---
title: Complete B-Categories Overview
url: https://www.emergentmind.com/topics/complete-b-categories
type: topic
---

# Complete B-Categories Overview

Searching arXiv for the core paper and closely related background papers on complete B-categories, bicategorical enrichment, and related completion notions.
Complete B-categories are categories enriched in a bicategory \(B\) in which every distributor that admits a right adjoint is representable. In the formulation developed for “sheaves on a bicategory,” this notion generalizes Cauchy-complete enriched categories and serves as the basis for an adjunction between complete \(B\)-categories and \(2\)-presheaves on the bicategory \(\Map(B)\) of left adjoints in \(B\) [2507.20820]. The terminology is not uniform across the literature: in closely related work, “completion” can also mean free cocompletion under weighted bicolimits, completion to a tensored braided enriched monoidal category, or completeness conditions in simplicial models of univalent \((2,1)\)-categories. Accordingly, “Complete B-Categories” denotes a family of structurally related but non-identical notions.

## 1. Enrichment over a bicategory

The bicategorical theory takes \(B\) to be locally cocomplete and closed; in many results an involution
\[
(-)^{\circ}: B^{\op}\to B
\]
is also assumed. The object-of-objects data is organized by a typed set: a set \(A\) equipped with a typing function \(t:A\to \Ob(B)\). A \(B\)-matrix \(M:A\to C\) between typed sets is a family of \(1\)-cells \(M(c,a): ta\to tc\) in \(B\), and a \(B\)-category is an endomatrix \(M:A\to A\) together with \(2\)-cells
\[
\iota_{abc}: M(a,b)\,M(b,c)\Rightarrow M(a,c),\qquad
\rho_a:\id_{ta}\Rightarrow M(a,a),
\]
satisfying associativity and unitality coherence. Thus the usual enriched composition and unit are replaced by bicategorical composition laws internal to the hom-categories of \(B\) [2507.20820].

A \(B\)-functor \(f:(M,A)\to (N,C)\) is a type-preserving function \(f:A\to C\) together with \(2\)-cells
\[
f_{aa'}: M(a,a')\Rightarrow N\big(f(a),f(a')\big)
\]
compatible with \(\iota\) and \(\rho\). These form a category \(\Cat(B)\). If \(B\) carries an involution, one may further impose symmetry:
\[
M(a,a')=M(a',a)^{\circ},\qquad \rho_a^{\circ}=\rho_a,\qquad \iota_{abc}^{\circ}=\iota_{cba},
\]
yielding a full subcategory \(\Cat_{\s}(B)\).

Distributors, also called modules or profunctors, are the bicategorical analogue of enriched profunctors. A distributor
\[
\phi:(M,A)\to (N,C)
\]
is a rectangular \(B\)-matrix \(\phi(c,a):ta\to tc\) equipped with “double action” \(2\)-cells
\[
\delta_{c,c',a',a}: N(c,c')\,\phi(c',a')\,M(a',a)\Rightarrow \phi(c,a)
\]
satisfying unitality and associativity. Composition is defined by the coend
\[
(\psi\phi)(d,a)=\int^{c\in C}_{N}\psi(d,c)\,\phi(c,a),
\]
and distributors form a bicategory \(\Dist(B)\). Representable distributors assemble into a pseudofunctor \(\Cat(B)\to \Dist(B)\), exhibiting \(\Dist(B)\) as an equipment over \(\Cat(B)\).

Two specializations are fundamental. If \(V\) is a closed monoidal category, then the delooping \(B_V\) is a one-object bicategory and \(B_V\)-categories are exactly \(V\)-enriched categories. If \(Q\) is a quantaloid, then \(Q\)-categories recover quantaloid enrichment, with all distributor axioms specializing to order inequalities. This places monoidal enrichment and quantaloid enrichment inside a single bicategorical framework.

## 2. Completeness as representability of adjointable distributors

In \(\Dist(B)\), a distributor \(\phi:(N,C)\to (M,A)\) is called a map if it has a right adjoint \(\phi^*\) with unit and counit satisfying the triangle identities. Representable distributors \(f_!\) are maps with right adjoint \(f^!\). The defining completeness condition is then:

> A \(B\)-category \((M,A)\) is complete if every distributor \(\phi:(N,C)\to (M,A)\) that has a right adjoint in \(\Dist(B)\) is representable, i.e. \(\phi\cong f_!\) for some \(B\)-functor \(f:(N,C)\to (M,A)\) [2507.20820].

This is explicitly presented as a generalization of Cauchy-complete enriched categories. The paper further shows that it suffices to test completeness on special maps called singletons. A presingleton on \((M,A)\) is a distributor
\[
\sigma:(\id_*,*)\to (M,A),
\]
equivalently a family of \(1\)-cells \(\sigma(a):*\to ta\) with action \(2\)-cells
\[
\delta_{ab}: M(a,b)\,\sigma(b)\Rightarrow \sigma(a)
\]
satisfying the expected unit and associativity axioms. A singleton is a presingleton admitting a right adjoint \(\sigma^*\).

For each \(a\in A\), the representable singleton \(M(-,a)\) has right adjoint \(M(a,-)\); its unit is \(\rho_a\) and its counit is \(\iota_{-\,a\,-}\). The Yoneda \(B\)-functor sends \(a\) to \(M(-,a)\). The \(B\)-category of singletons, denoted \((Sb,C A)\), is itself complete, and the co-Yoneda lemma states that a \(B\)-category is complete iff it is canonically the colimit of all its singletons in \(\Dist(B)\). The resulting completion functor
\[
C:\Cat(B)\to \Cat_k(B)
\]
sends \((M,A)\) to its singleton category, and is left adjoint to the inclusion \(i:\Cat_k(B)\to \Cat(B)\). In the monoidal case \(B=B_V\), this recovers the usual Cauchy-completion of \(V\)-enriched categories.

A common misunderstanding is to read “complete” here as ordinary bicategorical completeness by bilimits. The actual condition is representability of all adjointable distributors. This is a completeness of weights, not primarily a statement about having all limits.

## 3. The adjunction with \(2\)-presheaves on \(\Map(B)\)

The bicategory \(\Map(B)\) has the same objects as \(B\), \(1\)-cells given by left adjoints in \(B\), and \(2\)-cells those of \(B\) between maps. The sheaf-theoretic side uses pseudofunctors
\[
F:\Map(B)^{\mathrm{coop}}\to \Cat
\]
together with oplax natural transformations.

From such an \(F\), the Grothendieck construction \(\sigma F\) produces a \(B\)-category. Its objects are the disjoint union
\[
\sigma F=\bigsqcup_{x\in \Ob(B)}\Ob(F(x)),
\]
typed by the ambient object \(x\). For \(a,b\in \sigma F\), the hom-object \(N(a,b)\in B(tb,ta)\) is defined as the colimit of the diagram whose objects are pairs \((f,u)\) consisting of a map \(f:tb\to ta\) and a morphism \(u:b\to F(f)(a)\) in \(F(tb)\). The unit \(\rho_a\) is the canonical inclusion \(e_{\id}\), and composition
\[
\iota_{abc}: N(a,b)\,N(b,c)\Rightarrow N(a,c)
\]
is assembled from the universal property of these colimits and the pseudofunctoriality of \(F\) [2507.20820].

In the opposite direction, a complete \(B\)-category \((M,A)\) determines a pseudofunctor
\[
P_{(M,A)}:\Map(B)^{\mathrm{coop}}\to \Cat.
\]
For \(x\in \Ob(B)\), the category \(P_{(M,A)}(x)\) has as objects the elements \(a\in A\) of type \(x\), and as morphisms \(a\to b\) the \(2\)-cells
\[
\theta:\id_x\Rightarrow M(b,a).
\]
Composition is
\[
\chi\circ \theta:=\iota_{cba}\cdot (\chi * \theta),
\]
and identities are \(\rho_a\). For a map \(\gamma:x_1\to x_2\), the action on objects is
\[
P(\gamma)(a)=a\cdot \gamma,
\]
the unique element representing the singleton \(M(-,a)\gamma\). Completeness is essential here: it guarantees existence and uniqueness of \(a\cdot \gamma\).

The central result is the adjunction
\[
\mathcal C\,\sigma \dashv P.
\]
Explicitly, for any pseudofunctor \(F:\Map(B)^{\mathrm{coop}}\to \Cat\) and any complete \(B\)-category \((M,A)\), there is a natural bijection
\[
\Cat_\kappa\big(\mathcal C\,\sigma F,\,(M,A)\big)\cong [\Map(B)^{\mathrm{coop}},\Cat]\big(F,\,P_{(M,A)}\big).
\]
The construction uses explicit maps \(\mathsf A\) and \(\mathsf G\) between \(B\)-functors \(\sigma F\to (M,A)\) and oplax transformations \(F\Rightarrow P_{(M,A)}\). The use of oplax, rather than necessarily pseudo-, transformations is deliberate: the paper notes that \(\gamma^*\) need not be a map in \(B\), so invertibility is obstructed in general.

## 4. Quantaloids, sites, and classical sheaf theory

For quantaloids, the adjunction specializes to a sheaf-theoretic reflection. One obtains
\[
\mathcal C\,\sigma \dashv P:\quad \Cat_\kappa(Q)\rightleftarrows [\Map(Q)^{\op},\Poset],
\]
and in the involutive symmetric case,
\[
\mathcal C_{\s}\,\sigma \dashv P:\quad \Cat_{\kappa,\s}(Q)\rightleftarrows [\Map_{\s}(Q)^{\op},\Set].
\]
In this setting, \(P\) is always faithful; in the involutive symmetric case it is fully faithful, and the adjunction restricts to a left-exact reflection under the stated hypotheses [2507.20820].

The covering families used in the quantaloid case are defined by
\[
\id_q \le \bigvee_{i\in I} f_i f_i^{\circ}.
\]
For a presheaf \(F:\Map(Q)^{\op}\to \Set\), the symmetric completion \(Sc\,\sigma F\) is symmetrically complete iff two conditions hold. The first is local representability: for every symmetric singleton \(\sigma\) of \(Sc\,\sigma F\), there exists a covering family \((x_i,f_i)\) such that
\[
\sigma(a)\,f_i = M(a,x_i)
\]
for all \(a\in \sigma F\). The second is glueing along covers: for any covering family \((x_i,f_i)\) with
\[
f_i^{\circ}f_j \le M(x_i,x_j),
\]
there exists a unique \(x\in \sigma F\) with \(F(f_i)(x_i)=x\) for all \(i\). These are exactly the enriched restriction and glueing axioms specialized to quantaloids.

The worked topological example takes \(B=R(X)\), where objects are opens of a space \(X\), \(1\)-cells \(U\to V\) are opens \(W\subseteq U\wedge V\), and \(2\)-cells are inclusions. For a presheaf \(F\) on \(X\), \(\sigma F\) has as objects the disjoint union of sections over all opens, and homs
\[
N(s,t)=\text{the largest open where } s \text{ and } t \text{ are both defined and equal}.
\]
Then \(Sc\,\sigma F\) is complete iff \(F\) is a sheaf. The fiber pseudofunctor \(P_{Sc\,\sigma F}\) sends an open \(U\) to the discrete category of sections on \(U\), with inclusions acting by restriction.

The monoidal case behaves differently. For \(B=B_V\), the fiber category at the unique object satisfies
\[
P_{(\mathcal M,A)}(*)\cong \mathbf{VCat}(\mathcal I,-),
\]
so \(P\) recovers the underlying ordinary category functor. If every object of \(V\) is a small colimit of copies of the monoidal unit \(I\), as in \(V=\Set\) or \(V=\Ab\), then every complete \(B_V\)-category is a fixed point of the adjunction. The generalized metric-space example takes the quantale \(\overline{\mathbb R_+}\): then \(\Cat_k(B)\) is equivalent to complete metric spaces, while the adjunction records the underlying category whose arrows correspond to distance \(0\).

## 5. Other completion notions attached to bicategories and \(B\)-enrichment

The phrase “Complete B-Categories” is terminologically ambiguous because other papers use closely related language for different constructions. In the theory of bicategories enriched in a monoidal bicategory \(V\), the primary completion notion is free cocompletion under weighted bicolimits. A \(V\)-bicategory consists of objects, hom-objects \(B(x,y)\in V\), unit morphisms \(j_x:I\to B(x,x)\), composition morphisms
\[
m_{x,y,z}: B(y,z)\otimes B(x,y)\to B(x,z),
\]
and invertible associativity and unit \(2\)-cells. For a class of weights \(\Phi\), the construction \(\Phi(B)\) is the closure of representables in the \(V\)-bicategory \(M_B\) of right \(B\)-modules under \(\Phi\)-weighted colimits, and satisfies the universal property
\[
\Phi\text{-Cocts}(\Phi(B),C)\simeq V\text{-Bicat}(B,C),
\]
with weak inverse given by left Kan extension along the Yoneda-factorization \(Z:B\to \Phi(B)\) [1301.3191].

That theory explicitly emphasizes cocompletion rather than completeness. Indeed, the paper states that “the terminology in the paper focuses on cocompletion,” and that “complete \(B\)-category” in that sense is “\(\Phi\)-cocomplete \(V\)-bicategory.” This is therefore not the same notion as representability of adjointable distributors in \(\Dist(B)\); rather, it is a universal construction adjoining specified weighted bicolimits.

A second related notion appears for braided enriched monoidal categories. If \(\mathcal V\) is braided closed and \(\mathcal C\) is a \(\mathcal V\)-monoidal category, there is a completion operation
\[
\mathcal C \mapsto \overline{\mathcal C}
\]
whose objects are formal tensors \(a\blacktriangleleft u\), with hom-objects
\[
\overline{\mathcal C}(a\blacktriangleleft u\to b\blacktriangleleft v)
:=\widehat{\mathcal V}(u\to \mathcal C(a\to b)\,v).
\]
The completion is tensored, and for any \(\mathcal V\)-functor \(\mathcal F:\mathcal C\to \mathcal D\) with \(\mathcal D\) tensored, there is a tensored \(\mathcal V\)-functor
\[
\overline{\mathcal F}:\overline{\mathcal C}\to \mathcal D
\]
and a \(1_{\mathcal V}\)-graded \(\mathcal V\)-natural isomorphism \(\sigma:\mathcal F\Rightarrow \mathcal I\circ \overline{\mathcal F}\). In the monoidal setting, the corresponding classifying braided oplax monoidal functor \(\mathcal F^Z:\mathcal V\to Z(\mathcal T)\) is strong monoidal iff the original \(\mathcal V\)-monoidal category is tensored [1809.09782].

These constructions share a common theme: representables are embedded into a larger environment—modules, presheaves, or formal tensors—and “completion” means closure under a specified class of formally adjoined objects or weights. The exact property being added, however, differs substantially from the completeness of \(B\)-categories in the sheaf-theoretic sense.

## 6. Higher-categorical and homotopy-type-theoretic interpretations

A further, distinct use of “complete” arises in homotopy type theory through complete semi-Segal types. For \(n\in\{0,1,2\}\), a complete semi-Segal \(n\)-type is a semisimplicial type \((S_0,\dots,S_{n+2})\) equipped with: Segal conditions expressed as contractibility of inner horn fillers; a completeness property formulated via neutral edges; and an \((n,1)\)-categorical truncation condition on hom-types. The main theorem states that, for \(n\in\{0,1,2\}\), the type of complete semi-Segal \(n\)-types is equivalent to the type of univalent \((n,1)\)-categories [1707.03693].

The bicategorical case is \(n=2\). There, complete semi-Segal \(2\)-types have levels \(S_0,\dots,S_4\), Segal conditions for inner horns in dimensions \(2,3,4\), completeness, and \(S_1(x,y)\) a \(1\)-type. They are equivalent to univalent \((2,1)\)-categories: objects, \(1\)-morphisms as \(1\)-types, associators \(\alpha\), pentagonator, identities, unitors, triangle coherences, and univalence. Segal in dimension \(3\) yields the associator; Segal in dimension \(4\) yields the pentagonator; completeness reconstructs degeneracies and identities from neutral edges.

If “B-Categories” is read as “bicategories,” this provides a higher-categorical interpretation of “complete bicategorical structures.” The paper explicitly states that complete semi-Segal \(2\)-types model bicategories in the \((2,1)\)-sense: all \(2\)-morphisms are invertible and arise as equalities in hom \(1\)-types. This is aligned with the homotopical equality native to type theory, but it is not the same completion as the Cauchy-style representability condition of complete \(B\)-categories in bicategorical enrichment.

The comparison clarifies a recurrent source of confusion. In the semi-Segal setting, completeness recovers degeneracies and supports univalence; in the bicategory-enriched setting, completeness means that every adjointable distributor is representable; in free cocompletion, completion adjoins weighted bicolimits; and in braided enrichment, completion adjoins formal tensors to force tensoring. These notions are structurally adjacent, but they answer different categorical questions.

Source: https://www.emergentmind.com/topics/complete-b-categories