Thickened Smooth Sets in Synthetic Orbifold Theory

• Thickened smooth sets are defined as microlinear types with properly finite identifications, capturing internal symmetries in a synthetic framework.
• The methodology leverages the Kock–Lawvere axiom and infinitesimal reasoning to extend classical orbifold and groupoid presentations.
• This synthetic approach reconciles traditional differential geometry with homotopy type theory, paving the way for advances in higher geometry and moduli theory.

An orbifold in the setting of synthetic differential geometry (SDG) and homotopy type theory is defined as a microlinear type for which the type $(x = y)$ of identifications between any two points is properly finite. This approach, formalized in synthetic differential cohesive homotopy type theory, unifies classical concepts via internal models and extends orbifold theory beyond set-level foundations by directly encoding internal symmetries as higher identifications. The methodology leverages infinitesimal geometry, the Kock–Lawvere axiom, and a categorical framework to subsume both traditional manifold-based and groupoid-based presentations of orbifolds (Myers, 2022).

1. Microlinearity and Infinitesimal Varieties

A type $X$ is microlinear if, for every diagram of infinitesimal varieties (the duals of Weil algebras)—i.e., pushouts in the category of infinitesimal spaces—the corresponding diagram of function types into $X$ is a pullback. Formally, for any infinitesimal $\Rb$-pushout

$\begin{tikzcd}[row sep=small] V_1 \ar[r]^{i_2}\ar[d]_{i_1} & V_3 \ar[d]^{i_4}\ V_2 \ar[r]_{i_3} & V_4 \end{tikzcd}$

the induced square

$\begin{tikzcd} X^{V_4}\ar[r]\ar[d] & X^{V_3}\ar[d]\ X^{V_2}\ar[r] & X^{V_1}\pullbackcorner \end{tikzcd}$

is a pullback in the type universe. The tangent space at $x$,

$T_xX = \{\,v : D \to X \mid v(0) = x \,\}$

then naturally acquires a commutative $\Rb$-module structure. Microlinearity generalizes the behavior of smooth manifolds to a broader class of objects within SDG and type theory (Myers, 2022).

2. The Kock–Lawvere Axiom and Weil Algebras

The Kock–Lawvere axiom is fundamental in SDG, asserting for first-order infinitesimals $D = \{\epsilon: \Rb \mid \epsilon^2 = 0 \}$ that every map $f: D \to \Rb$ uniquely decomposes as $f(\epsilon) = f(0) + b\epsilon$ for $b \in \Rb$. This axiom, in type-theory form,

$\Rb[x]/(x^2) \cong (D \to \Rb)$

ensures the internal logic is compatible with differentiation and infinitesimal reasoning. Microlinearity extends this by requiring the type $X$ to satisfy analogous lifting properties for all Weil algebras, not just for $D$ (Myers, 2022).

3. Orbifolds as Microlinear Types with Properly Finite Identification Types

An orbifold is defined as a microlinear type $X$ such that each identification type $(x = y)$ is properly finite: discrete and a subquotient of a finite set. Explicitly,

$$(\forall x, y : X.\; (x = y)\text{ is properly finite}) \wedge X\;\text{microlinear}$$

This encodes the classical intuition that orbifold points may have finite local symmetry groups, now interpreted as finite path spaces or automorphism types present intrinsically in the homotopy theory rather than externally as groupoid data (Myers, 2022).

Proper finiteness is characterized via existence of an association $r: S \times \{0, ..., n-1\} \to \Prop$ providing single-valued coverings; this ensures direct compatibility with the topological and stack-theoretic foundations of orbifolds.

A key theorem establishes that any proper étale Lie groupoid's Rezk completion (i.e., the stack or higher groupoid object it presents) is a microlinear type with properly finite isotropy, hence an orbifold in the synthetic sense. Microlinearity descends along surjective Im-étale maps and isotropy fibers are properly finite by Dubuc–Penon compactness and discreteness results (Myers, 2022).

4. Mapping Spaces and Internal Symmetries

In the synthetic framework, mapping spaces between orbifolds are simply function types—maps are just dependent functions on points. For homotopy quotients $X/G$,

$\Map(A, X/G) \cong \left\{ \pi: C \to A \mid C\ \text{is a %%%%16%%%%-torsor} \right\}_{/\cong} \times_G \Map_G(C, X)$

This internalizes mapping stacks and ensures the synthetic definition recovers the standard orbit stack mapping spaces for global quotient orbifolds. For configuration spaces $X^n / n!$, maps classify branched covers, aligning with classical moduli problems (Myers, 2022).

5. Dubuc–Penon Compactness Versus Open Cover Compactness

Dubuc–Penon compactness is a synthetic axiom stating that for any $A : \Prop$ and $B : K \to \Prop$,

$$\left( \forall k : K.\; A \vee B(k) \right) \implies A \vee (\forall k : K.\; B(k))$$

A set $K$ is Dubuc–Penon compact if this holds; a map is proper if preimages of DP-compact subtypes are DP-compact. In second-countable manifolds, any discrete, DP-compact subset is refinement-subcompact and subfinitely enumerable, thus yielding proper finiteness of isotropy fibers in proper étale groupoids. The synthetic and open-cover notions of compactness are compared via countable and finite subcovers, with DP-compactness implying strong compactness properties in the context of orbifolds (Myers, 2022).

6. Illustrative Examples

All classical orbifolds, including finite-quotient tori, weighted projective spaces, moduli of elliptic curves, and configuration spaces, arise as global quotients in the synthetic theory. For instance:

• Finite-quotient torus: $\Rb^n / \Gamma$, with $\Gamma$ a finite group.
• Weighted projective space: Described via homotopy quotients involving $U(1)$-lines and their tensor powers.
• Moduli of elliptic curves: Modeled as $\mathfrak{h}/SL_2(\mathbb{Z})$.
• Configuration space: $X^n / n!$ as the homotopy quotient representing unordered points.

Each example is microlinear (by discrete group quotient) and possesses properly finite automorphism types by group-theoretic arguments. The synthetic formalism enables direct computation and manipulation of such orbifolds as first-class objects within type theory (Myers, 2022).

7. Impact and Prospects for Future Research

This synthetic framework for orbifolds via microlinear types and finite identification types generalizes previous presentations based on groupoids and stacks. The approach reconciles internal and external symmetries, supports pointwise maps, and connects the Dubuc–Penon and open-cover notions of compactness to proper finiteness. The theory paves the way for generalizing SDG tools to higher-categorical and homotopical settings, and it facilitates future developments in higher geometry, moduli theory, and generalized differential structures within homotopy type theory (Myers, 2022).