Display Map 2-Categories

• Display map 2-categories are enhanced 2-categories featuring a strict subcategory of display maps within a broader weak setting.
• The framework formalizes structure-preserving morphisms via 2-monads, using enriched limit theory and rigged weights to detect strictness.
• Applications span monoidal categories, finite product structures, and higher 2-fibrations, linking traditional display maps with modern higher-categorical tools.

“Display map 2-categories” is best read as a useful slogan, not as a standardized term of art. In the relevant literature, it denotes a recurring 2-categorical pattern in which one studies categories, or more generally algebras for a 2-monad, equipped with extra structure; a large class of weak structure-preserving morphisms; and a distinguished subclass of strict morphisms that behave like display maps. In the formulation developed by Lack and Shulman, this pattern is formalized as an enhanced 2-category, equivalently an $\mathcal{F}$-category, and its central technical content is a limit theory in which designated projections are strict and jointly detect strictness of other morphisms (Lack et al., 2011).

1. Terminological status and the basic pattern

The expression “display map 2-category” does not originate with Lack and Shulman; it functions as a slogan for a situation in which strict maps sit inside a larger 2-category of weak maps. A typical instance has objects given by categories with extra structure, or strict algebras for a 2-monad $T$; 1-cells given by weak structure-preserving morphisms of some flavour $w\in\{p,l,c\}$ together with a distinguished subclass of strict ones; and 2-cells given by structure-respecting natural transformations (Lack et al., 2011).

For a 2-monad $T$ on a 2-category $\mathcal K$, the standard notation is

$T\text{-}\mathbf{Alg}_s$

for strict $T$-algebras and strict $T$-morphisms, and

$T\text{-}\mathbf{Alg}_w$

for the same strict algebras but weak morphisms of flavour $w$. There is then a canonical inclusion

$T$0

which is identity on objects, faithful, and locally fully faithful. This inclusion is exactly the extra datum that records which weak morphisms are in fact strict.

Unwound, the pattern is “strict inside weak.” The weak side carries the ambient 2-category of lax, pseudo, or colax structure-preservers. The strict side is a full sub-2-category on the same objects, and its 1-cells are the designated “display-like” maps. A common misconception is that the weak 2-category alone contains all relevant information. In this framework it does not: the choice of strict morphisms is additional structure, and the theory of limits depends essentially on that choice (Lack et al., 2011).

For example, if $T$1 and $T$2 are strict $T$3-algebras, then a lax $T$4-morphism $T$5 consists of a 1-cell $T$6 together with a 2-cell

$T$7

satisfying associativity and unit coherence. Strict $T$8-morphisms are the special case in which the defining square commutes strictly. The display-map interpretation is precisely that these strict morphisms form the chosen tight class inside the loose world of weak morphisms (Lack et al., 2011).

2. Enhanced 2-categories and $T$9-enrichment

Lack and Shulman formalize the preceding pattern using a cartesian closed base category $w\in\{p,l,c\}$0. Let $w\in\{p,l,c\}$1 be the arrow category of $w\in\{p,l,c\}$2. The category $w\in\{p,l,c\}$3 is the full sub-2-category of $w\in\{p,l,c\}$4 whose objects are full embeddings

$w\in\{p,l,c\}$5

that is, functors that are fully faithful and injective on objects (Lack et al., 2011).

Each object of $w\in\{p,l,c\}$6 has a tight part $w\in\{p,l,c\}$7 and a loose part $w\in\{p,l,c\}$8. A morphism in $w\in\{p,l,c\}$9 is a commutative square

$T$0

and a 2-cell is a commuting square of natural transformations. Because $T$1 is monic and fully faithful, $T$2 and the 2-cell are determined by $T$3. The category $T$4 is complete, cocomplete, and cartesian closed. Its internal hom $T$5 has loose part $T$6 and tight part the full subcategory of functors preserving tightness (Lack et al., 2011).

An $T$7-category $T$8 consists of objects together with, for each pair $T$9, a hom-object

$\mathcal K$0

so each hom is itself a full embedding

$\mathcal K$1

Composition and identities are morphisms in $\mathcal K$2. Equivalently, the tight homs assemble to a 2-category $\mathcal K$3, the loose homs assemble to a 2-category $\mathcal K$4, and the inclusions assemble to a 2-functor

$\mathcal K$5

that is identity on objects, faithful, and locally fully faithful (Lack et al., 2011).

This is the precise sense in which an enhanced 2-category is a 2-category with a distinguished class of 1-cells. Any ordinary 2-category may be regarded as chordate, where all morphisms are tight, or inchordate, where only identities are tight. The central examples are the $\mathcal K$6-categories $\mathcal K$7, whose objects are strict $\mathcal K$8-algebras, whose tight morphisms are strict $\mathcal K$9-morphisms, and whose loose morphisms are $T\text{-}\mathbf{Alg}_s$0-weak $T\text{-}\mathbf{Alg}_s$1-morphisms (Lack et al., 2011).

3. Limits, rigged weights, and display-like projections

The decisive structural feature of these enhanced 2-categories is their limit theory. If $T\text{-}\mathbf{Alg}_s$2 is an $T\text{-}\mathbf{Alg}_s$3-weight and $T\text{-}\mathbf{Alg}_s$4 an $T\text{-}\mathbf{Alg}_s$5-functor, a $T\text{-}\mathbf{Alg}_s$6-weighted limit $T\text{-}\mathbf{Alg}_s$7 is characterized by an isomorphism of hom-objects

$T\text{-}\mathbf{Alg}_s$8

natural in $T\text{-}\mathbf{Alg}_s$9 (Lack et al., 2011).

Unwinding this yields the display-map behavior. A $T$0-limit is, first, a 2-categorical weighted limit $T$1 in the loose 2-category $T$2. But it is not merely that. For each object $T$3 and each element $T$4, the corresponding projection

$T$5

must be tight, and the family $T$6 must jointly detect tightness: a loose morphism into the limit is tight if and only if all of its composites with these designated projections are tight (Lack et al., 2011).

This is the most literal sense in which tight maps behave like display maps. The limit exists in the weak world, but its structural projections belong to the strict world, and strictness of arbitrary cones can be tested projectionwise. The paper’s characterization of which weights lift to $T$7 is given in terms of rigged weights. The theory uses weak-transformation classifiers $T$8, first in the 2-categorical setting and then in the $T$9-enriched setting, to encode weak naturality data. An $T$0-weight $T$1 is $T$2-rigged when it is a $T$3-coalgebra and the induced map

$T$4

is pointwise surjective on objects; equivalently, precomposing any modification with $T$5 reflects identities (Lack et al., 2011).

The main lifting theorem states that for an $T$6-weight $T$7, the following are equivalent: $T$8 is $T$9-rigged; for every $T\text{-}\mathbf{Alg}_w$0-monad $T\text{-}\mathbf{Alg}_w$1 on an $T\text{-}\mathbf{Alg}_w$2-category $T\text{-}\mathbf{Alg}_w$3, the forgetful $T\text{-}\mathbf{Alg}_w$4-functor

$T\text{-}\mathbf{Alg}_w$5

creates $T\text{-}\mathbf{Alg}_w$6-weighted limits; and, in particular, for every 2-monad $T\text{-}\mathbf{Alg}_w$7 on a 2-category $T\text{-}\mathbf{Alg}_w$8, the forgetful $T\text{-}\mathbf{Alg}_w$9 creates such limits when $w$0 is viewed as chordate (Lack et al., 2011).

In the pseudo case $w$1, this recovers the classical theory of PIE-limits: the Cat-weights that are $w$2-coalgebras are exactly the PIE-weights, and these are precisely the weights whose limits always lift to $w$3 for any 2-monad $w$4 (Lack et al., 2011). This identifies the classical pseudoalgebra lifting theorem with the enriched strict-inside-weak formalism.

The standard examples make the display-map interpretation concrete. For monoidal categories, strict monoidal functors are tight and strong or lax monoidal functors are loose. For categories with chosen finite products, tight maps strictly preserve chosen products and loose maps preserve them only up to coherent comparison maps. For categories with finite limits, tight maps are strictly limit-preserving and loose maps are weakly limit-preserving. In such cases, certain inserters, comma objects, and related limits have projections that are strict structure-preservers and detect when a weak morphism is actually strict (Lack et al., 2011).

4. Displayed categories, fibrations, and the 1-categorical root

A closely related 1-categorical background is provided by displayed categories. A displayed category $w$5 over a category $w$6 consists of displayed objects $w$7 over each $w$8, displayed morphisms $w$9 over each $T$00, displayed identities, displayed composition, and dependent associativity and unit laws (Ahrens et al., 2017). Its total category $T$01 has objects $T$02 and morphisms $T$03, together with a projection functor

$T$04

A displayed category over $T$05 is equivalent to a category equipped with a functor to $T$06 (Ahrens et al., 2017).

This reformulation is not merely notational. It provides a fiberwise presentation of categories “over a base” that avoids equality on objects and makes fibrational structure direct. A displayed morphism

$T$07

is cartesian when every displayed arrow over $T$08 factors uniquely through it; a cleaving chooses cartesian lifts; and displayed fibrations, opfibrations, isofibrations, and discrete fibrations correspond to the classical notions for the projection functor $T$09 (Ahrens et al., 2017).

The display-map connection is explicit in the treatment of comprehension. A comprehension category is given by a fibration $T$10 together with a comprehension functor

$T$11

over $T$12 preserving cartesian arrows. In the displayed formulation, a type over $T$13 produces a display map

$T$14

and substitution is modeled by cartesian liftings. Categories with attributes are converted into comprehension categories by combining the equivalence between presheaves and discrete fibrations with displayed slice constructions (Ahrens et al., 2017).

This suggests a 1-dimensional root of display map 2-categories. Displayed categories encode “types over contexts” as indexed families, cartesian morphisms encode substitution, and displayed functors into slices encode context extension. The same paper explicitly points toward displayed bicategories, displayed 2-functors, and a “2-dimensional analogue of a comprehension category,” while noting that displayed categories, displayed functors, and displayed natural transformations over a fixed base should form a bicategory (Ahrens et al., 2017). A plausible implication is that display map 2-categories can be understood as a higher-categorical synthesis of the enhanced-2-categorical strict/weak pattern and the displayed-category comprehension pattern.

5. Higher-dimensional generalizations

At the $T$15-level, cartesian 2-fibrations provide a direct higher analogue of display maps. In the scaled-simplicial-set framework of $T$16-bicategories, one studies a map

$T$17

together with $T$18-cartesian and $T$19-cocartesian arrows, and with higher lifting conditions for triangles. The theory distinguishes four variance flavours: 2-inner cartesian, 2-inner cocartesian, 2-outer cartesian, and 2-outer cocartesian fibrations (Gagna et al., 2021).

The 2-dimensional content lies in the notion of inner and outer triangles. A triangle in $T$20 can be left or right $T$21-inner, or left or right $T$22-outer, according to whether associated arrows in slice $T$23-bicategories are strongly cartesian or strongly cocartesian. A 2-inner fibration is a weak fibration with enough left and right inner lifts for triangles; a 2-outer fibration has enough outer lifts together with closure of outer triangles under whiskering (Gagna et al., 2021). This is precisely the kind of higher lifting-and-stability data that a display-map formalism requires.

Several structural results align with the display-map viewpoint. If $T$24 is a 2-inner fibration, then for every $T$25 the induced map on mapping $T$26-categories

$T$27

is a cartesian fibration; if $T$28 is 2-outer, then each $T$29 is a cocartesian fibration (Gagna et al., 2021). Equivalences between such fibrations can be tested fiberwise: under an equivalence of bases, a morphism between 2-inner or 2-outer cartesian fibrations is an equivalence if and only if it is an equivalence on each fiber (Gagna et al., 2021).

The prototypical example is the domain projection

$T$30

where $T$31 is the $T$32-bicategory of functors, lax natural transformations, and modifications. This map is a 2-outer cartesian fibration (Gagna et al., 2021). As in ordinary arrow fibrations, objects are arrows, cartesian edges model reindexing by precomposition, and outer cartesian triangles express the higher coherence of lax squares. This provides a concrete higher-dimensional candidate for a display map 2-category.

6. Homotopy-coherent refinements and ambient frameworks

Two further directions clarify the scope of the subject. First, when mapping spaces are 2-types, two-track categories provide an algebraic model for categories enriched in 2-type mapping spaces. A two-track category is a category enriched in two-typical double groupoids, equivalently a category enriched in 2-types up to weak equivalence. The associated Baues–Wirsching type cohomology $T$33 classifies two-track extensions of a track category $T$34 by a natural system $T$35, with equivalence classes of two-track extensions corresponding to elements of

$T$36

(Blanc et al., 2010). This suggests that display map 2-categories with homotopy-theoretic mapping data admit genuinely higher coherence refinements whose obstruction theory is cohomological rather than merely bicategorical.

Second, accessibility theory indicates that weak rather than strict ambient 2-categories are usually the correct setting. In Bourke’s framework, a 2-category in LP is accessible with filtered colimits, has flexible limits, and has finite flexible limits commuting with filtered colimits. If $T$37 LPM, then the 2-category of fibrations

$T$38

is in LP; similarly, accessibility of retract equivalences controls the accessibility of isofibrations and equivalences in $T$39 (Bourke, 2020). The same paper shows that 2-categories of weak structures and pseudomorphisms, such as monoidal categories with strong monoidal functors or bicategories with pseudofunctors, lie in LPM, whereas sufficiently strict variants often fail even to be accessible (Bourke, 2020). This suggests that display map 2-categories should generally be sought among weak structures with pseudomorphisms, rather than among overly strict algebraic 2-categories.

Taken together, these developments support a broad understanding of display map 2-categories. At the strict/weak level, they are enhanced 2-categories or $T$40-categories in which tight morphisms are singled out inside a loose 2-category and are controlled by enriched limit theory (Lack et al., 2011). At the fibrational level, they are organized by displayed-category and comprehension ideas (Ahrens et al., 2017). At the higher level, they are modeled by inner or outer cartesian 2-fibrations of $T$41-bicategories (Gagna et al., 2021). And at the homotopy-coherent and accessibility levels, they are shaped by cohomological refinement and by the preference for weak, flexible ambient 2-categories (Blanc et al., 2010, Bourke, 2020).