---
title: Bicategorical Enrichment
url: https://www.emergentmind.com/topics/enrichment-over-a-bicategory
type: topic
---

# Bicategorical Enrichment

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** [2211.12122] [2507.20820] [1301.3191].

## 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 [2211.12122]. 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 [2211.12122].

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 \(V\) is regarded as a one-object bicategory, then a \(V\)-category in this sense is exactly an ordinary \(V\)-enriched category, so bicategory enrichment is a strict generalization of monoidal enrichment [2211.12122]. 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 \(2\)-cells rather than only order relations [2507.20820].

The associated \(2\)-category \(\mathscr B\mbox{-}\mathbf{Cat}\) has \(\mathscr B\)-categories as objects, \(\mathscr B\)-functors as \(1\)-cells, and \(\mathscr B\)-natural transformations as \(2\)-cells. A \(\mathscr B\)-functor preserves extents and carries comparison \(2\)-cells on homs; a \(\mathscr B\)-natural transformation \(\alpha:T\to S\) is given by \(2\)-cells
\[
\alpha_x:1_{|x|}\Rightarrow \mathbb D(Tx,Sx),
\]
or equivalently by a family
\[
\alpha_{x,x'}:\mathbb C(x,x')\Rightarrow \mathbb D(Tx,Sx')
\]
compatible with composition in two evident ways [2211.12122].

## 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 \(\mathscr B\) and every \(\mathscr B\)-category \(\mathbb X\), there is a bicategory \(\mathscr B/\mathbb X\) such that
\[
\mathscr B\mbox{-}\mathbf{Cat}/\mathbb X \cong (\mathscr B/\mathbb X)\mbox{-}\mathbf{Cat}.
\]
Here the left-hand side is the strict slice \(2\)-category over \(\mathbb X\), so objects are \(\mathscr B\)-functors \(F:\mathbb Y\to\mathbb X\), and the theorem says that functors into \(\mathbb X\) can themselves be regarded as categories enriched over a new bicategory \(\mathscr B/\mathbb X\) [2211.12122].

The explicit construction shows why monoidal bases are not stable under this operation. The bicategory \(\mathscr B/\mathbb X\) has
\[
ob(\mathscr B/\mathbb X)=ob(\mathbb X),
\]
and hom-categories
\[
(\mathscr B/\mathbb X)(x,x')=\mathscr B(|x|,|x'|)/\mathbb X(x,x').
\]
Thus a \(1\)-cell \(x\to x'\) in \(\mathscr B/\mathbb X\) is a \(1\)-cell \(S:|x|\to |x'|\) in \(\mathscr B\) together with a \(2\)-cell
\[
s:S\Rightarrow \mathbb X(x,x').
\]
Composition is induced by the composition in \(\mathbb X\): if \(s:S\to \mathbb X(x,x')\) and \(t:T\to \mathbb X(x',x'')\), then their composite is the pasting
\[
TS \xRightarrow{t\, s} \mathbb X(x',x'')\,\mathbb X(x,x') \xRightarrow{M_{x,x',x''}} \mathbb X(x,x'').
\]
When \(\mathscr B\) and \(\mathbb X\) each have only one object, this recovers the familiar slice monoidal category \(V/M\) over a monoid \(M\) in a monoidal category \(V\) [2211.12122].

The conceptual explanation is oplax-limit-theoretic. Writing \(\mathbb X:X_c\to \mathscr B\) as a lax functor, \(\mathscr B/\mathbb X\) is characterized as its oplax limit in the \(2\)-category \(\mathbf{BICAT}\) of bicategories, lax functors and icons. The proof passes through the enrichment \(2\)-functor
\[
Enr_1:\mathbf{BICAT}\to CAT/Set,
\]
which sends \(\mathscr B\) to \(\mathscr B\mbox{-}\mathbf{Cat}\) equipped with the object-of-objects functor, and preserves all weighted limits which happen to exist in \(\mathbf{BICAT}\) [2211.12122]. 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 \(\mathscr B(a,b)\) has pullbacks, a \(\mathscr B\)-functor
\[
F:\mathbb Y\to\mathbb X
\]
is a fibration in \(\mathscr B\mbox{-}\mathbf{Cat}\) iff for each \(y\in\mathbb Y\) and each morphism \(g:x\to Fy\) in the underlying ordinary category \(\mathbb X_0\), there is a cartesian lifting
\[
\overline g:g^*y\to y
\]
in \(\mathbb Y\) with \(F\overline g=g\). Here cartesianness is characterized by pullback squares on hom-objects in the ambient hom-categories of \(\mathscr B\) [2211.12122].

The slice bicategory makes this fibrational condition an enriched universal property. In \(\mathscr B/\mathbb X\), a \(1\)-cell
\[
(w:v\to \mathbb X(x,x')):x\to x'
\]
is called **singleton** if \(|x|=|x'|\) and \(v=1_{|x|}\); equivalently it comes from a morphism \(w:x\to x'\) in the underlying category \(\mathbb X_0\). If \(F:\mathbb Y\to\mathbb X\) corresponds under the slice equivalence to a \(\mathscr B/\mathbb X\)-category \(\overline{\mathbb Y}\), then the power of \(y\in\overline{\mathbb Y}\) by a singleton \(w:x\to Fy\) amounts exactly to a lifting
\[
\overline w:w\pitchfork y\to y
\]
with \(F\overline w=w\), and the power universal property is precisely the pullback condition defining a cartesian lifting [2211.12122].

Accordingly, if each hom-category of \(\mathscr B\) has pullbacks, then
\[
F:\mathbb Y\to\mathbb X \text{ is a fibration }
\]
iff the corresponding \(\mathscr B/\mathbb X\)-category has powers by singleton \(1\)-cells; moreover the canonical isomorphism
\[
(\mathscr B/\mathbb X)\mbox{-}\mathbf{Cat}\cong \mathscr B\mbox{-}\mathbf{Cat}/\mathbb X
\]
restricts to an isomorphism between \(\mathscr B/\mathbb X\)-categories with powers by singleton \(1\)-cells and fibrations over \(\mathbb X\) with fibration morphisms [2211.12122]. In the special case \(\mathscr B=Set\) viewed as a one-object bicategory, this yields
\[
(Set/\mathbb X)\mbox{-}\mathbf{Cat}\cong Cat/\mathbb X,
\]
and Grothendieck fibrations correspond to those \(Set/\mathbb X\)-categories admitting powers by singleton \(1\)-cells [2211.12122].

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

A different development starts not from categories enriched in a bicategory, but from a monoidal bicategory \(B\) that canonically enriches itself over a scalar base extracted from the tensor unit. The scalar category is the hom-category
\[
B(I,I),
\]
monoidal under composition, and in the setting of the thesis it is braided monoidal. The main theorem states:
\[
\text{Every left-composition-closed, right-monoidal-closed bicategory } B \text{ is the underlying bicategory of a } Cat_{B(I,I)}\text{-enriched bicategory.}
\]
Thus each hom-category \(B(A,B)\) becomes a \(B(I,I)\)-enriched category, and composition, identities, associators, and unitors all acquire enriched structure [2403.14475].

The enriched hom-object is defined by a cotrace construction. If \(f,g:A\to B\), then
\[
\underline B(A,B)(f,g):=(f g),
\]
where the right-hand side is the cotrace of the right lift of \(g\) through \(f\). If \(f\) has a right adjoint \(f^\dagger\), then the lift is \(f^\dagger\circ g\), so one gets
\[
\underline B(A,B)(f,g)=(f^\dagger\circ g),
\]
matching the Frobenius-inner-product analogy described in the thesis [2403.14475]. The scalar action on \(B(A,B)\) arises from the spread functor
\[
Spr_B:B(I,I)\to B(B,B),
\]
and its right adjoint is the cotrace
\[
\overline{Tr}_A:B(A,A)\to B(I,I).
\]

This scalar enrichment also reinterprets trace-like constructions. The thesis shows that the cotrace is an enriched version of the categorical trace or \(2\)-trace studied by Ganter–Kapranov and Bartlett: the scalar enrichment proves that the \(2\)-trace is the underlying set of the cotrace [2403.14475]. 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 [2403.14475].

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 \(V\), 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
\[
\int_a F(a,a),\qquad \int^a F(a,a)
\]
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 [2509.05070].

## 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 \(V\) to the equipment of \(V\)-enriched categories, functors, transformations, and modules [1301.3191]. Their paper stresses that this is not the same notion as categories enriched in a bicategory: the former is a genuinely \(2\)-dimensional enrichment theory, whereas the latter is the classical many-object generalization of monoidal enrichment [1301.3191].

In the free-cocompletion theorem, if \(\Phi\) is a class of weights and \(\mathcal B\) a \(\mathcal V\)-bicategory, then \(\Phi(\mathcal B)\) is obtained as the closure of the representables inside the module bicategory \(\mathcal M\mathcal B\) under \(\Phi\)-weighted colimits, and the Yoneda embedding
\[
Z:\mathcal B\to \Phi(\mathcal B)
\]
exhibits \(\Phi(\mathcal B)\) as the free completion of \(\mathcal B\) under \(\Phi\)-colimits [1301.3191]. Applied to bicategories and equipments, this yields universal properties for constructions such as \(\mathrm{Mod}_\kappa(\mathcal C)\), categories enriched in an equipment, and module bicategories [1301.3191].

The sheaf-theoretic direction returns to the classical notion of categories enriched in a bicategory \(\Bc\). In that setting a \(\Bc\)-category is defined via a typed set and a \(\Bc\)-matrix with composition and unit \(2\)-cells, and the theory is used to define **complete \(\Bc\)-categories**, a generalization of Cauchy-complete enriched categories. Completeness means that every distributor into the \(\Bc\)-category that has a right adjoint is representable; equivalently, in skeletal form, every singleton is representable [2507.20820]. The paper then proves an adjunction between complete \(\Bc\)-categories and \(2\)-presheaves on the category \(\Map(\Bc)\) of left adjoints in \(\Bc\), and gives conditions under which this adjunction becomes a left-exact reflection, yielding back the usual results linking sheaves on sites and enriched categories [2507.20820].

This sheaf-theoretic theory simultaneously recovers the monoidal and quantaloidal cases. If \(\V\) is a closed monoidal category, then a \(\Bc_\V\)-category is exactly a \(\V\)-enriched category; if \(\Q\) is a quantaloid, then a \(\Q\)-category in the bicategorical sense is exactly a category enriched in the quantaloid \(\Q\) [2507.20820]. 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 [2507.20820].

## 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 \(1\)-cells [2507.05529]. 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
\[
Enr:VDBL\to 2
\]
is familial, while
\[
Enr:BICAT\to 2
\]
is not a parametric right \(2\)-adjoint [2507.05529]. 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 \(\infty\)-category rather than an arbitrary bicategory, but the resulting theory constructs a double \(\infty\)-category whose objects are enriched \(\infty\)-categories, one class of \(1\)-morphisms are enriched functors, the other class are bimodules, and the underlying \((\infty,2)\)-category has \(2\)-morphisms identified with natural transformations [1506.07341]. In the algebraic-higher-categorical framework of \(T\)-multicategories, bicategories appear as representable \(0\)-discrete \(fc\)-multicategories, and more generally enrichment of algebraic higher categories is encoded by a \(T\)-multifunctor
\[
Hom:Mu_*A\to V
\]
from a lower-dimensional algebraic object into a multicategorical base [2205.12235]. A related tricategorical program defines categories enriched over a \(1\)-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 \(Cat_2\)” is one of its lower-dimensional models [2101.01460].

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
\[
sk:A\to B
\]
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 [1903.06982]. 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 \(2\)-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 [2204.01032].

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 \(1\)-cells [2211.12122]. In universal-property and sheaf-theoretic directions it interacts with modules, distributors, Cauchy completion, and \(2\)-presheaves on \(\Map(\Bc)\) [1301.3191] [2507.20820]. In higher-dimensional directions it leads to scalar enrichment, enriched bi(co)ends, virtual-double-category enrichment, and \(\infty\)-categorical bimodule formalisms [2403.14475] [2509.05070] [2507.05529] [1506.07341]. 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.

Source: https://www.emergentmind.com/topics/enrichment-over-a-bicategory