Truncated Disc-Presheaf in Embedding Calculus

• The paper introduces truncated disc-presheaves as presheaves restricted to discs of size at most k, capturing finite configuration data in manifold embeddings.
• It establishes a cardinality filtration analogous to a Taylor polynomial, where each stage refines local embedding and smooth structure information.
• Low-stage truncations reveal tangible geometric data, such as tangent bundle properties, while higher stages encode increasingly complex configuration and normal data.

Searching arXiv for papers on disc-presheaves, truncation, and embedding calculus. In the setting of manifolds and embedding calculus, a truncated disc-presheaf is best understood as a Disc-presheaf restricted to discs of size at most $k$, that is, a $k$-truncated presheaf on the truncated disc category ${}_{d,\le k}^{}$. A Disc-presheaf itself is a presheaf on the category of finite disjoint unions of open $d$-discs, with values in spaces; truncation is therefore categorical and cardinality-based, not a truncation of homotopy groups. In the presentation developed in "Manifolds and Disc-presheaves" (Kupers, 30 Sep 2025), these truncations are the finite stages of the cardinality tower and coincide with the Goodwillie–Weiss polynomial approximations that organize manifold- and embedding-valued functors.

1. Formal setting

The underlying framework begins with a category whose objects are $d$-dimensional smooth manifolds and whose mapping spaces are spaces of smooth embeddings. Inside it sits the full subcategory on those objects diffeomorphic to $S \times {}^d$ for $S$ a finite set. For any category $\mathcal C$, the associated presheaf category is

$$PSh(\mathcal C)\coloneqq \mathrm{Fun}(\mathcal C^{op}, \mathbb S),$$

where $\mathbb S$ denotes the $k$0-category of spaces. A Disc-presheaf is then an object of the presheaf category on the disc subcategory, equivalently a collection of spaces $k$1 for $k$2, contravariantly natural in embeddings of discs into discs (Kupers, 30 Sep 2025).

A smooth manifold $k$3 determines a canonical Disc-presheaf $k$4, called the Disc-presheaf associated to $k$5. It sends a finite disjoint union of discs to the corresponding embedding space into $k$6, and it arises from the restricted Yoneda embedding. This makes the passage from manifolds to Disc-presheaves structural rather than ad hoc: the presheaf records, for each finite configuration of discs, how that configuration embeds in $k$7.

This formulation shifts attention from manifolds as objects to the functorial pattern of their local embedding data. A plausible implication is that truncation should be interpreted as limiting how much finite-configuration information is retained, rather than as imposing a local geometric simplification on the manifold itself.

2. Cardinality truncation

The paper introduces a cardinality filtration of the disc category,

$k$8

Here $k$9 consists of those objects with at most ${}_{d,\le k}^{}$0 discs. A presheaf on ${}_{d,\le k}^{}$1 is called a ${}_{d,\le k}^{}$2-truncated presheaf. The exact phrase "truncated disc-presheaf" is not introduced as a separate formal definition, but this is the notion used throughout the discussion of the cardinality tower and embedding calculus (Kupers, 30 Sep 2025).

The paper is explicit that this truncation is not homotopy ${}_{d,\le k}^{}$3-truncation. It is a truncation by allowing only inputs of cardinality ${}_{d,\le k}^{}$4. Accordingly, the object under study is not a space whose higher homotopy groups vanish above some degree; it is a presheaf whose domain has been restricted to configurations involving at most ${}_{d,\le k}^{}$5 discs.

This categorical meaning is central. Truncating a Disc-presheaf means only remembering embeddings of at most ${}_{d,\le k}^{}$6 discs, and the paper presents this as a finite-stage approximation analogous to a Taylor polynomial of degree ${}_{d,\le k}^{}$7. In that analogy, the hierarchy of stages measures progressively finer interaction data among configurations of points and discs rather than successive derivatives in an algebraic sense.

3. The approximation tower

For a manifold ${}_{d,\le k}^{}$8 and a Disc-presheaf ${}_{d,\le k}^{}$9, the associated tower has stages $d$0, obtained by restricting from the full disc category to the subcategory of at most $d$1 discs. The paper states

$d$2

with initial terms

$d$3

It also identifies the first stage $d$4 in terms of a space $d$5 equipped with a $d$6-plane bundle (Kupers, 30 Sep 2025).

The passage from $d$7 to $d$8 is described concretely. The fiber of

$d$9

is given in terms of fillers in a diagram involving configuration spaces $d$0 and $d$1, together with equivariance under $d$2. The paper emphasizes that this is a concrete description of what the $d$3-th truncation sees: it is controlled by $d$4-point configuration data and the relevant symmetry action.

This description is significant because it identifies the successive layers of truncation with increasingly high-order configuration information. The first stages depend only on low-cardinality embeddings; higher stages encode compatibility data for larger finite configurations. This suggests a precise sense in which the tower resolves manifold information by arity.

4. Descent, goodness, and polynomiality

The finite stages are not merely restrictions of domain. The paper emphasizes that $d$5 satisfies a sheaf condition for a Grothendieck topology of $d$6-covers. An open cover $d$7 of $d$8 is a $d$9-cover if every collection of $S \times {}^d$0 points of $S \times {}^d$1 lies in some $S \times {}^d$2, and it is complete if the same holds for all finite intersections. For complete $S \times {}^d$3-covers,

$S \times {}^d$4

The paper further states that the $S \times {}^d$5 are the initial presheaves under $S \times {}^d$6 that are good and polynomial of degree $S \times {}^d$7 (Kupers, 30 Sep 2025).

The two defining conditions are also made explicit. Being good means taking increasing unions $S \times {}^d$8 to limits. Being polynomial of degree $S \times {}^d$9 means sending certain $S$0-cube diagrams to cartesian cubes. The equivalence between these conditions and the $S$1-descent statement identifies the truncated stages with the polynomial approximations of embedding calculus rather than with arbitrary finite-domain truncations.

This is one of the main structural points of the theory. A truncated disc-presheaf is not merely a partial record of values on small configurations; when promoted to $S$2, it becomes the universal degree-$S$3 approximation satisfying the relevant descent and excision properties.

5. Low stages and geometric information

The first two truncation levels admit especially concrete descriptions. A $S$4-truncated presheaf on $S$5 is the same as a space with a $S$6-dimensional vector bundle. For $S$7, this is just $S$8 with its tangent bundle. At this stage, only tangent-bundle-level information is retained (Kupers, 30 Sep 2025).

A $S$9-truncated presheaf contains additional data organized by a square involving $\mathcal C$0, $\mathcal C$1, and the diagonal map on $\mathcal C$2. For $\mathcal C$3, the corresponding square is

$\mathcal C$4

The paper interprets this as encoding how pairs of points in the manifold can be separated, and relates it to normal data and Poincaré duality.

Higher truncations detect subtler smooth information. The paper notes that the $\mathcal C$5-truncations of $\mathcal C$6 and $\mathcal C$7 are equivalent for any homotopy sphere $\mathcal C$8, but higher truncations can distinguish them. Specifically, if

$\mathcal C$9

then

$$PSh(\mathcal C)\coloneqq \mathrm{Fun}(\mathcal C^{op}, \mathbb S),$$0

for $$PSh(\mathcal C)\coloneqq \mathrm{Fun}(\mathcal C^{op}, \mathbb S),$$1; a partial converse is also given under the condition $$PSh(\mathcal C)\coloneqq \mathrm{Fun}(\mathcal C^{op}, \mathbb S),$$2 and $$PSh(\mathcal C)\coloneqq \mathrm{Fun}(\mathcal C^{op}, \mathbb S),$$3. This shows that finite truncations can already detect nontrivial smooth-structure phenomena.

6. Relation to embedding calculus and neighboring notions

Disc-presheaves first arose in embedding calculus, and the tower

$$PSh(\mathcal C)\coloneqq \mathrm{Fun}(\mathcal C^{op}, \mathbb S),$$4

is presented as the Goodwillie–Weiss tower of approximations to $$PSh(\mathcal C)\coloneqq \mathrm{Fun}(\mathcal C^{op}, \mathbb S),$$5, obtained by restricting to $$PSh(\mathcal C)\coloneqq \mathrm{Fun}(\mathcal C^{op}, \mathbb S),$$6. For embedding spaces, the same formalism gives polynomial approximations

$$PSh(\mathcal C)\coloneqq \mathrm{Fun}(\mathcal C^{op}, \mathbb S),$$7

through mapping spaces between the associated Disc-presheaves $$PSh(\mathcal C)\coloneqq \mathrm{Fun}(\mathcal C^{op}, \mathbb S),$$8 and $$PSh(\mathcal C)\coloneqq \mathrm{Fun}(\mathcal C^{op}, \mathbb S),$$9. The paper also highlights the pro-object

$\mathbb S$0

indicating that finite truncations may be studied coherently as a pro-system when direct passage to the limit is difficult (Kupers, 30 Sep 2025).

The paper also insists on a hierarchy of notions. A Disc-presheaf is an arbitrary presheaf on the disc category. A $\mathbb S$1-truncated presheaf is its restriction to the objects of cardinality at most $\mathbb S$2. A polynomial degree $\mathbb S$3 object is one satisfying the $\mathbb S$4-descent or cartesian-cube condition. A sheaf-like object in this setting is therefore defined relative to the truncated topology, not by ordinary sheaf conditions on the full category.

This should be distinguished from perverse sheaf theory on a single topological disc. "New description of perverse sheaves on a disc" studies $\mathbb S$5 for the stratification $\mathbb S$6, proving an equivalence with a category $\mathbb S$7 built from a vector space and four complementary subspaces (Olechowski, 2022). That subject concerns constructible and perverse sheaves on a stratified disc, whereas truncated disc-presheaves belong to the embedding-calculus study of presheaves on a category of finite disjoint unions of discs. The shared word "disc" therefore masks two different mathematical settings.

Taken together, these results present truncated disc-presheaves as finite-cardinality stages in a manifold-recognition machine. At $\mathbb S$8 one sees tangent-bundle data; at $\mathbb S$9, pairwise separation and normal information; at higher $k$00, increasingly delicate configuration-space and smooth-structure data. The full Disc-presheaf is recovered only in the limit, but the finite stages are already structured, descent-theoretic, and calculable approximations in the sense of embedding calculus (Kupers, 30 Sep 2025).