm{BAut}}(F)$ , providing us with a notion of $F$ -fiber bundles, which is directly a proposition. However, in those definitions, while it is possible to construct a surjective trivializing map, it is unclear how we may require that this map has additional properties. One example, where we are interested in special surjections, is the definition of a $V$ -manifold, where we will use formally étale surjections.

We review the type ${\mathrm{BAut}}(F)$ now, which will be used to give the alternative definition of fiber bundles mentioned above:

We are now ready to give yet another definition of fiber bundles:

Definition 4.9. A map $p: E\to B$ is an $F$ -fiber bundle if and only if there is a map $\chi \colon B\to{\mathrm{BAut}}(F)$ , such that there is a pullback square

In this case, $\chi$ is called the classifying map of $p$ .

This definition also has a surprisingly easy dependent variant, which is obviously a mere proposition:

Definition 4.10. Let $E \colon B\to \mathcal U$ be a dependent type. We say $E$ is an $F$ -fiber bundle if

\begin{equation*} \prod _{b \colon\! B}\| E(b)\simeq F\| \text {.}\end{equation*}

Again, we will switch between the dependent and non-dependent versions by taking fibers of $p$ and the sum, respectively. To arrive at the dependent version, we can directly use the classifying morphism $\chi$ of an $F$ -fiber bundle $p \colon E \to B$ to construct a term of

\begin{equation*} \prod _{b \colon\! B}\| p^{-1}(b)\simeq F\|, \end{equation*}

since all points $\chi (b) \colon{\mathrm{BAut}}(F)$ are of the form $(F^{\prime},\gamma )$ , with $F^{\prime}\simeq p^{-1}(b)$ by the pullback square and $\gamma$ a proof that $F^{\prime}$ is merely equivalent to $F$ .

Now, for the converse, let

\begin{equation*} E \colon B\to \mathcal U \end{equation*}

be an $F$ -fiber bundle by $t \colon \prod _{b \colon B}\| E(b)\simeq F\|$ . Then the classifying map is given by $(x \colon B)\mapsto (E(b), t_x)$ and the pullback square is given by pasting:Footnote ²⁰

We will conclude this section by showing that all our definitions of fiber bundles are equivalent and discuss some examples. The equivalence is most efficiently proven by establishing the equivalence of the two dependent definitions first:

Theorem 4.11. Let $F$ be a type and $E \colon B\to \mathcal U$ be a dependent type, then

\begin{equation*} \prod _{b \colon\! B}\| E(b)\simeq F\| \end{equation*}

if and only if there is a type $W$ and a surjective $w \colon W\to B$ such that

\begin{equation*} \prod _{x \colon\! W} E(w(x))\simeq F.\end{equation*}

For the proof, we need to construct a trivializing cover at some point.Footnote ²¹ The construction we use is similar to the universal cover and interesting on its own:

Definition 4.12. Let $E \colon B\to \mathcal U$ be an $F$ -fiber bundle by $t \colon \prod _{b \colon B}\| E(b)\simeq F\|$ , then

\begin{equation*} W \colon \equiv \sum _{b \colon\! B} E(b) \simeq F \end{equation*}

together with its projection to $B$ is the canonical trivializing cover of $p$ .

The given $t$ directly proves that this projection is surjective. Let us denote this projection by $w \colon W\to B$ , then for all $x \colon W$ , with $x=(b,e)$ , we have

\begin{equation*} E(w(x))\simeq E(\pi _1(b,e))\simeq F\end{equation*}

by transport and $e \colon E(b)\simeq F$ itself.

4.2 $V$ -Manifolds

A smooth $n$ -manifold is a space that is locally diffeomorphic to $\mathbb{R}^n$ , Hausdorff, and second countable. A detailed comparison between the notion of $V$ -manifold, which will be introduced below, and other notions of manifold may be found in (Khavkine and Schreiber, Reference Khavkine and Schreiber2017)[3.3, 3.4] and (Myers, Reference Jaz Myers2022b)[Section 5, p. 40 ff].

The definition of $V$ -manifolds just mimics the property of being locally diffeomorphic to a fixed space, which we will only require to be homogenous (as defined in Definition 3.9). A covering $(U_i)_{i\in I}$ with $U_i\simeq \mathbb{R}^n$ of an $n$ -manifold $M$ yields a surjective local diffeomorphism

\begin{equation*} \coprod _{i\in I}U_i \to M.\end{equation*}

By projecting, there is also a local diffeomorphism from $U:\equiv \coprod _{i\in I}U_i$ to $\mathbb{R}^n$ . So in total, we have a span of local diffeomorphisms where the right one is surjective.

In applications, more general vector spaces might take the role of $\mathbb{R}^n$ – so we will follow the literature and use the letter $V$ in the more abstract definition of a $V$ -manifold below. Instead of local diffeomorphisms, we will use formally étale maps. This is justified by the external calculation (Khavkine and Schreiber, Reference Khavkine and Schreiber2017) [Proposition 3.2], which shows that formally étale maps between two smooth manifolds are exactly the local diffeomorphisms.

Definition 4.14. Let $V$ be a homogeneous type. A type $M$ is a $V$ -manifold if there is a span

where the left map is formally étale and the right map is formally étale and surjective.

There is one trivial example:

Examples 4.15. Let $V$ be a homogeneous type, then $V$ is a $V$ -manifold witnessed by the span:

Less obvious are the following two ways of producing new $V$ -manifolds. However, without adding anything to our type theory to make the modality $\Im$ more specific, we cannot hope for examples that are not given as homogeneous types. What could be added will be discussed at the beginning of the next section.

The statement in (a) is a variant of the classical fact that the tangent bundle of a manifold is a manifold, but in our case, the infinitesimal information is kept separate.

One important special case of part (b) of the lemma is that any formal disk $\mathbb{D}_x$ of $M$ is a $V$ -manifold.

In the following, let $V$ be homogeneous and $M$ be a fixed $V$ -manifold. The definition of $V$ -manifolds entails a stronger local triviality condition on the formal disk bundle of $M$ than was discussed in the last section about $F$ -fiber bundles since there has to be a formally étale trivializing cover.Footnote ²⁴ This property of the trivializing cover will not be used in the following lemma.

The classifying morphism $\tau _M$ is compatible with formally étale maps in the sense of the following remark.

Remark 4.18. Let $\varphi :N\to M$ be formally étale, then $N$ is also a $V$ -manifold by 4.16 . There is a 2-cell given by the differential of $\varphi$ :

Proof. We proved in Lemma 3.16 (a) that the differential of a formally étale map is an equivalence at all points. Applied to $\varphi$ , this fact may be expressed in the following way:

\begin{equation*} d\varphi :\prod _{x:N}\mathbb {D}_x\simeq \mathbb {D}_{\varphi (x)} \end{equation*}

This yields a 2-cell of the desired type since the formal disks $\mathbb{D}_x$ and $\mathbb{D}_{\varphi (x)}$ are merely equivalent to $\mathbb{D}_e$ for all $x$ .

This will be useful when we work with $G$ -jet-structures in the next section.

4.3 $G$ -Jet-structures

Intuitively, the classifying morphism $\tau _M:M\to{\mathrm{BAut}}(\mathbb{D}_e)$ of a $V$ -manifold $M$ describes how the formal disk bundle is glued together using automorphisms of $\mathbb{D}_e$ . Lifts of $\tau _M$ along the delooping $\mathrm{B}G\to{\mathrm{BAut}}(\mathbb{D}_e)$ of a morphism from a group $G$ will be called $G$ -jet-structures.

Some simple, classical $G$ -structures on $\mathbb{R}^n$ -manifolds (or $n$ -manifolds) only consider automorphisms of the tangent space, so their delooped automorphism group $\mathrm{B}{\mathrm{GL}}_n(\mathbb{R})$ Footnote ²⁵ takes the role of ${\mathrm{BAut}}(\mathbb{D}_e)$ in our $G$ -jet-structures. For an $\mathbb{R}^n$ -manifold, the type ${\mathrm{BAut}}(\mathbb{D}_e)$ will be a delooping of the infinite jet group $\mathrm{J}^\infty _n(\mathbb{R})$ . It is known (see e.g., (Kolař and Slovák, Reference Peter, Ivan Kolař and Slovák1993, p. 131)), that the kernel of the projection $\mathrm{J}^\infty _n(\mathbb{R})\to{\mathrm{GL}}_n(\mathbb{R})$ is contractible. The projection also has a section given by extending linear automorphisms to the formal disk. This situation is nice enough, that we expect no problems with lifting our general classifying map $\tau _M:M\to{\mathrm{BAut}}(\mathbb{D}_e)$ , to a classical classifying map $M\to \mathrm{B}{\mathrm{GL}}_n(\mathbb{R})$ in the case of $\mathbb{R}^n$ -manifolds – which would admit reusing the classical examples.

There are lots of interesting classical examples of structures on manifolds that can be encoded as $G$ -structures. We give a list of examples of what group morphisms – which are almost always inclusions of subgroups – encode structures on a smooth $n$ -manifold as $G$ -structures. Some of the examples assume $n=2d$ .

For a definition of $\mathrm{O}(n)$ - and ${\mathrm{GL}}(d,\mathbb{C})$ -structures, see (Chern, Reference Chern1966). Note that in all of the above examples, $G$ is a 1-group,Footnote ²⁶ yet our theory also supports higher groups. The string 2-group and the fivebrane 6-group are examples of higher $G$ -structures of interest in physics. See (Sati et al., Reference Sati, Schreiber and Stasheff2009) for details and references. In the classical theory, torsion-free $G$ -structures are to $G$ -structures what symplectic structures are to almost symplectic structures. We will give a candidate analog of torsion-freeness for $G$ -jet-structures at the end of this section.

We will now turn to the formal treatment of $G$ -jet-structures on $V$ -manifolds and the construction of the moduli spaces of these structures. From now on, let $V$ be a homogeneous type. As we learned in the last section in 4.17, the formal disk bundle of a $V$ -manifold $M$ is always classified by a morphism $\tau _M:M\to{\mathrm{BAut}}(\mathbb{D}_e)$ , where $\mathbb{D}_e$ is the formal disk at the unit $e:V$ . Since this is the only feature of a $V$ -manifold that we need for the constructions in this section, we will work with the following more general class of spaces, where $D$ is an arbitrary type which takes the role of $\mathbb{D}_e$ .

We are interested in the case $D\equiv \mathbb{D}_e$ for $e:V$ , meaning that $M$ is a formal $\mathbb{D}_e$ -space if $\prod _{x:M}\|\mathbb{D}_x\simeq \mathbb{D}_e\|$ . In 4.16, we saw that we can “pullback” the structure of a $V$ -manifold along a formally étale map. Formal $\mathbb{D}_e$ -spaces behave the same way by virtue of the 2-cell we already saw in 4.18.

Lemma 4.21. Let $M$ be a formal $\mathbb{D}_e$ -space. For any formally étale $\varphi :N\to M$ , $N$ is also a formal $\mathbb{D}_e$ -space and there is the triangle:

Proof. First, the triangle in the statement exists for a formally étale map between any types if ${\mathrm{BAut}}(\mathbb{D}_e)$ is replaced with the universe:

By assumption, we know that $(x:M)\mapsto \mathbb{D}_x$ lands in ${\mathrm{BAut}}(\mathbb{D}_e)$ . But $\varphi$ is formally étale, so we have $d\varphi :\prod _{x:N}\mathbb{D}_x\simeq \mathbb{D}_{f(x)}$ . The latter may be truncated and composed with $\tau _M:\prod _{x:M}\|\mathbb{D}_x\simeq \mathbb{D}_e\|$ to get $\tau _N:\prod _{x:N}\|\mathbb{D}_x\simeq \mathbb{D}_e\|$ . So both maps to $\mathcal U$ factor over ${\mathrm{BAut}}(\mathbb{D}_e)$ .

From now on, we assume that $\mathrm{BG}$ is a connected, pointed type and $(\ast =_{{\mathrm{BG}}}\ast )\simeq \mathrm{G}$ . We will define $G$ -jet-structuresFootnote ²⁸ or reductions of the structure group, a synonym hinting that in a lot of cases, $G$ is a subgroup of $\mathrm{Aut}(\mathbb{D}_e)$ . We will not restrict ourselves to reductions to subgroups and look at general pointed maps ${\mathrm{BG}}\to{\mathrm{BAut}}(\mathbb{D}_e)$ , which correspond to general group homomorphisms $\mathrm{G}\to \mathrm{Aut}(\mathbb{D}_e)$ .

Definition 4.22. Let $\chi :{\mathrm{BG}}\to{\mathrm{BAut}}(\mathbb{D}_e)$ be a pointed map and $M$ be a formal $\mathbb{D}_e$ -space. A $G$ -jet-structure on $M$ is a map $\varphi :M\to{\mathrm{BG}}$ together with a 2-cell $\eta :\chi \circ \varphi \Rightarrow \tau _M$ :

We write

\begin{equation*} {G\text {-str}}(M):\equiv \sum _{\varphi :M\to {\mathrm {BG}}}(\chi \circ \varphi \Rightarrow \tau _M) \end{equation*}

for the type of $G$ -jet-structures on $M$ .

The special case $G=1$ turns out to be interesting – a $1$ -jet-structure on a formal $\mathbb{D}_e$ -space is nothing else than a trivialization of the formal disk bundle, like we produced in 3.12 for any homogeneous type. This provides us with an example of a $1$ -jet-structure, whose construction is, in spite of the name we will give below, not entirely trivial.

Definition 4.23. The trivial $1$ -jet-structure on $V$ is the trivialization $\psi :\prod _{x:V}\mathbb{D}_e\simeq \mathbb{D}_x$ constructed in 3.12 :

Since we have pointed maps, there is a triangle for any $\chi :{\mathrm{BG}}\to{\mathrm{BAut}}(\mathbb{D}_e)$ :

So we can define a trivial structure in the same way as above for arbitrary $G$ . Let us fix a pointed map $\chi :{\mathrm{BG}}\to{\mathrm{BAut}}(\mathbb{D}_e)$ from now on.

Definition 4.24. Let $\mathcal T:\mathbb{D}_e\simeq \chi (\ast )$ be the transport along the equality witnessing that $\chi$ is pointed. The trivial $G$ -jet-structure on $V$ is given by $\psi^{\prime}_{x}:\equiv \psi _x\circ \mathcal T$ :

An important notion that we will introduce at the end of this section is a torsion-free $G$ -structure. In some sense to be precise, these $G$ -jet-structures will be trivial on all formal disks. Before we can do this, we need to be able to restrict $G$ -jet-structures to formal disks, or more generally, to pull them back along formally étale maps.

Construction 1 (of $f^\ast$ ). Let $\Theta \equiv (\varphi, \eta ):{G\text{-str}}(M)$ . Then we can paste the triangle constructed in 4.18 to the triangle given by $(\varphi, \eta )$ :

We define the result of the pasting to be $f^\ast (\varphi, \eta ):{G\text{-str}}(N)$ . Or, put differently:

\begin{equation*} f^\ast (\varphi, \eta ):\equiv (\varphi \circ f,(y:N)\mapsto \eta _{f(y)}{\,\bullet \,} df_y^{-1}). \end{equation*}

Pulling back $G$ -jet-structures is 1-functorial in the following sense.

Remark 4.26. Let $f:N\to M$ , $g:L\to N$ be formally étale, and $M$ a formal $\mathbb{D}_e$ -space. Then there is a triangle

Proof. By 3.6, we have

\begin{equation*} d(f\circ g)_x=(df)_{g(x)}\circ dg_x.\end{equation*}

In diagrams, this yields a 3-cell between the pasting of

and

This means the 2-cells we paste when applying $(f\circ g)^\ast$ or $g^\ast \circ f^\ast$ are equal, so the functions must be equal, too.

Let $M$ be a fixed formal $\mathbb{D}_e$ -space from now on. The final definition of this article is that of a torsion-free $G$ -jet-structure. The aim is to ask if a $G$ -jet-structure “looks like the trivial $G$ -jet-structure everywhere on an infinitesimal scale.” To do this, we restrict a $G$ -jet-structure to the formal disk at a point and compare it to the trivial $G$ -jet-structure on $\mathbb{D}_e$ . So let us fix a notation for this structure:

Definition 4.27. Let $\xi :{G\text{-str}}(V)$ be the trivial $G$ -jet-structure from 4.24 and $ u_e:\mathbb{D}_e\to V$ the formal disk inclusion. Then

\begin{equation*} \xi _e:\equiv u_e^* \xi \end{equation*}

is the trivial $G$ -jet-structure on $\mathbb{D}_e$ .

But a priori, we have no means of comparing $G$ -jet-structures on formal disks with this trivial structure, so we need formally étale maps from all formal disks to $\mathbb{D}_e$ . For formal $\mathbb{D}_e$ -spaces, we merely have an equivalence from any formal disk to $\mathbb{D}_e$ . More precisely, by 4.10, we have

\begin{equation*} \tau _M:\prod _{x:M}\|\mathbb {D}_x\simeq \mathbb {D}_e\|. \end{equation*}

And by pulling back to the canonical cover $w:W\to M$ from 4.12, we get

\begin{equation*} \omega _M:\prod _{x:W}\mathbb {D}_{w(x)}\simeq \mathbb {D}_e\end{equation*}

This is enough to make the indicated comparison.

Definition 4.28. A $G$ -jet-structure $\Theta$ on $M$ is torsion-free if

\begin{equation*} \prod _{x:W} \| (\omega _{M,w(x)}^{-1})^*u_{w(x)}^*\Theta = \xi _e\|\equiv :\text {torsion-free}(\Theta ) \end{equation*}

It turns out that even for the trivial 1-jet-structure on $V$ , torsion-freeness is non-trivial. If the trivial 1-jet-structure is left-invariant as defined below, it is an example of a torsion-free 1-jet-structure. To match classic notions, we assume that the equivalences of the homogeneous structure are left translations.

Definition 4.29. The trivial $G$ -jet-structure $\xi$ on $V$ is called left-invariant, if the following condition holds:

\begin{align*} \prod _{x:V}t_x^\ast \xi =\xi \end{align*}

If our homogeneous space $V$ is a Lie-Group, the trivial 1-jet-structure is constructed the same way as the Maurer–Cartan form, which satisfies the equation above. Turning this around, we get the following example:Footnote ²⁹

Proof. We will use the following equation given by the group structure:

\begin{equation*} t_{t_x(y)}=t_{xy}=t_x\circ t_y\end{equation*}

Evaluating at $e$ and using the chain rule 3.6 yields:

\begin{equation*} d(t_{t_x(y)})_e=\left (dt_x\right )_{t_y(e)}\circ \left (dt_y\right )_e=\left (dt_x\right )_y\circ (dt_y)_e=(dt_y)_e{\,\bullet \,}\left (dt_x\right )_y \end{equation*}

The latter equality is just moving our equation to ${\mathrm{BAut}}(\mathbb{D}_e)$ .

Now for the trivial $G$ -jet-structure $\xi \equiv \left (\_\mapsto \ast, y\mapsto \left (dt_y\right )_e\right )$ , we can calculate

\begin{align*} t_x^\ast \xi &=\left (\_\mapsto \ast, y\mapsto \left (dt_{t_x(y)}\right )_e{\,\bullet \,} \left (dt_x\right )_y^{-1}\right ) \\ &=\left (\_\mapsto \ast, y\mapsto \left ( \left ( dt_y \right )_e{\,\bullet \,}\left ( dt_x\right )_y \right ){\,\bullet \,} \left (dt_x\right )_y^{-1}\right ) \\ &=\left (\_\mapsto \ast, y\mapsto \left (dt_y \right )_e\right ) \\ &=\xi \end{align*}

Proof. Let $t_x$ be the translation to $x:V$ given by the homogeneous structure on $V$ and $\xi \equiv (\_\mapsto \ast, x\mapsto \left (dt_x\right )_e)$ the trivial $G$ -jet-structure on $V$ . Then for all $x:V$ , we have a square of formally étale maps:

By 4.26, we get the following formula:

\begin{equation*} u_e^\ast t_x^\ast \xi =dt_x^\ast u_{x}^\ast \xi \end{equation*}

By 4.30, we can simplify the left-hand side:

\begin{equation*} u_e^\ast \xi =dt_x^\ast u_{x}^\ast \xi \end{equation*}

The left-hand side is the trivial structure on $\mathbb{D}_e$ and we have to identify the right-hand side with the term $(\omega _{M,x}^{-1})^\ast u_{\omega (x)}^\ast \xi$ from 4.28, where $\omega =\mathrm{id}_V$ . This amounts to an identification $dt_x^\ast u_{x}^\ast \xi =u^\ast _x\xi$ , which is given by 3.12.

Since torsion-freeness – as we defined it – is a proposition, the type of torsion-free $G$ -jet-structures is a subtype of the type of $G$ -jet-structures. The latter should be distinguished from the moduli space of $G$ -jet-structures on $M$ , which is the quotient of the type of $G$ -jet-structures by the action of the automorphism group of $M$ . If $M$ is a 0-type, we could just build this quotient as a higher inductive type, but this is a bit unsatisfactory and not the most pleasant definition to work with. A more promising approach is to use that the quotient of an action given as a dependent type $\rho :{\mathrm{BG}}\to \mathcal U$ is just $\sum _{x:{\mathrm{BG}}}\rho (x)$ . To make this approach work, the author reformulated a lot of the original theory in (Wellen, Reference Wellen2017). With the present version, we will see that this construction works without considerable effort.

To realize the construction of the moduli space as a dependent sum, we need to note that the definition of $G$ -jet-structures is actually a dependent type over ${\mathrm{BAut}}(M)$ .

This means that we can now construct the moduli spaces of $G$ -jet-structures and torsion-free $G$ -jet-structures in a nice way: