Bicategorical Enrichment

• Bicategorical enrichment is a framework where enriched objects carry extents and homs become heter-homs across different base categories, generalizing monoidal enrichment.
• It leverages slicing, oplax limits, and powers by singleton 1-cells to formulate precise fibrational and universal properties within higher categorical structures.
• The theory further interconnects with scalar enrichment and enriched bi(co)ends, enabling enriched trace constructions and Cauchy-completion processes in bicategorical settings.

Enrichment over a bicategory is the extension of ordinary enrichment over a monoidal category in which the base has many objects, so that enriched objects carry extents and homs are heter-homs in different hom-categories of the base bicategory rather than endomorphisms of a single object. In this sense it is a strict generalization of monoidal enrichment, and recent work treats it as a two-fold generalization of enrichment over both monoidal categories and quantaloids; in parallel, a higher-categorical literature studies the distinct but related notion of bicategories enriched in a monoidal bicategory (Fujii et al., 2022, Caramello et al., 28 Jul 2025, Garner et al., 2013).

1. Classical bicategorical enrichment and typed homs

A category enriched in a bicategory $\mathscr B$ consists of a set $ob(\mathbb X)$, a function

$|{-}| : ob(\mathbb X)\to ob(\mathscr B),$

a 1-cell

$\mathbb X(x,x') : |x|\to |x'|$

for each pair of objects, unit 2-cells

$j_x:1_{|x|}\Rightarrow \mathbb X(x,x),$

and composition 2-cells

$$M_{x,x',x''}:\mathbb X(x',x'')\,\mathbb X(x,x')\Rightarrow \mathbb X(x,x''),$$

subject to associativity and identity axioms generalizing the usual enriched-category axioms (Fujii et al., 2022). This is the familiar notion of enrichment over a bicategory, going back at least to Street, and a $\mathscr B$-category may equivalently be presented as a lax functor $\mathbb X:X_c\to \mathscr B$ from the chaotic category on its object set; the paper notes that such lax functors were studied by Bénabou as polyads (Fujii et al., 2022).

The decisive new feature, compared with ordinary enrichment over a monoidal category $V$, is the presence of extents. Objects do not all live over a single base object, and the homs are heter-homs in $\mathscr B$, not all endomorphisms of one object. If $ob(\mathbb X)$0 is regarded as a one-object bicategory, then a $ob(\mathbb X)$1-category in this sense is exactly an ordinary $ob(\mathbb X)$2-enriched category, so bicategory enrichment is a strict generalization of monoidal enrichment (Fujii et al., 2022). This typed character is also what makes bicategorical enrichment a two-fold generalization of the monoidal and quantaloidal cases: one gains many extents as in quantaloid enrichment, while retaining genuine $ob(\mathbb X)$3-cells rather than only order relations (Caramello et al., 28 Jul 2025).

The associated $ob(\mathbb X)$4-category $ob(\mathbb X)$5 has $ob(\mathbb X)$6-categories as objects, $ob(\mathbb X)$7-functors as $ob(\mathbb X)$8-cells, and $ob(\mathbb X)$9-natural transformations as $|{-}| : ob(\mathbb X)\to ob(\mathscr B),$0-cells. A $|{-}| : ob(\mathbb X)\to ob(\mathscr B),$1-functor preserves extents and carries comparison $|{-}| : ob(\mathbb X)\to ob(\mathscr B),$2-cells on homs; a $|{-}| : ob(\mathbb X)\to ob(\mathscr B),$3-natural transformation $|{-}| : ob(\mathbb X)\to ob(\mathscr B),$4 is given by $|{-}| : ob(\mathbb X)\to ob(\mathscr B),$5-cells

$|{-}| : ob(\mathbb X)\to ob(\mathscr B),$6

or equivalently by a family

$|{-}| : ob(\mathbb X)\to ob(\mathscr B),$7

compatible with composition in two evident ways (Fujii et al., 2022).

2. Slice stability, oplax limits, and the closure of the theory

A central structural result is that bicategory-enriched category theory is closed under slicing. For every bicategory $|{-}| : ob(\mathbb X)\to ob(\mathscr B),$8 and every $|{-}| : ob(\mathbb X)\to ob(\mathscr B),$9-category $\mathbb X(x,x') : |x|\to |x'|$0, there is a bicategory $\mathbb X(x,x') : |x|\to |x'|$1 such that

$\mathbb X(x,x') : |x|\to |x'|$2

Here the left-hand side is the strict slice $\mathbb X(x,x') : |x|\to |x'|$3-category over $\mathbb X(x,x') : |x|\to |x'|$4, so objects are $\mathbb X(x,x') : |x|\to |x'|$5-functors $\mathbb X(x,x') : |x|\to |x'|$6, and the theorem says that functors into $\mathbb X(x,x') : |x|\to |x'|$7 can themselves be regarded as categories enriched over a new bicategory $\mathbb X(x,x') : |x|\to |x'|$8 (Fujii et al., 2022).

The explicit construction shows why monoidal bases are not stable under this operation. The bicategory $\mathbb X(x,x') : |x|\to |x'|$9 has

$j_x:1_{|x|}\Rightarrow \mathbb X(x,x),$0

and hom-categories

$j_x:1_{|x|}\Rightarrow \mathbb X(x,x),$1

Thus a $j_x:1_{|x|}\Rightarrow \mathbb X(x,x),$2-cell $j_x:1_{|x|}\Rightarrow \mathbb X(x,x),$3 in $j_x:1_{|x|}\Rightarrow \mathbb X(x,x),$4 is a $j_x:1_{|x|}\Rightarrow \mathbb X(x,x),$5-cell $j_x:1_{|x|}\Rightarrow \mathbb X(x,x),$6 in $j_x:1_{|x|}\Rightarrow \mathbb X(x,x),$7 together with a $j_x:1_{|x|}\Rightarrow \mathbb X(x,x),$8-cell

$j_x:1_{|x|}\Rightarrow \mathbb X(x,x),$9

Composition is induced by the composition in $$M_{x,x',x''}:\mathbb X(x',x'')\,\mathbb X(x,x')\Rightarrow \mathbb X(x,x''),$$0: if $$M_{x,x',x''}:\mathbb X(x',x'')\,\mathbb X(x,x')\Rightarrow \mathbb X(x,x''),$$1 and $$M_{x,x',x''}:\mathbb X(x',x'')\,\mathbb X(x,x')\Rightarrow \mathbb X(x,x''),$$2, then their composite is the pasting

$$M_{x,x',x''}:\mathbb X(x',x'')\,\mathbb X(x,x')\Rightarrow \mathbb X(x,x''),$$3

When $$M_{x,x',x''}:\mathbb X(x',x'')\,\mathbb X(x,x')\Rightarrow \mathbb X(x,x''),$$4 and $$M_{x,x',x''}:\mathbb X(x',x'')\,\mathbb X(x,x')\Rightarrow \mathbb X(x,x''),$$5 each have only one object, this recovers the familiar slice monoidal category $$M_{x,x',x''}:\mathbb X(x',x'')\,\mathbb X(x,x')\Rightarrow \mathbb X(x,x''),$$6 over a monoid $$M_{x,x',x''}:\mathbb X(x',x'')\,\mathbb X(x,x')\Rightarrow \mathbb X(x,x''),$$7 in a monoidal category $$M_{x,x',x''}:\mathbb X(x',x'')\,\mathbb X(x,x')\Rightarrow \mathbb X(x,x''),$$8 (Fujii et al., 2022).

The conceptual explanation is oplax-limit-theoretic. Writing $$M_{x,x',x''}:\mathbb X(x',x'')\,\mathbb X(x,x')\Rightarrow \mathbb X(x,x''),$$9 as a lax functor, $\mathscr B$0 is characterized as its oplax limit in the $\mathscr B$1-category $\mathscr B$2 of bicategories, lax functors and icons. The proof passes through the enrichment $\mathscr B$3-functor

$\mathscr B$4

which sends $\mathscr B$5 to $\mathscr B$6 equipped with the object-of-objects functor, and preserves all weighted limits which happen to exist in $\mathscr B$7 (Fujii et al., 2022). The theorem therefore identifies slicing as an internal operation of bicategory-based enrichment rather than an external accident.

3. Fibrations, singleton powers, and concrete cartesianness

Under a mild local completeness condition, slicing interacts tightly with fibrational structure. If each hom-category $\mathscr B$8 has pullbacks, a $\mathscr B$9-functor

$\mathbb X:X_c\to \mathscr B$0

is a fibration in $\mathbb X:X_c\to \mathscr B$1 iff for each $\mathbb X:X_c\to \mathscr B$2 and each morphism $\mathbb X:X_c\to \mathscr B$3 in the underlying ordinary category $\mathbb X:X_c\to \mathscr B$4, there is a cartesian lifting

$\mathbb X:X_c\to \mathscr B$5

in $\mathbb X:X_c\to \mathscr B$6 with $\mathbb X:X_c\to \mathscr B$7. Here cartesianness is characterized by pullback squares on hom-objects in the ambient hom-categories of $\mathbb X:X_c\to \mathscr B$8 (Fujii et al., 2022).

The slice bicategory makes this fibrational condition an enriched universal property. In $\mathbb X:X_c\to \mathscr B$9, a $V$0-cell

$V$1

is called singleton if $V$2 and $V$3; equivalently it comes from a morphism $V$4 in the underlying category $V$5. If $V$6 corresponds under the slice equivalence to a $V$7-category $V$8, then the power of $V$9 by a singleton $\mathscr B$0 amounts exactly to a lifting

$\mathscr B$1

with $\mathscr B$2, and the power universal property is precisely the pullback condition defining a cartesian lifting (Fujii et al., 2022).

Accordingly, if each hom-category of $\mathscr B$3 has pullbacks, then

$\mathscr B$4

iff the corresponding $\mathscr B$5-category has powers by singleton $\mathscr B$6-cells; moreover the canonical isomorphism

$\mathscr B$7

restricts to an isomorphism between $\mathscr B$8-categories with powers by singleton $\mathscr B$9-cells and fibrations over $ob(\mathbb X)$00 with fibration morphisms (Fujii et al., 2022). In the special case $ob(\mathbb X)$01 viewed as a one-object bicategory, this yields

$ob(\mathbb X)$02

and Grothendieck fibrations correspond to those $ob(\mathbb X)$03-categories admitting powers by singleton $ob(\mathbb X)$04-cells (Fujii et al., 2022).

4. Scalar enrichment, cotraces, and self-enrichment phenomena

A different development starts not from categories enriched in a bicategory, but from a monoidal bicategory $ob(\mathbb X)$05 that canonically enriches itself over a scalar base extracted from the tensor unit. The scalar category is the hom-category

$ob(\mathbb X)$06

monoidal under composition, and in the setting of the thesis it is braided monoidal. The main theorem states: $ob(\mathbb X)$07 Thus each hom-category $ob(\mathbb X)$08 becomes a $ob(\mathbb X)$09-enriched category, and composition, identities, associators, and unitors all acquire enriched structure (Reader, 2024).

The enriched hom-object is defined by a cotrace construction. If $ob(\mathbb X)$10, then

$ob(\mathbb X)$11

where the right-hand side is the cotrace of the right lift of $ob(\mathbb X)$12 through $ob(\mathbb X)$13. If $ob(\mathbb X)$14 has a right adjoint $ob(\mathbb X)$15, then the lift is $ob(\mathbb X)$16, so one gets

$ob(\mathbb X)$17

matching the Frobenius-inner-product analogy described in the thesis (Reader, 2024). The scalar action on $ob(\mathbb X)$18 arises from the spread functor

$ob(\mathbb X)$19

and its right adjoint is the cotrace

$ob(\mathbb X)$20

This scalar enrichment also reinterprets trace-like constructions. The thesis shows that the cotrace is an enriched version of the categorical trace or $ob(\mathbb X)$21-trace studied by Ganter–Kapranov and Bartlett: the scalar enrichment proves that the $ob(\mathbb X)$22-trace is the underlying set of the cotrace (Reader, 2024). In compact-closed bicategories, the trace and cotrace live in the same scalar category and share properties including duality invariance, linearity, cyclicity, and tensor preservation (Reader, 2024).

The same higher-categorical line continues in the theory of enriched bi(co)ends. In that setting, one works with bicategories enriched in a monoidal bicategory $ob(\mathbb X)$23, following Garner–Shulman, then strictifies the base to a Gray monoid or a semi-strict braided monoidal bicategory so that opposites, tensor products, and self-enrichment can be constructed. The resulting enriched biends and bicoends

$ob(\mathbb X)$24

are representing objects for categories of extra-pseudonatural transformations, giving an enriched version of the bi(co)end theory of Corner and a bicategorical version of classical enriched (co)ends (Carissimi, 5 Sep 2025).

5. Universal properties, cocompletion, and sheaf theory

A separate but closely related strand studies bicategories enriched in a monoidal bicategory rather than categories enriched in a bicategory. Garner–Shulman develop this higher-dimensional theory up to a description of the free cocompletion of an enriched bicategory under a class of weighted bicolimits, and then use it to analyze the process sending a monoidal category $ob(\mathbb X)$25 to the equipment of $ob(\mathbb X)$26-enriched categories, functors, transformations, and modules (Garner et al., 2013). Their paper stresses that this is not the same notion as categories enriched in a bicategory: the former is a genuinely $ob(\mathbb X)$27-dimensional enrichment theory, whereas the latter is the classical many-object generalization of monoidal enrichment (Garner et al., 2013).

In the free-cocompletion theorem, if $ob(\mathbb X)$28 is a class of weights and $ob(\mathbb X)$29 a $ob(\mathbb X)$30-bicategory, then $ob(\mathbb X)$31 is obtained as the closure of the representables inside the module bicategory $ob(\mathbb X)$32 under $ob(\mathbb X)$33-weighted colimits, and the Yoneda embedding

$ob(\mathbb X)$34

exhibits $ob(\mathbb X)$35 as the free completion of $ob(\mathbb X)$36 under $ob(\mathbb X)$37-colimits (Garner et al., 2013). Applied to bicategories and equipments, this yields universal properties for constructions such as $ob(\mathbb X)$38, categories enriched in an equipment, and module bicategories (Garner et al., 2013).

The sheaf-theoretic direction returns to the classical notion of categories enriched in a bicategory $ob(\mathbb X)$39. In that setting a $ob(\mathbb X)$40-category is defined via a typed set and a $ob(\mathbb X)$41-matrix with composition and unit $ob(\mathbb X)$42-cells, and the theory is used to define complete $ob(\mathbb X)$43-categories, a generalization of Cauchy-complete enriched categories. Completeness means that every distributor into the $ob(\mathbb X)$44-category that has a right adjoint is representable; equivalently, in skeletal form, every singleton is representable (Caramello et al., 28 Jul 2025). The paper then proves an adjunction between complete $ob(\mathbb X)$45-categories and $ob(\mathbb X)$46-presheaves on the category $ob(\mathbb X)$47 of left adjoints in $ob(\mathbb X)$48, and gives conditions under which this adjunction becomes a left-exact reflection, yielding back the usual results linking sheaves on sites and enriched categories (Caramello et al., 28 Jul 2025).

This sheaf-theoretic theory simultaneously recovers the monoidal and quantaloidal cases. If $ob(\mathbb X)$49 is a closed monoidal category, then a $ob(\mathbb X)$50-category is exactly a $ob(\mathbb X)$51-enriched category; if $ob(\mathbb X)$52 is a quantaloid, then a $ob(\mathbb X)$53-category in the bicategorical sense is exactly a category enriched in the quantaloid $ob(\mathbb X)$54 (Caramello et al., 28 Jul 2025). The conceptual point is that sheaf conditions are recast as representability conditions for certain distributors or singletons, so that completion becomes a bicategorical Cauchy-completion or sheafification process (Caramello et al., 28 Jul 2025).

6. Extensions, higher dimensions, and conceptual boundaries

Bicategory enrichment now sits inside a broader landscape. Categories, functors, and natural transformations enriched over a bicategory are a special case of enrichment over a virtual double category, where the bicategory is regarded as a virtual double category whose vertical category is discrete and whose horizontal morphisms correspond to the $ob(\mathbb X)$55-cells (Fujii et al., 7 Jul 2025). The point of this generalization is not only to include more examples, but also to improve the formal behavior of enrichment itself: the paper proves that

$ob(\mathbb X)$56

is familial, while

$ob(\mathbb X)$57

is not a parametric right $ob(\mathbb X)$58-adjoint (Fujii et al., 7 Jul 2025). This suggests that bicategories are an important intermediate generalization of ordinary monoidal enrichment, but that the strongest formal properties emerge only after passing to virtual double categories.

Higher-categorical analogues continue this pattern. In Haugseng’s setting, enrichment is over a monoidal $ob(\mathbb X)$59-category rather than an arbitrary bicategory, but the resulting theory constructs a double $ob(\mathbb X)$60-category whose objects are enriched $ob(\mathbb X)$61-categories, one class of $ob(\mathbb X)$62-morphisms are enriched functors, the other class are bimodules, and the underlying $ob(\mathbb X)$63-category has $ob(\mathbb X)$64-morphisms identified with natural transformations (Haugseng, 2015). In the algebraic-higher-categorical framework of $ob(\mathbb X)$65-multicategories, bicategories appear as representable $ob(\mathbb X)$66-discrete $ob(\mathbb X)$67-multicategories, and more generally enrichment of algebraic higher categories is encoded by a $ob(\mathbb X)$68-multifunctor

$ob(\mathbb X)$69

from a lower-dimensional algebraic object into a multicategorical base (Shapiro, 2022). A related tricategorical program defines categories enriched over a $ob(\mathbb X)$70-strict tricategory and proves, under explicit hypotheses, that such enriched categories can be made into categories internal in that tricategory; the bicategorical paradigm “bicategory = category enriched over $ob(\mathbb X)$71” is one of its lower-dimensional models (Femić, 2021).

The recent literature also clarifies what does not count as the standard notion. The paper on enriched sets weakens associativity and functoriality by a skeleton functor

$ob(\mathbb X)$72

from a tensor category rather than by allowing extents and heter-homs in a bicategory, so it is adjacent to weak higher enrichment but not enrichment over a bicategory in the Bénabou–Street sense (Willocks, 2019). Likewise, enrichment over an oplax monoidal category is a nearby one-object weakening of monoidal enrichment; it has a distributor/module correspondence and a good $ob(\mathbb X)$73-category of enriched categories, but it does not introduce many extents or hom-categories of a bicategory, and is therefore not bicategory enrichment in the standard sense (Basile et al., 2022).

The modern picture is therefore stratified. In its classical form, enrichment over a bicategory is typed enrichment with extents and heter-homs. It is closed under slicing and admits a precise fibrational interpretation through powers by singleton $ob(\mathbb X)$74-cells (Fujii et al., 2022). In universal-property and sheaf-theoretic directions it interacts with modules, distributors, Cauchy completion, and $ob(\mathbb X)$75-presheaves on $ob(\mathbb X)$76 (Garner et al., 2013, Caramello et al., 28 Jul 2025). In higher-dimensional directions it leads to scalar enrichment, enriched bi(co)ends, virtual-double-category enrichment, and $ob(\mathbb X)$77-categorical bimodule formalisms (Reader, 2024, Carissimi, 5 Sep 2025, Fujii et al., 7 Jul 2025, Haugseng, 2015). The common thread is that bicategorical bases absorb operations—most notably slicing, distributor calculus, and trace-like or profunctorial constructions—that exceed the stability of ordinary monoidal enrichment.