Complete B-Categories Overview

• Complete B-Categories are enriched bicategories defined by the criterion that every distributor admitting a right adjoint is representable.
• They generalize Cauchy-complete categories and connect bicategorical structures with sheaf-theoretic adjunctions through constructions like the Yoneda functor.
• Alternative completions, including free cocompletion under weighted bicolimits and braided monoidal completions, illustrate varied applications in enriched category theory.

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 $B$ (Caramello et al., 28 Jul 2025). 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$0-matrix $B$1 between typed sets is a family of $B$2-cells $B$3 in $B$4, and a $B$5-category is an endomatrix $B$6 together with $B$7-cells

$B$8

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$9 (Caramello et al., 28 Jul 2025).

A $2$0-functor $2$1 is a type-preserving function $2$2 together with $2$3-cells

$2$4

compatible with $2$5 and $2$6. These form a category $2$7. If $2$8 carries an involution, one may further impose symmetry: $2$9 yielding a full subcategory $\Map(B)$0.

Distributors, also called modules or profunctors, are the bicategorical analogue of enriched profunctors. A distributor

$\Map(B)$1

is a rectangular $\Map(B)$2-matrix $\Map(B)$3 equipped with “double action” $\Map(B)$4-cells

$\Map(B)$5

satisfying unitality and associativity. Composition is defined by the coend

$\Map(B)$6

and distributors form a bicategory $\Map(B)$7. Representable distributors assemble into a pseudofunctor $\Map(B)$8, exhibiting $\Map(B)$9 as an equipment over $B$0.

Two specializations are fundamental. If $B$1 is a closed monoidal category, then the delooping $B$2 is a one-object bicategory and $B$3-categories are exactly $B$4-enriched categories. If $B$5 is a quantaloid, then $B$6-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 $B$7, a distributor $B$8 is called a map if it has a right adjoint $B$9 with unit and counit satisfying the triangle identities. Representable distributors $(2,1)$0 are maps with right adjoint $(2,1)$1. The defining completeness condition is then:

A $(2,1)$2-category $(2,1)$3 is complete if every distributor $(2,1)$4 that has a right adjoint in $(2,1)$5 is representable, i.e. $(2,1)$6 for some $(2,1)$7-functor $(2,1)$8 (Caramello et al., 28 Jul 2025).

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 $(2,1)$9 is a distributor

$B$0

equivalently a family of $B$1-cells $B$2 with action $B$3-cells

$B$4

satisfying the expected unit and associativity axioms. A singleton is a presingleton admitting a right adjoint $B$5.

For each $B$6, the representable singleton $B$7 has right adjoint $B$8; its unit is $B$9 and its counit is $(-)^{\circ}: B^{\op}\to B$0. The Yoneda $(-)^{\circ}: B^{\op}\to B$1-functor sends $(-)^{\circ}: B^{\op}\to B$2 to $(-)^{\circ}: B^{\op}\to B$3. The $(-)^{\circ}: B^{\op}\to B$4-category of singletons, denoted $(-)^{\circ}: B^{\op}\to B$5, is itself complete, and the co-Yoneda lemma states that a $(-)^{\circ}: B^{\op}\to B$6-category is complete iff it is canonically the colimit of all its singletons in $(-)^{\circ}: B^{\op}\to B$7. The resulting completion functor

$(-)^{\circ}: B^{\op}\to B$8

sends $(-)^{\circ}: B^{\op}\to B$9 to its singleton category, and is left adjoint to the inclusion $A$0. In the monoidal case $A$1, this recovers the usual Cauchy-completion of $A$2-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 $A$3-presheaves on $A$4

The bicategory $A$5 has the same objects as $A$6, $A$7-cells given by left adjoints in $A$8, and $A$9-cells those of $t:A\to \Ob(B)$0 between maps. The sheaf-theoretic side uses pseudofunctors

$t:A\to \Ob(B)$1

together with oplax natural transformations.

From such an $t:A\to \Ob(B)$2, the Grothendieck construction $t:A\to \Ob(B)$3 produces a $t:A\to \Ob(B)$4-category. Its objects are the disjoint union

$t:A\to \Ob(B)$5

typed by the ambient object $t:A\to \Ob(B)$6. For $t:A\to \Ob(B)$7, the hom-object $t:A\to \Ob(B)$8 is defined as the colimit of the diagram whose objects are pairs $t:A\to \Ob(B)$9 consisting of a map $B$00 and a morphism $B$01 in $B$02. The unit $B$03 is the canonical inclusion $B$04, and composition

$B$05

is assembled from the universal property of these colimits and the pseudofunctoriality of $B$06 (Caramello et al., 28 Jul 2025).

In the opposite direction, a complete $B$07-category $B$08 determines a pseudofunctor

$B$09

For $B$10, the category $B$11 has as objects the elements $B$12 of type $B$13, and as morphisms $B$14 the $B$15-cells

$B$16

Composition is

$B$17

and identities are $B$18. For a map $B$19, the action on objects is

$B$20

the unique element representing the singleton $B$21. Completeness is essential here: it guarantees existence and uniqueness of $B$22.

The central result is the adjunction

$B$23

Explicitly, for any pseudofunctor $B$24 and any complete $B$25-category $B$26, there is a natural bijection

$B$27

The construction uses explicit maps $B$28 and $B$29 between $B$30-functors $B$31 and oplax transformations $B$32. The use of oplax, rather than necessarily pseudo-, transformations is deliberate: the paper notes that $B$33 need not be a map in $B$34, 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

$B$35

and in the involutive symmetric case,

$B$36

In this setting, $B$37 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 (Caramello et al., 28 Jul 2025).

The covering families used in the quantaloid case are defined by

$B$38

For a presheaf $B$39, the symmetric completion $B$40 is symmetrically complete iff two conditions hold. The first is local representability: for every symmetric singleton $B$41 of $B$42, there exists a covering family $B$43 such that

$B$44

for all $B$45. The second is glueing along covers: for any covering family $B$46 with

$B$47

there exists a unique $B$48 with $B$49 for all $B$50. These are exactly the enriched restriction and glueing axioms specialized to quantaloids.

The worked topological example takes $B$51, where objects are opens of a space $B$52, $B$53-cells $B$54 are opens $B$55, and $B$56-cells are inclusions. For a presheaf $B$57 on $B$58, $B$59 has as objects the disjoint union of sections over all opens, and homs

$B$60

Then $B$61 is complete iff $B$62 is a sheaf. The fiber pseudofunctor $B$63 sends an open $B$64 to the discrete category of sections on $B$65, with inclusions acting by restriction.

The monoidal case behaves differently. For $B$66, the fiber category at the unique object satisfies

$B$67

so $B$68 recovers the underlying ordinary category functor. If every object of $B$69 is a small colimit of copies of the monoidal unit $B$70, as in $B$71 or $B$72, then every complete $B$73-category is a fixed point of the adjunction. The generalized metric-space example takes the quantale $B$74: then $B$75 is equivalent to complete metric spaces, while the adjunction records the underlying category whose arrows correspond to distance $B$76.

5. Other completion notions attached to bicategories and $B$77-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 $B$78, the primary completion notion is free cocompletion under weighted bicolimits. A $B$79-bicategory consists of objects, hom-objects $B$80, unit morphisms $B$81, composition morphisms

$B$82

and invertible associativity and unit $B$83-cells. For a class of weights $B$84, the construction $B$85 is the closure of representables in the $B$86-bicategory $B$87 of right $B$88-modules under $B$89-weighted colimits, and satisfies the universal property

$B$90

with weak inverse given by left Kan extension along the Yoneda-factorization $B$91 (Garner et al., 2013).

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$92-category” in that sense is “$B$93-cocomplete $B$94-bicategory.” This is therefore not the same notion as representability of adjointable distributors in $B$95; rather, it is a universal construction adjoining specified weighted bicolimits.

A second related notion appears for braided enriched monoidal categories. If $B$96 is braided closed and $B$97 is a $B$98-monoidal category, there is a completion operation

$B$99

whose objects are formal tensors $2$00, with hom-objects

$2$01

The completion is tensored, and for any $2$02-functor $2$03 with $2$04 tensored, there is a tensored $2$05-functor

$2$06

and a $2$07-graded $2$08-natural isomorphism $2$09. In the monoidal setting, the corresponding classifying braided oplax monoidal functor $2$10 is strong monoidal iff the original $2$11-monoidal category is tensored (Morrison et al., 2018).

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 $2$12-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 $2$13, a complete semi-Segal $2$14-type is a semisimplicial type $2$15 equipped with: Segal conditions expressed as contractibility of inner horn fillers; a completeness property formulated via neutral edges; and an $2$16-categorical truncation condition on hom-types. The main theorem states that, for $2$17, the type of complete semi-Segal $2$18-types is equivalent to the type of univalent $2$19-categories (Capriotti et al., 2017).

The bicategorical case is $2$20. There, complete semi-Segal $2$21-types have levels $2$22, Segal conditions for inner horns in dimensions $2$23, completeness, and $2$24 a $2$25-type. They are equivalent to univalent $2$26-categories: objects, $2$27-morphisms as $2$28-types, associators $2$29, pentagonator, identities, unitors, triangle coherences, and univalence. Segal in dimension $2$30 yields the associator; Segal in dimension $2$31 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$32-types model bicategories in the $2$33-sense: all $2$34-morphisms are invertible and arise as equalities in hom $2$35-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 $2$36-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.