Co‑Nerve: A Bicategory Cellular Nerve

• Co‑Nerve is a homotopy coherent cellular nerve that uses normal pseudofunctors to embed bicategories fully into presheaves over Θ₂.
• It encodes bicategorical structure by explicitly capturing objects, 1‑cells, and 2‑cells, along with coherence via pasting compositions.
• The construction establishes Quillen equivalences linking bicategories, 2‑quasi‑categories, Rezk’s Θ‑spaces, and enriched Segal categories.

Co‑Nerve for bicategories, in the sense of Leinster’s construction, is the homotopy coherent cellular nerve: a presheaf on $\Theta_2$ defined by normal pseudofunctors out of Joyal’s two‑dimensional cell category. In the formulation developed in "A homotopy coherent cellular nerve for bicategories" (Campbell, 2019), it defines a fully faithful functor from the category of bicategories and normal pseudofunctors to the category of presheaves over $\Theta_2$, the nerve of a bicategory is a $2$‑quasi‑category in Ara’s sense, and the resulting adjunctions place bicategories, $2$‑quasi‑categories, Rezk’s $(2,2)$‑$\Theta$‑spaces, and quasi‑category‑enriched Segal categories into a common Quillen‑equivalence framework.

1. Definition on $\Theta_2$

The category $\Theta_2$ (also denoted $O_2$) has objects given by free strict $2$‑categories

$\Theta_2$0

generated by the $\Theta_2$1‑graph with objects $\Theta_2$2 and, for $\Theta_2$3, a hom‑graph $\Theta_2$4 with $\Theta_2$5 composable edges. Concretely, $\Theta_2$6 has hom‑categories

$\Theta_2$7

for $\Theta_2$8, with $\Theta_2$9 the ordinal category. Morphisms $2$0 consist of a simplicial map $2$1 and, for each $2$2, a functor

$2$3

equivalently a family of simplicial maps indexed by the slots between $2$4 and $2$5. The category carries a Reedy structure with degree

$2$6

and representables $2$7 together with their boundaries $2$8 play the role of cells and shells.

The homotopy coherent cellular nerve is the singular functor induced by the full inclusion $2$9. For any bicategory $2$0 and any cell $2$1,

$2$2

that is, the set of normal pseudofunctors $2$3. In this sense, the co‑nerve is not merely a record of objects and arrows; it is a $2$4‑diagram of normal pseudofunctors whose shape already carries the relevant $2$5‑dimensional composition data.

The inclusion $2$6 is dense, and the nerve $2$7 is fully faithful: the natural map

$2$8

is bijective. The paper also proves that $2$9 is determined by its restriction to the subcategory $(2,2)$0 on objects of degree $(2,2)$1 and monomorphisms from degree $(2,2)$2 to degree $(2,2)$3: the truncated nerve $(2,2)$4 is already fully faithful, and

$(2,2)$5

via right Kan extension (Campbell, 2019).

2. Low‑dimensional cells and coherence encoding

The low‑dimensional evaluations of $(2,2)$6 make explicit how the construction encodes objects, $(2,2)$7‑cells, $(2,2)$8‑cells, and bicategorical coherence. On $(2,2)$9, $\Theta$0 is the set of objects of $\Theta$1. On $\Theta$2, elements are $\Theta$3‑morphisms $\Theta$4 in $\Theta$5, with faces encoding source and target. On $\Theta$6, elements are $\Theta$7‑cells $\Theta$8 in $\Theta$9, with faces encoding source and target of $\Theta_2$0‑cells and degeneracies giving identities.

On $\Theta_2$1, elements are invertible $\Theta_2$2‑simplices $\Theta_2$3 witnessing a composite $\Theta_2$4; the nerve sends them to pasting composites in $\Theta_2$5 that combine the $\Theta_2$6‑cell data with the composition constraints of the pseudofunctor shape $\Theta_2$7. The formula recorded in the paper is that $\Theta_2$8 on $\Theta_2$9 is given by

$\Theta_2$0

On $\Theta_2$1, elements are commutative pasting equations involving the associator

$\Theta_2$2

in $\Theta_2$3; in the nerve they assert the axioms expressing preservation of associativity pasting by normal pseudofunctors.

The point is that composition and coherence are encoded intrinsically via pseudofunctorial constraints rather than imposed externally. This is the distinguishing feature of the homotopy coherent cellular nerve. A plausible implication is that the co‑nerve is adapted to weak rather than strict $\Theta_2$4‑categorical input precisely because the $\Theta_2$5‑cells are evaluated by normal pseudofunctors, not by strict $\Theta_2$6‑functors.

3. Fibrancy as a $\Theta_2$7‑quasi‑category

Ara’s model structure on $\Theta_2$8 takes cofibrations to be monomorphisms and generating weak equivalences to be all spine inclusions

$\Theta_2$9

together with

$O_2$0

where $O_2$1 is the strict nerve of the free $O_2$2‑category with a single invertible $O_2$3‑cell. Its fibrant objects are called $O_2$4‑quasi‑categories; they satisfy inner horn‑filling properties in $O_2$5. A horn‑filling style characterization states that $O_2$6 is fibrant iff $O_2$7 has right lifting for the set consisting of

$O_2$8

$O_2$9

and

$2$0

for all $2$1 and $2$2.

The main fibrancy theorem is that the nerve functor $2$3 participates as right adjoint in a Quillen adjunction

$2$4

between bicategories with Lack’s model structure and $2$5 with Ara’s model structure. Since every bicategory is fibrant in Lack’s model, $2$6 preserves fibrant objects; hence $2$7 is a $2$8‑quasi‑category for every bicategory $2$9. The proof strategy described in the paper is to show that $\Theta_2$00 sends the generators, including spines and $\Theta_2$01, to biequivalences in $\Theta_2$02, while $\Theta_2$03 preserves trivial fibrations and reflects them via lifting against boundary inclusions (Campbell, 2019).

This result identifies the co‑nerve as a fibrant $\Theta_2$04‑cellular model of bicategorical input. It succeeds exactly where the strict cellular nerve fails, because the coherent construction records associators and unitors through pseudofunctorial data.

4. Homotopy bicategory and equivalence criterion

For a $\Theta_2$05‑quasi‑category $\Theta_2$06, the paper constructs the homotopy bicategory $\Theta_2$07. The underlying bisimplicial set

$\Theta_2$08

obtained by restriction along

$\Theta_2$09

is a Joyal‑enriched Segal category: $\Theta_2$10 is discrete, the Segal maps for $\Theta_2$11 are weak categorical equivalences, and the $\Theta_2$12 are quasi‑categories. Applying $\Theta_2$13 levelwise, where $\Theta_2$14 is the left adjoint of $\Theta_2$15, yields a simplicial category $\Theta_2$16 which is a Tamsamani $\Theta_2$17‑category. Bicategory reflection $\Theta_2$18 then produces a bicategory $\Theta_2$19 with objects the elements of $\Theta_2$20 and hom‑categories

$\Theta_2$21

the homotopy categories of the hom‑quasi‑categories.

The universal property is

$\Theta_2$22

Moreover, the unit

$\Theta_2$23

is bijective on objects and an equivalence on hom‑quasi‑categories iff $\Theta_2$24 is $\Theta_2$25‑truncated. Consequently, $\Theta_2$26‑truncated $\Theta_2$27‑quasi‑categories are exactly those equivalent to coherent nerves of bicategories.

A central structural theorem gives an intrinsic criterion for weak equivalences of $\Theta_2$28‑quasi‑categories: a morphism $\Theta_2$29 is a weak equivalence iff it is essentially surjective on objects and fully faithful. Essential surjectivity means that

$\Theta_2$30

is essentially surjective on objects, equivalently

$\Theta_2$31

Fully faithfulness means that for all $\Theta_2$32, the induced map

$\Theta_2$33

is an equivalence of Kan complexes, that is, a weak homotopy equivalence between fibrant mapping spaces (Campbell, 2019).

5. Quillen equivalences and model comparisons

Lack’s model structure on $\Theta_2$34 has weak equivalences given by biequivalences, fibrations given by equifibrations, and trivial fibrations given by morphisms that are surjective on objects and surjective‑on‑objects equivalences on hom‑categories. Against this background, the adjunction

$\Theta_2$35

is Quillen, the derived right adjoint is fully faithful, and Lack’s structure is right‑induced along $\Theta_2$36.

The Quillen equivalence statement requires a Bousfield localisation on the $\Theta_2$37 side. Restricting to $\Theta_2$38‑truncated $\Theta_2$39‑quasi‑categories, namely fibrant objects local with respect to

$\Theta_2$40

the adjunction $\Theta_2$41 becomes a Quillen equivalence between bicategories and $\Theta_2$42. The paper then combines this with Ara’s comparison adjunctions

$\Theta_2$43

between $\Theta_2$44 and simplicial presheaves on $\Theta_2$45 to deduce that the composite

$\Theta_2$46

is a Quillen equivalence. Here $\Theta_2$47‑$\Theta_2$48‑spaces are Rezk $\Theta_2$49‑spaces $\Theta_2$50 such that each hom complete Segal space $\Theta_2$51 is $\Theta_2$52‑truncated.

The paper also proves a Quillen equivalence

$\Theta_2$53

between quasi‑category‑enriched Segal categories, denoted $\Theta_2$54 in the Hirschowitz–Simpson–Pellissier model, and Ara’s $\Theta_2$55‑quasi‑categories. The left adjoint $\Theta_2$56 takes a $\Theta_2$57‑set to its underlying bisimplicial set, and the right adjoint $\Theta_2$58 is right Kan extension along $\Theta_2$59. This validates $\Theta_2$60‑quasi‑categories as a robust $\Theta_2$61‑cellular model for $\Theta_2$62‑categories, compatible with enriched Segal models (Campbell, 2019).

6. Variants, examples, and significance

The paper contrasts the homotopy coherent cellular nerve with the strict cellular nerve $\Theta_2$63, defined using strict $\Theta_2$64‑functors $\Theta_2$65. The strict cellular nerve is fully faithful into presheaves but fails fibrancy in general unless $\Theta_2$66 is rigid, meaning it has no non‑identity invertible $\Theta_2$67‑cells. Nevertheless, the inclusion

$\Theta_2$68

is a weak equivalence. The paper also situates the construction relative to Street, Duskin, and Roberts nerves, which are simplicial nerves of bicategories or $\Theta_2$69‑categories, and to Lurie’s homotopy coherent nerve of simplicial categories. The analogy with Lurie’s construction is explicit: both encode homotopy coherent composition, but the target and cell shapes differ, since the present construction uses $\Theta_2$70‑cells rather than $\Theta_2$71‑simplices with enrichment.

A worked example is provided by a monoidal category $\Theta_2$72 regarded as a one‑object bicategory $\Theta_2$73. The single object is $\Theta_2$74; the $\Theta_2$75‑morphisms $\Theta_2$76 are objects of $\Theta_2$77; composition is tensor $\Theta_2$78; identity is $\Theta_2$79; and $\Theta_2$80‑morphisms are morphisms in $\Theta_2$81, with associator $\Theta_2$82 and unitors $\Theta_2$83 as the bicategory constraints. Then

$\Theta_2$84

$\Theta_2$85

$\Theta_2$86

Moreover, $\Theta_2$87 consists of coherent triangles $\Theta_2$88, and $\Theta_2$89 encodes the associativity pentagon. Inner horns for $\Theta_2$90 of $\Theta_2$91‑dimensional cells are filled in $\Theta_2$92 by the associator $\Theta_2$93 and unitors $\Theta_2$94 of $\Theta_2$95, matching Ara’s inner horn‑lifting notion via spines and $\Theta_2$96.

The significance of the co‑nerve is therefore twofold. First, it provides a fully faithful embedding of bicategories into $\Theta_2$97‑cellular presheaves, landing in Ara’s $\Theta_2$98‑quasi‑categories. Second, through the adjunctions and Quillen equivalences above, it connects bicategories to Rezk $\Theta_2$99‑spaces and enriched Segal categories as models of $2$00. This suggests that the homotopy coherent cellular nerve is not only a representation theorem for bicategories but also a comparison mechanism across several established higher‑categorical formalisms (Campbell, 2019).