Tangentads: Ads, AI & Category Theory

• Tangentads in advertising use deep CNN embeddings and visual similarity metrics to seamlessly integrate native ads within image searches.
• In AI web contexts, Tangentads employ semantic overlays and DOM strategies to create machine-readable ads that preserve user experience and trust.
• In category theory, Tangentads formalize differential-geometric constructs via a strict 2-categorical framework unifying various tangent-like notions.

“Tangentads” is a polysemous research term used for distinct constructions in separate literatures. In advertising and human–computer interaction, it denotes ad systems that are made visually congruent with surrounding content or, in a later usage, machine-readable for AI web agents (Kalantidis et al., 2016, Nitu et al., 17 Jul 2025). In category theory, it denotes a formal 2-categorical framework that internalizes tangent structures and their geometric constructions across tangent categories, tangent monads, tangent fibrations, restriction categories, and related settings (Lanfranchi, 24 Mar 2025, Lanfranchi, 21 Jan 2026, Lanfranchi, 19 Sep 2025). Because these usages are technically unrelated, the term requires domain qualification in scholarly discussion.

1. Visually congruent Tangentads in image search

In the advertising literature, Tangentads are visually congruent, native advertisements designed to “blend” with the look and feel of the image search results they accompany. The defining criterion is not only semantic or topical relevance to the query or user, but also visual similarity to the returned image set, so that the ad is perceptually coherent in color, composition, texture, and content style while remaining clearly disclosed as sponsored (Kalantidis et al., 2016).

The system assumes a text query $q$ from user $u$, an image result set $X=\{x_i\}$, and a pool of relevant ads $A=\{a_j\}$ already filtered for semantic or topical relevance. Tangentads then perform a second-stage visual selection and placement step. In the practical implementation, images and ads are represented by deep CNN embeddings: AlexNet fc7 features of dimension $4096$, reduced by PCA trained on YFCC100M to $d=128$, and $L^2$-normalized before similarity computation. For web-scale operation, the image corpus is compressed with Multi-LOPQ product quantization, splitting each 128-dimensional vector into $M=16$ sub-vectors with 1 byte per sub-code and two coarse quantization indices of vocabulary size $2^{13}$. Total storage is approximately $154$ bits, or about $u$0 bytes per image. Ads remain uncompressed, and asymmetric distance computation is used between uncompressed ads and compressed images.

The core selection objective is vector-to-set visual congruence. Using Euclidean distance,

$u$1

the robust aggregate used in the paper is

$u$2

and the selected ad is

$u$3

The paper also notes cosine similarity, a nearest-neighbor alternative, and weighted variants using weights $u$4 to reflect text relevance rank, freshness, or importance. A reciprocity check can then be imposed: if $u$5 is the nearest image to $u$6 in $u$7, and $u$8 is the nearest ad in $u$9 to $X=\{x_i\}$0, require $X=\{x_i\}$1.

Placement is formulated as a local neighborhood optimization on a 2D image grid. For position $X=\{x_i\}$2 with neighbor set $X=\{x_i\}$3, the cost is

$X=\{x_i\}$4

and the placement rule is

$X=\{x_i\}$5

The implemented strategies are: preserving original ordering by inserting the ad next to its nearest image and choosing left or right based on similarity; local reordering by placing the ad between its two nearest neighbors and clustering the immediate $X=\{x_i\}$6- or $X=\{x_i\}$7-neighborhood by ascending proximity; and a global visual layout that projects images and the ad into 2D with t-SNE, clusters them with fast Mean Shift, and tiles the grid greedily by cluster. In the global strategy, an ad is rejected if it forms a singleton cluster.

The empirical evaluation used 896 qualified US workers on Amazon Mechanical Turk, with eight queries across five topics—animals, cars, fashion, movies, and TV series—and three conditions per query: Most Similar, Random, and Least Similar. The primary user-experience measure was the Likert item “The ad experience was of very high quality” (Q8). Agree/Strongly Agree responses were $X=\{x_i\}$8 for Most Similar, $X=\{x_i\}$9 for Random, and $A=\{a_j\}$0 for Least Similar, with $A=\{a_j\}$1 for Most Similar versus Random and $A=\{a_j\}$2 for Most Similar versus Least Similar. For perceived relevance to the image set (Q5), the reported $A=\{a_j\}$3-values were $A=\{a_j\}$4 for Most Similar versus Random and $A=\{a_j\}$5 for Most Similar versus Least Similar. Ad visibility remained stable: “Saw an ad” was approximately $A=\{a_j\}$6 across all conditions, specifically $A=\{a_j\}$7, $A=\{a_j\}$8, and $A=\{a_j\}$9. Brand recall for the correct ad brand was $4096$0 for Most Similar, $4096$1 for Random, and $4096$2 for Least Similar. Time on page for workers who saw an ad was approximately $4096$3 seconds across conditions, with means $4096$4s, $4096$5s, and $4096$6s and standard deviations of about $4096$7–$4096$8s.

The result is a native advertising mechanism specialized to image-heavy interfaces: semantically targeted ads are reranked by visual congruence and inserted so as to minimize perceptual discontinuity. A plausible implication is that Tangentads treat image search as an explicitly visual ranking-and-layout problem rather than a purely keyword-matching problem.

2. Machine-readable Tangentads for AI web agents

A later, separate usage defines Tangentads as a system blueprint for making online ads reliably detectable, interpretable, and actionable for AI web agents without degrading human UX or trust (Nitu et al., 17 Jul 2025). This usage is distilled from experiments on a faithful React-based clone of the Tiroler Tageszeitung site with realistic DOM complexity and ad inventory, including static banners, GIFs, carousels, videos, cookie dialogues, and paywalls.

The evaluation used DOM-centric Browser Use with Playwright version $4096$9 and GPT-4o, Claude 3.7 Sonnet, and Gemini 2.0 Flash, together with the pixel-centric OpenAI Operator. Ten realistic user tasks were run ten times per model, yielding 300 baseline trials overall, with additional follow-ups on sticky CTA removal, CTA implementation variants, hidden-text and ARIA strategies, deep-scroll coupon hunts, dynamic media, and cookie banner variants. Logging included per-run JSON action logs for DOM agents, final answers, animated GIF screen recordings, and human annotation for Operator outcomes.

The reported behavioral regularities are highly specific. The agents displayed severe satisficing and rarely explored beyond two viewports. Quantitatively, average scroll_down actions per run were mean $d=128$0 for Gemini 2.0 Flash, mean $d=128$1 for GPT-4o, and mean $d=128$2 for Claude 3.7 Sonnet, with maxima $d=128$3, $d=128$4, and $d=128$5, respectively. In an infinite-scroll test where a coupon was placed on page 10, none of the 30 runs reached sufficient depth. Purely visual CTA text embedded in banners did not elicit clicks from DOM-centric agents, and Operator also did not click such banners in the tested setup. By contrast, real semantic button overlays or off-screen text labels in the DOM did induce interaction: with a real <button> overlay on side banners, Claude and GPT-4o clicked banners in $d=128$6 runs and Gemini in $d=128$7; with an off-screen <span> not marked aria-hidden, banner interaction was near-perfect at $d=128$8, $d=128$9, and $L^2$0. ARIA-only labels helped GPT-4o in $L^2$1 runs and Gemini in $L^2$2, but not Claude; data-LLM-description had $L^2$3 effect across models.

Five design principles constitute the Tangentads blueprint. Semantic overlays add real DOM controls such as <button> or <a> with role="button", aria-label, and descriptive text. Hidden labels use off-screen text linked to the CTA target so that agents can parse clickability and offer details; the blueprint specifically recommends avoiding aria-hidden on labels intended for agents. Top-left placement exploits the observed top-to-bottom, left-to-right traversal and the empirical left-banner bias: when identical banners appeared left and right, agents interacted exclusively with the left one. Static frames provide DOM-text equivalents for dynamic offers because DOM agents rarely wait for videos or carousels; dialogue replacement converts native prompts into accessible HTML modals with real “Accept,” “Reject,” and “Manage settings” buttons.

The same study reports trust and safety failures. When sweepstake participation required a purchase, GPT-4o and Claude 3.7 Sonnet subscribed in $L^2$4 of trials, and Gemini 2.0 Flash in $L^2$5. Cookie-handling behavior varied by banner type: for a non-essential sticky footer, GPT-4o accepted $L^2$6, Claude $L^2$7, and Gemini $L^2$8; for a modal blocker, GPT-4o accepted $L^2$9, Claude $M=16$0, and Gemini $M=16$1; for a predatory modal, acceptance was $M=16$2 for all models. The proposed safeguard layer therefore includes explicit cost disclosure in hidden labels, intermediate confirmation pages for paid actions, server-side enforcement preventing auto-subscription without an explicit “Yes” click, and logging of CTA element IDs, aria-labels, hidden-label use, scroll depth, viewport visibility, and conversion outcomes.

This version of Tangentads is not about visual congruence. It is an accessibility- and markup-oriented ad design framework for autonomous browsing systems. The paper did not report $M=16$3-values, confidence intervals, or effect sizes, so the quantitative guidance is observational rather than inferential.

3. Tangentads as a formal 2-categorical notion

In category theory, a tangentad is a 2-categorical formalization of tangent categories that generalizes many tangent-like notions across multiple ambient settings (Lanfranchi, 24 Mar 2025). The motivating problem is that tangent category theory has acquired many “flavours”—tangent monads, tangent fibrations, restriction tangent categories, reverse tangent categories, strong display tangent categories, infinitesimal objects—and the interaction of these flavours with constructions such as vector fields or bundles is more naturally studied at the 2-categorical level.

Formally, in a strict 2-category $M=16$4, a tangentad on an object $M=16$5 can be given in Leung form as a strong monoidal functor

$M=16$6

preserving the fundamental tangent limits pointwise. Unpacked internally, this is equivalent to an endo-1-cell $M=16$7 equipped with 2-cells

$M=16$8

satisfying the tangent-category coherence data: $M=16$9, $2^{13}$0, coassociativity of $2^{13}$1, the hexagon identities for $2^{13}$2, compatibility of $2^{13}$3 and $2^{13}$4, and local linearity via the relevant pointwise pullback. An optional negative structure is a 2-cell $2^{13}$5 with $2^{13}$6.

Morphisms are likewise internalized. A lax tangent 1-morphism $2^{13}$7 consists of $2^{13}$8 and a 2-cell $2^{13}$9 satisfying additivity and compatibility with lift and flip. Colax tangent morphisms reverse the direction of the distributive law; strong morphisms have invertible $154$0; strict morphisms have $154$1. Tangent 2-morphisms $154$2 are 2-cells $154$3 satisfying the tangent compatibility equation $154$4.

This formalization makes several structural results possible. For any tangentad $154$5, the endo-1-cell $154$6 carries a canonical monad structure with unit $154$7 and multiplication $154$8, and $154$9 is a distributive law of the monad with itself. The assignment $u$00 carries a canonical 2-comonad whose counit is the forgetful 2-functor and whose comultiplication sends $u$01 to $u$02. Special classes include Cartesian tangentads, adjunctable tangentads for which $u$03 and all $u$04 admit left adjoints, and representable tangentads where $u$05.

The framework captures ordinary tangent categories as tangentads in $u$06, tangent monads as tangentads in $u$07, tangent fibrations as tangentads in $u$08, tangent indexed categories as tangentads in $u$09, split restriction tangent categories as tangentads in $u$10, and infinitesimal objects as tangentads in a display-span setting. It also yields a genuine Grothendieck construction in the tangent setting and shows that every tangent monad admits Eilenberg–Moore algebras whenever the underlying monad does.

4. Hom tangent categories, vector fields, and corepresentability

A central refinement of the formal theory is the pointwise tangent structure on Hom-categories (Lanfranchi, 19 Sep 2025). For tangentads $u$11 and $u$12, the Hom-category $u$13 carries a tangent structure by postcomposition with $u$14; if $u$15 is Cartesian, then this Hom-category is a Cartesian tangent category. This pointwise construction is the mechanism through which universal properties can be stated uniformly across all tangentads.

Vector fields provide the canonical example. In an ordinary tangent category, a vector field on $u$16 is a section $u$17 of $u$18. In the formal theory, there is a tangentad $u$19 with a strict tangent morphism $u$20 and a canonical natural transformation

$u$21

which is a vector field in the Hom tangent category $u$22. The induced map

$u$23

is invertible for every tangentad $u$24. This expresses that vector fields are corepresented by the universal vector field $u$25. The same framework yields a 2-functor $u$26, a commutative monoid bundle on the forgetful map $u$27, an abelian group bundle when negatives exist, a Lie algebra bundle with bracket

$u$28

and a 2-monad structure on $u$29 with unit $u$30 and multiplication $u$31.

The article also develops differential objects, differential bundles, and connections by the same Hom-wise corepresentability method and shows that vector fields and connections can be constructed by PIE limits—products, inserters, and equifiers—whenever the relevant axioms are algebraic. This gives the framework a strong syntactic component: once the universal object is identified in the Hom tangent category, functorial transport across flavours follows automatically.

5. Differential objects, bundles, and connections in tangentads

Part II extends the formal theory from vector fields to Euclidean objects, vector bundles, and Koszul-like connections, again by universal properties internal to tangentads (Lanfranchi, 21 Jan 2026). The point of departure is a Cartesian tangentad $u$32 and the pointwise Hom tangent structure described above.

A differential object in a Cartesian tangent category is a commutative monoid $u$33 with a differential projection $u$34 satisfying additivity, linearity, and the universal property that $u$35 is a product diagram. In the tangentad setting, one obtains a tangentad $u$36 of differential objects together with a forgetful strict tangent morphism $u$37 and pointwise natural transformations $u$38, $u$39, and $u$40. The main theorem states that for any tangentad $u$41,

$u$42

Thus differential objects in a Hom tangent category are precisely lax tangent morphisms into $u$43.

Display differential bundles are treated analogously. A tangentad $u$44 carries Base and Tot morphisms to $u$45 and a universal pointwise bundle

$u$46

Again there is a universal equivalence

$u$47

A canonical restriction of $u$48 to bundles over the terminal object yields $u$49, and the formal theory proves the canonical equivalence

$u$50

This is the tangentad-level extension of the standard tangent-category equivalence between differential objects and differential bundles over the terminal object.

Connections are organized in several tangentads: $u$51 for vertical linear connections, $u$52 for horizontal linear connections, $u$53 for linear connections, and affine variants $u$54, $u$55, and $u$56. For a differential bundle $u$57, a vertical linear connection is a retraction $u$58 of the vertical lift; a horizontal linear connection is a section $u$59 of the relevant projection; a linear connection is a pair $u$60 satisfying orthogonality and direct-sum identities. The formal theory then internalizes the classical constructions

$u$61

$u$62

for curvature, and in the affine case

$u$63

for torsion. These become tangent morphisms $u$64, $u$65, and $u$66 defined by universal pullbacks.

The theory also establishes liftings of $u$67, $u$68, and $u$69 along strong Cartesian tangent monads and tangent fibrations, extends these constructions to tangent indexed categories via the Grothendieck 2-equivalence, and develops restriction-category versions by suitable 2-pullbacks selecting total structure maps. Connections admit an algebraic presentation by PIE limits: vertical and horizontal connections arise from inserters and equifiers imposing the connection axioms, and linear connections are then obtained by a 2-pullback plus additional equifiers. Concrete examples include smooth manifolds, where $u$70 recovers Euclidean spaces $u$71, $u$72 recovers smooth vector bundles, and $u$73 recovers Koszul connections; synthetic differential geometry; tangent restriction categories; and reverse tangent categories.

6. Distinctions, search collisions, and usage discipline

The three usages described above are technically independent. The image-search system is a visual retrieval and layout method for native advertising; the AI-agent blueprint is an accessibility and trust framework for machine-readable ads; the categorical theory is a formal language for differential-geometric structure in 2-categories. No shared theorem, algorithm, or ontology connects them directly.

A common search collision is “Tangent Attack,” a hard-label black-box adversarial attack that identifies an optimal tangent point of a virtual hemisphere on the decision boundary to reduce $u$74 distortion, with a semi-ellipsoid generalization for curved boundaries (Ma et al., 2021). That work is not a tangentad theory; it concerns decision-boundary geometry, binary-query attacks, and query efficiency in adversarial machine learning. Its relevance here is terminological rather than conceptual.

This suggests that bibliographic practice should disambiguate the term by domain. In current research usage, “Tangentads” may refer to visually congruent ads in image search, machine-readable ad layers for AI web agents, or formal tangent structures in category theory, and these meanings should not be conflated.