Parametric Right 2-Adjoint: Fibered and Categorical Views

• Parametric Right 2-Adjoint is a concept defining families of adjunctions in a fibrational setting with coherent base change, formalized via local fibered right adjoints.
• It is classified by a polynomial representation theorem where functors between fibered slices are exactly polynomial functors, ensuring uniform right-adjoint behavior.
• Different 2-categorical approaches, such as Hopf parametric adjoint objects and indexed families in algebra and type theory, illustrate its broad applicability across categorical structures.

“Parametric right 2-adjoint” does not have a single uniform meaning across the literature represented here. The most precise categorical formulation is fibrational: for functors between fibered slices over a locally cartesian closed category, the correct parameterized notion of right adjoint is a local fibered right adjoint, and these are exactly the fibered polynomial functors (Kock et al., 2010). Other works use different languages while exhibiting closely related structures: Hopf parametric adjoint objects across the 2-adjunction $$\operatorname{Adj}_R(\mathbf{Cat}) \leftrightarrows \operatorname{Mnd}(\mathbf{Cat})$$ (Vazquez-Marquez, 2017), indexed families of right adjoints arising from operadic change in equivariant algebra (Blumberg et al., 2017), families of adjunctions indexed by parabolic data in $p$-adic representation theory (Bezrukavnikov et al., 2011), dependent right adjoints internalized in multimode type theory (Nuyts et al., 2020), half 2-adjoint equivalences in Homotopy Type Theory (Carranza et al., 2020), and a free $(\infty,2)$-categorical construction that adds right adjoints to all $1$-morphisms (Riva et al., 6 Oct 2025).

1. Local fibered right adjoints as the precise fibrational notion

In the formulation of Kock–Kock, the ambient structure is a locally cartesian closed category $\mathcal E$ with terminal object, together with the fibered slices

$$\mathcal E\!\mid I \to \mathcal E,\qquad \mathcal E\!\mid J \to \mathcal E,$$

whose fiber over $K\in \mathcal E$ is canonically

$$(\mathcal E\!\mid I)_K \cong \mathcal E/(I\times K),\qquad (\mathcal E\!\mid J)_K \cong \mathcal E/(J\times K).$$

This packages all slices $\mathcal E/(I\times K)$ uniformly in the parameter $K$, and that uniformity is the source of the “parametric” interpretation (Kock et al., 2010).

A fibered functor

$p$0

is called a local fibered right adjoint if, after the canonical factorization

$p$1

the induced $p$2 is a fibered right adjoint. The paper stresses the difference between an ordinary local right adjoint and a local fibered right adjoint: the latter is a right adjoint in the 2-category of fibrations over $p$3, so the right-adjoint behavior is required coherently with respect to base change, not merely fiberwise (Kock et al., 2010).

This yields the most explicit 2-categorical reading in the cited corpus. The relevant 2-category is

$p$4

whose objects are categories fibered over $p$5, whose $p$6-cells are fibered functors, and whose $p$7-cells are fibered natural transformations. In that setting, a local fibered right adjoint is precisely a right adjoint appearing after slicing over the image of the terminal object, but now in a parameterized 2-categorical environment (Kock et al., 2010).

2. Polynomial representation theorem

The central classification theorem is that local fibered right adjoints between fibered slices are exactly polynomial functors. A polynomial in $p$8 is a diagram

$p$9

and it determines the fibered polynomial functor

$(\infty,2)$0

On the $(\infty,2)$1-fiber, this recovers the ordinary polynomial functor

$(\infty,2)$2

so

$(\infty,2)$3

Kock–Kock prove that if

$(\infty,2)$4

is a local fibered right adjoint, then

$(\infty,2)$5

for some polynomial diagram $(\infty,2)$6, and conversely every fibered polynomial functor is a local fibered right adjoint (Kock et al., 2010).

The proof factors $(\infty,2)$7 through the slice over $(\infty,2)$8. Writing

$(\infty,2)$9

with $1$0 of the form $1$1, one obtains

$1$2

where

$1$3

is a fibered right adjoint. A separate classification theorem for fibered right adjoints then shows that

$1$4

for a span $1$5, hence

$1$6

The converse is immediate from the factorization

$1$7

since $1$8 has fibered left adjoint $1$9 (Kock et al., 2010).

This theorem is the strongest precise answer in the cited literature to what should count as a parameterized or parametric right adjoint for slice-like constructions. It also explains why plain local right adjointness is insufficient outside $\mathcal E$0: the missing ingredient is the fibered coherence across parameters $\mathcal E$1 (Kock et al., 2010).

3. 2-adjunctions, Hopf operators, and parametric adjoint objects

A different 2-categorical approach appears in the study of the 2-adjunction

$\mathcal E$2

where objects of $\mathcal E$3 are adjunctions and objects of $\mathcal E$4 are monads. In this setting, the paper does not define a literal “parametric right 2-adjoint,” because the naive right-parametric object is obstructed by variance and opposition issues. Instead it introduces Hopf parametric adjoint objects (Vazquez-Marquez, 2017).

On the adjunction side, an adjoint object exists for a $\mathcal E$5-cell

$\mathcal E$6

precisely when the structural natural transformation $\mathcal E$7 is invertible. On the monad side, an adjoint object for

$\mathcal E$8

exists precisely when $\mathcal E$9 is invertible. The paper proves that these characterizations correspond across the 2-adjunction, so adjoint objects are transported between $$\mathcal E\!\mid I \to \mathcal E,\qquad \mathcal E\!\mid J \to \mathcal E,$$0 and $$\mathcal E\!\mid I \to \mathcal E,\qquad \mathcal E\!\mid J \to \mathcal E,$$1 by invertibility of the relevant structural $$\mathcal E\!\mid I \to \mathcal E,\qquad \mathcal E\!\mid J \to \mathcal E,$$2-cell (Vazquez-Marquez, 2017).

The parametric extension begins with an ordinary parametric adjunction

$$\mathcal E\!\mid I \to \mathcal E,\qquad \mathcal E\!\mid J \to \mathcal E,$$3

in the classical sense, indexed by a category $$\mathcal E\!\mid I \to \mathcal E,\qquad \mathcal E\!\mid J \to \mathcal E,$$4. Because the naive right-parametric object does not remain a $$\mathcal E\!\mid I \to \mathcal E,\qquad \mathcal E\!\mid J \to \mathcal E,$$5-cell of the same 2-category, the paper imposes a left Hopf condition: a certain Hopf operator must be invertible. Under this condition, the left part of the parametric data induces, for each parameter, an adjoint object, and the right-side structure assembles dinaturally. The result is denoted a Hopf parametric adjoint object rather than a parametric right 2-adjoint (Vazquez-Marquez, 2017).

The principal comparison theorem states that the 2-adjunction $$\mathcal E\!\mid I \to \mathcal E,\qquad \mathcal E\!\mid J \to \mathcal E,$$6 induces a bijection between Hopf parametric adjoint objects on the adjunction side and Hopf parametric adjoint objects on the monad side. A further lifting theorem identifies Hopf parametric adjunctions with parametric adjoint liftings to Eilenberg–Moore categories. This makes the paper one of the most direct 2-categorical sources for the topic, even though its terminology is deliberately more specialized (Vazquez-Marquez, 2017).

4. Indexed families of right adjoints in concrete algebra and geometry

Several cited works do not formulate a theory of parametric right 2-adjoints, but they exhibit families of adjunctions indexed by an external parameter and are therefore structurally suggestive.

In equivariant algebra, for $$\mathcal E\!\mid I \to \mathcal E,\qquad \mathcal E\!\mid J \to \mathcal E,$$7 operads $$\mathcal E\!\mid I \to \mathcal E,\qquad \mathcal E\!\mid J \to \mathcal E,$$8, there is a forgetful functor

$$\mathcal E\!\mid I \to \mathcal E,\qquad \mathcal E\!\mid J \to \mathcal E,$$9

that forgets norm maps present in $K\in \mathcal E$0 but absent from $K\in \mathcal E$1. This functor has both a left adjoint and a right adjoint, and the right adjoint is characterized objectwise by

$K\in \mathcal E$2

naturally in the finite $K\in \mathcal E$3-set $K\in \mathcal E$4. The construction depends on the inclusion of indexing systems $K\in \mathcal E$5, so there is a family of adjunctions varying over the poset of indexing systems. The paper explicitly states that it does not formulate a 2-category of indexing systems or a right 2-adjoint theorem, but it also notes that one may read the assignment $K\in \mathcal E$6 as suggestive of an indexed-category or pseudofunctor interpretation (Blumberg et al., 2017).

In the representation theory of reductive $K\in \mathcal E$7-adic groups, for each parabolic $K\in \mathcal E$8 with opposite $K\in \mathcal E$9, one has normalized parabolic induction $$(\mathcal E\!\mid I)_K \cong \mathcal E/(I\times K),\qquad (\mathcal E\!\mid J)_K \cong \mathcal E/(J\times K).$$0, normalized Jacquet functors $$(\mathcal E\!\mid I)_K \cong \mathcal E/(I\times K),\qquad (\mathcal E\!\mid J)_K \cong \mathcal E/(J\times K).$$1 and $$(\mathcal E\!\mid I)_K \cong \mathcal E/(I\times K),\qquad (\mathcal E\!\mid J)_K \cong \mathcal E/(J\times K).$$2, and Bernstein’s second adjointness

$$(\mathcal E\!\mid I)_K \cong \mathcal E/(I\times K),\qquad (\mathcal E\!\mid J)_K \cong \mathcal E/(J\times K).$$3

The geometric construction uses a $$(\mathcal E\!\mid I)_K \cong \mathcal E/(I\times K),\qquad (\mathcal E\!\mid J)_K \cong \mathcal E/(J\times K).$$4-equivariant bimodule map

$$(\mathcal E\!\mid I)_K \cong \mathcal E/(I\times K),\qquad (\mathcal E\!\mid J)_K \cong \mathcal E/(J\times K).$$5

that induces a natural transformation

$$(\mathcal E\!\mid I)_K \cong \mathcal E/(I\times K),\qquad (\mathcal E\!\mid J)_K \cong \mathcal E/(J\times K).$$6

while the canonical map

$$(\mathcal E\!\mid I)_K \cong \mathcal E/(I\times K),\qquad (\mathcal E\!\mid J)_K \cong \mathcal E/(J\times K).$$7

provides the complementary adjunction datum. The paper emphasizes that this is not packaged as a theorem on parametric right 2-adjoints, but the construction is indexed by parabolic or Levi data $$(\mathcal E\!\mid I)_K \cong \mathcal E/(I\times K),\qquad (\mathcal E\!\mid J)_K \cong \mathcal E/(J\times K).$$8, so it gives a family of adjunctions that is naturally interpreted bicategorically in terms of objects, $$(\mathcal E\!\mid I)_K \cong \mathcal E/(I\times K),\qquad (\mathcal E\!\mid J)_K \cong \mathcal E/(J\times K).$$9-morphisms, and $\mathcal E/(I\times K)$0-morphisms (Bezrukavnikov et al., 2011).

A thin 1-categorical analogue occurs for Pultr functors on digraphs. For a Pultr template $\mathcal E/(I\times K)$1, one always has

$\mathcal E/(I\times K)$2

and in many cases $\mathcal E/(I\times K)$3 itself admits a further right adjoint

$\mathcal E/(I\times K)$4

yielding chains

$\mathcal E/(I\times K)$5

The paper does not use 2-categorical language, and it works largely in the thin category of digraphs, but it provides a substantial body of explicit iterated right-adjoint phenomena that function as concrete lower-dimensional analogues (Foniok et al., 2013).

5. Internal and homotopical reformulations

In type theory, the most closely related construction is the transpension type former. The paper introduces

$\mathcal E/(I\times K)$6

and states that, in cartesian settings, it is right adjoint to quantification over a shape variable,

$\mathcal E/(I\times K)$7

while in the general multimodal system the relevant adjoint chains are

$\mathcal E/(I\times K)$8

and, in cartesian settings,

$\mathcal E/(I\times K)$9

Semantically, this is expressed by the theorem

$K$0

The ambient multimode type theory is parametrized by a strict 2-category of modes, modalities, and $K$1-cells, but the paper is explicit that the transpension itself is an ordinary right adjoint functor or dependent right adjoint modality, not a higher-categorical right 2-adjoint in the usual bicategorical sense (Nuyts et al., 2020).

Homotopy Type Theory supplies a different use of “2-adjoint.” For a function $K$2, the paper defines the right-flavored half 2-adjoint equivalence structure

$K$3

and the left-flavored variant

$K$4

The defining higher coherence for $K$5 is

$K$6

while the left variant uses

$K$7

Both are propositions, and the paper proves

$K$8

By contrast, the full $K$9-adjoint equivalence structure

$p$00

is generally not a proposition. This is not a theory of parametric right 2-adjoints, but it gives a precise internal account of right- and left-flavored 2-dimensional adjunction coherence in HoTT (Carranza et al., 2020).

These type-theoretic developments show that the expression “right 2-adjoint” can refer either to an external 2-categorical universal property or to an internal coherence structure on equivalences. The cited papers make that distinction explicit by reserving ordinary adjunction language for the semantics, while using “2-adjoint equivalence” for higher path coherence (Nuyts et al., 2020, Carranza et al., 2020).

6. Free addition of right adjoints and higher-categorical scope

A recent $p$01-categorical construction makes the adjoint-adding aspect entirely explicit. There is a functor

$p$02

such that, for an $p$03-category $p$04, the objects of $p$05 are the objects of $p$06, the $p$07-morphisms are formal zigzags of $p$08-morphisms of $p$09, and the $p$10-morphisms are generated under vertical composition by formal zigzags of commutative squares, with the first and last vertical legs invertible. For every $p$11-morphism $p$12 of $p$13, its image in $p$14 has a chosen right adjoint, and the paper proves the universal property

$p$15

Equivalently,

$p$16

The example

$p$17

identifies the zigzag construction on the walking arrow with the walking adjunction (Riva et al., 6 Oct 2025).

This is not presented under the label “parametric right 2-adjoint,” but it has the expected universal mapping-space behavior of a higher-categorical adjunction. A plausible implication is that it gives a free completion under right adjoints in a form that is close in spirit to a 2-categorical parametric construction, even though the paper works with complete globular double Segal spaces rather than strict 2-categories (Riva et al., 6 Oct 2025).

Taken together, the cited works support a layered understanding of the subject. The most exact formulation of a parameterized right adjoint is the fibrational notion of local fibered right adjoint, classified by polynomial functors (Kock et al., 2010). The most explicit 2-categorical transport mechanism is given by Hopf parametric adjoint objects under the Adj–Mnd 2-adjunction (Vazquez-Marquez, 2017). Several major examples then realize indexed families of adjunctions without formalizing a single global right 2-adjoint theorem, while type theory and $p$18-category theory show how right-adjoint data can be internalized or freely generated [(Blumberg et al., 2017); (Bezrukavnikov et al., 2011); (Nuyts et al., 2020); (Carranza et al., 2020); (Riva et al., 6 Oct 2025)]. In that sense, “parametric right 2-adjoint” names a cluster of related ideas rather than a universally fixed definition.