Squares Functor in (∞,2)-Categories

• Squares functor is a canonical construction that maps (∞,2)-categories to double ∞-categories, encoding lax commutative squares and companion structures.
• It underpins universal properties such as the span–squares adjunction and Gray tensor product equivalence, unifying diagrammatic approaches in higher category theory.
• Recent research leverages the squares functor for derived algebraic geometry and squares K-theory, linking categorical exactness and homotopy-coherent algebra.

Searching arXiv for papers on the "Squares Functor" and closely related constructions to ground the article in current literature. First, I’ll look for papers explicitly titled around “squares functor,” then broaden to adjacent constructions like span–squares adjunctions and squares K-theory. The term squares functor refers to several mathematically distinct constructions that organize, encode, or classify square-shaped data. In current usage, the most explicit and foundational meaning arises in the theory of $(\infty,2)$-categories, where the squares functor $Sq$ assigns to an $(\infty,2)$-category $C$ a double $\infty$-category whose objects are those of $C$, whose horizontal and vertical 1-cells are 1-cells of $C$, and whose 2-cells are lax commutative squares in $C$ (Loubaton et al., 10 Jul 2025). Closely related work identifies a right adjoint to the span construction as a double $\infty$-category of squares, thereby exhibiting a universal property of square-shaped diagrams in $\infty$-categorical algebra and algebraic $Sq$0-theory (Raptis et al., 8 Jun 2026). In other areas, the phrase is also used informally or analogically for quadratic or squaring constructions, including the squaring operation on derived categories of commutative DG rings (Yekutieli, 2014), square-type classifications of quadratic functors (Hartl et al., 2008), and square-producing mechanisms in Goodwillie calculus and equivariant homotopy theory (Konovalov, 2020). This suggests that the phrase does not designate a single invariant object across all mathematics, but rather a family of constructions whose common feature is the functorial production of square-shaped or quadratic structure.

1. Terminological scope and major usages

In the most direct sense, the squares functor is the construction

$Sq$1

from $Sq$2-categories to complete double $Sq$3-categories, developed to encode lax commutative squares and their companion structure (Loubaton et al., 10 Jul 2025). This is the usage most closely associated with Gaitsgory–Rozenblyum’s $Sq$4-categorical foundations for derived algebraic geometry, where the squares functor interacts coherently with the Gray tensor product and with double $Sq$5-categorical formulations of correspondences and Beck–Chevalley phenomena (Loubaton et al., 10 Jul 2025).

A second, closely related usage appears in the span–squares adjunction. There the right adjoint to the span construction

$Sq$6

is the functor $Sq$7, where $Sq$8 is the double $Sq$9-category of commutative grids in $(\infty,2)$0, and $(\infty,2)$1 denotes horizontal opposite (Raptis et al., 8 Jun 2026). In that setting, squares are not merely diagrams internal to an $(\infty,2)$2-category; they are the universal targets for encoding functors out of span $(\infty,2)$3-categories (Raptis et al., 8 Jun 2026).

A broader use of the phrase occurs in algebraic $(\infty,2)$4-theory. Recent work shows that the $(\infty,2)$5-theory spectra of many assemblers are equivalent to the $(\infty,2)$6-theory of a squares category, yielding the slogan that “all K-theory is squares K-theory” for a substantial class of examples (Kuijper, 1 Dec 2025). Here a squares category is a structure with horizontal and vertical morphisms and distinguished squares satisfying closure axioms, and the relevant functoriality lies in the passage from covering-family data to square-based $(\infty,2)$7-theoretic models (Kuijper, 1 Dec 2025).

Other appearances are more analogical. In Goodwillie calculus, certain canonical homotopy cartesian fracture squares are produced functorially from a homotopy functor, and for the norm functor these squares agree with classical equivariant fracture squares (Konovalov, 2020). In commutative DG algebra, the squaring operation

$(\infty,2)$8

is a quadratic endofunctor needed for rigid complexes and Grothendieck duality (Yekutieli, 2014). In algebraic functor theory, quadratic functors are modeled by square-type algebraic data such as square groups and quadratic modules (Hartl et al., 2008). These constructions share structural affinities with the higher-categorical squares functor, but they are not instances of the same definition.

2. The $(\infty,2)$9-categorical squares functor

The paper “On the squares functor and the Gaitsgory-Rozenblyum conjectures” gives the most precise modern formulation of the squares functor (Loubaton et al., 10 Jul 2025). In that framework, $C$0-categories are modeled as presheaves on $C$1 satisfying Segal-like and completeness conditions, while double $C$2-categories are modeled as bisimplicial spaces whose rows and columns are Segal spaces and whose rows are complete (Loubaton et al., 10 Jul 2025).

For an $C$3-category $C$4, the squares functor $C$5 is characterized by the mapping-space formula

$C$6

This identifies $C$7 as the double $C$8-category whose objects are those of $C$9, whose horizontal and vertical arrows are 1-cells of $\infty$0, and whose 2-cells are lax commutative squares in $\infty$1 (Loubaton et al., 10 Jul 2025).

The construction is obtained by first defining a directed Čech nerve for filtrations $\infty$2, namely

$\infty$3

and then restricting along the “universal” filtration $\infty$4 (Loubaton et al., 10 Jul 2025). This produces an adjunction

$\infty$5

with $\infty$6 the right adjoint (Loubaton et al., 10 Jul 2025).

A basic interpretive point is that $\infty$7 packages both directions of 1-morphism in $\infty$8 and records 2-dimensional laxity explicitly. This suggests that $\infty$9 is a canonical device for passing from a genuinely 2-dimensional category to a double environment where horizontal and vertical composition coexist but remain distinguishable.

3. Universal property and companions

A central achievement of the $C$0-categorical theory is the universal property of the squares functor (Loubaton et al., 10 Jul 2025). The relevant notion is that of a companion in a double $C$1-category: a vertical arrow $C$2 has a companion if there is a horizontal arrow $C$3 and 2-cells $C$4 and $C$5 satisfying triangle identities analogous to those of an adjunction (Loubaton et al., 10 Jul 2025). In $C$6, every vertical arrow has a companion and every horizontal arrow is a companion (Loubaton et al., 10 Jul 2025).

The paper proves that $C$7 is the free completion of the vertical inclusion $C$8 under companions. More precisely, the canonical map $C$9 induces a monomorphism

$C$0

whose image consists of those double functors $C$1 sending every arrow of $C$2 to a vertical arrow in $C$3 admitting a companion (Loubaton et al., 10 Jul 2025). Under a local completeness hypothesis there is also a horizontal analogue using $C$4 (Loubaton et al., 10 Jul 2025).

This universal property generalizes a theorem of Grandis–Paré for strict double categories and supplies a conceptual explanation for why $C$5 is the natural receptacle for lax squares, base-change diagrams, and Beck–Chevalley structures (Loubaton et al., 10 Jul 2025). A plausible implication is that many double-categorical constructions in derived algebraic geometry can be reduced to verifying companion conditions in a target double $C$6-category.

4. Relation to Gray tensor product and the Gaitsgory–Rozenblyum conjectures

The squares functor is tightly linked to the Gray tensor product on $C$7-categories (Loubaton et al., 10 Jul 2025). The paper proves a comparison theorem stating that for $C$8-categories $C$9,

$C$0

equivalently,

$C$1

naturally in $C$2 and $C$3 (Loubaton et al., 10 Jul 2025).

This theorem resolves the last remaining open conjecture among eight foundational claims formulated by Gaitsgory and Rozenblyum concerning the Gray tensor product, the squares functor, and related constructions (Loubaton et al., 10 Jul 2025). The significance is twofold. First, it shows that the Gray product is the $C$4-categorical reflection of a double $C$5-category with horizontal direction $C$6 and vertical direction $C$7. Second, it confirms that the squares construction is not an auxiliary gadget but a structural counterpart to the Gray tensor product (Loubaton et al., 10 Jul 2025).

The same paper records additional structural facts: $C$8 is complete for every $C$9, $\infty$0 is fully faithful, and the Gray tensor product has no nontrivial natural endomorphisms (Loubaton et al., 10 Jul 2025). These results collectively establish the squares functor as a rigid and foundational operation in the homotopy-coherent 2-dimensional algebra underlying derived algebraic geometry.

5. The span–squares adjunction

The paper “The span-squares adjunction” develops a different but closely allied universal characterization (Raptis et al., 8 Jun 2026). Here double $\infty$1-categories are modeled as double Segal spaces, and the span construction is defined as

$\infty$2

where

$\infty$3

For an $\infty$4-category $\infty$5, the associated squares object is

$\infty$6

the double $\infty$7-category of commutative grids in $\infty$8 (Raptis et al., 8 Jun 2026).

The main theorem is the adjunction

$\infty$9

with $\infty$0 fully faithful (Raptis et al., 8 Jun 2026). Equivalently, for $\infty$1 and $\infty$2,

$\infty$3

This gives a universal property of span $\infty$4-categories: to define a functor out of $\infty$5 is the same as to define a double functor from $\infty$6 to the double $\infty$7-category of squares in $\infty$8 (Raptis et al., 8 Jun 2026).

The construction is leveraged to recover equivalences between several models of algebraic $\infty$9-theory, including Quillen’s $Sq$00-construction, Waldhausen’s $Sq$01-construction, cobordism models, and squares $Sq$02-theory (Raptis et al., 8 Jun 2026). This suggests that squares provide a common target language for correspondences and exactness phenomena in higher category theory.

6. Squares categories and squares K-theory

A separate but compatible direction appears in “All K-theory is squares K-theory” (Kuijper, 1 Dec 2025). There a squares category $Sq$03 consists of vertical and horizontal categories with the same objects, a basepoint object $Sq$04, and a class of distinguished squares closed under horizontal and vertical composition, together with identity-square and initial-object axioms (Kuijper, 1 Dec 2025).

From such a squares category one builds simplicial categories $Sq$05 and $Sq$06, leading to the squares $Sq$07-theory space

$Sq$08

For squares categories with complements, the paper proves

$Sq$09

and uses this to compare squares $Sq$10-theory with assembler $Sq$11-theory (Kuijper, 1 Dec 2025).

The main comparison theorem states that under axioms relating a squares category with complements $Sq$12 and a category with covering families $Sq$13, there is an equivalence

$Sq$14

(Kuijper, 1 Dec 2025). In particular, for a category with covering families satisfying hypotheses (C1)–(C5), one constructs a minimal squares category $Sq$15 and obtains

$Sq$16

(Kuijper, 1 Dec 2025).

The paper applies this to assemblers of varieties, definable sets in o-minimal structures, and polytopes in Euclidean or hyperbolic geometry (Kuijper, 1 Dec 2025). In the definable-set case, the squares model is then used to lift the definable Euler characteristic to a map of $Sq$17-theory spectra (Kuijper, 1 Dec 2025). This suggests that square-based formalisms are sufficiently expressive to absorb both exact-sequence and scissors-congruence types of additivity.

7. Related square-producing and quadratic constructions

Several additional bodies of work use “square” language in ways that illuminate, but do not define, the higher-categorical squares functor.

In equivariant Goodwillie calculus, the norm functor $Sq$18 has a Goodwillie tower whose canonical fracture squares agree with classical equivariant fracture squares indexed by families of subgroups (Konovalov, 2020). The paper proves

$Sq$19

and then identifies the Goodwillie fracture square with the equivariant localization square (Konovalov, 2020). This does not define a functor named $Sq$20, but it does exhibit a functorial assignment of canonical squares to a homotopy functor.

In the paper on absolutely homotopy-cartesian squares, a square is called absolutely cartesian if it remains homotopy cartesian after applying every homotopy functor (Eldred, 2013). The classification theorem states that a square of spaces is absolutely cartesian iff it is a map of two absolutely cartesian 1-cubes, equivalently of the form

$Sq$21

(Eldred, 2013). This again concerns the behavior of squares under functorial passage, though not through a specific functor called $Sq$22.

In commutative DG algebra, the squaring operation

$Sq$23

is defined via DG algebra resolutions and gives a quadratic endofunctor characterized by

$Sq$24

for $Sq$25 (Yekutieli, 2014). It is used to define rigid complexes and underlies a rigid approach to Grothendieck duality (Yekutieli, 2014). The shared theme with the squares functor is functorial square formation; the mathematical object, however, is different.

Finally, in algebraic functor theory, quadratic functors on pointed categories are classified by quadratic $Sq$26-modules involving square-type structure maps $Sq$27, extending square groups and quadratic $Sq$28-modules (Hartl et al., 2008). This is best understood as a theory of “degree-two” functors rather than of square-shaped diagrams.

8. Conceptual significance

Across these usages, a consistent pattern emerges. The squares functor in the strict sense provides a canonical passage from 2-dimensional categorical structure to double structure, making both directions of morphisms visible and recording 2-cells as squares (Loubaton et al., 10 Jul 2025). In the span–squares adjunction, squares are the right adjoint counterpart to correspondences, hence the natural language for describing functors out of span categories (Raptis et al., 8 Jun 2026). In squares $Sq$29-theory, distinguished squares encode additivity and complement data in a way broad enough to recover assembler and Waldhausen $Sq$30-theories (Kuijper, 1 Dec 2025).

This suggests that the mathematical importance of squares lies in their role as the minimal 2-dimensional unit in which covariance, contravariance, base change, complements, and excision can all be expressed simultaneously. A plausible implication is that square-based formalisms serve as a bridge between categorical exactness, correspondence formalisms, and homotopy-coherent higher algebra.

9. Outlook

Current work places the squares functor at the center of several active programs. In $Sq$31-category theory it clarifies the foundations of derived algebraic geometry and completes the proof of the Gaitsgory–Rozenblyum conjectures concerning squares and Gray products (Loubaton et al., 10 Jul 2025). In higher categorical algebra it yields the span–squares adjunction and new derivations of equivalences between competing algebraic $Sq$32-theory models (Raptis et al., 8 Jun 2026). In scissors-congruence and assembler $Sq$33-theory it supports the claim that many $Sq$34-theory spectra are naturally squares $Sq$35-theory spectra (Kuijper, 1 Dec 2025).

Further directions explicitly proposed include higher-dimensional analogues such as cubes and hypercubes, extensions to $Sq$36-categories, and the use of square-based universal properties to classify higher Beck–Chevalley structures and other lax or oplax phenomena (Loubaton et al., 10 Jul 2025). This suggests that the squares functor is likely to remain a foundational construction wherever two-dimensional homotopy-coherent algebra is organized through universal properties rather than ad hoc diagrammatics.