nLab geometric nerve of a bicategory

Idea

The (unitary) geometric nerve is a natural nerve operation on bicategories. It is a functor from BiCat to sSet. This is also sometimes called the Duskin nerve. The notion is implicit in work by R. Street (1987). The direct approach was used by Duskin in work at about the same time, as explained in both articles. (Duskin’s article directly on the idea was published in 2002.)

The construction, thus, yields a functor:

$$N : BiCat_{NLax} \to sSet \,.$$

extending the ordinary nerve construction on the category of small categories, where morphisms of BiCat are normal lax 2-functors: these are the lax 2-functors which strictly preserve identities.

Special cases of the construction relate to earlier constructions relating to the homotopy coherent nerve, see below for more detail.

Definition

We may think of the simplex category $\Delta$ as the full subcategory of Cat on the categories free on non-empty finite linear graphs. This gives the canonical inclusion $\Delta \hookrightarrow Cat$ that defines the ordinary nerve of categories.

There is also the canonical embedding of categories into bicategories. Combined this gives the inclusion

$$\Delta \hookrightarrow Cat \hookrightarrow BiCat \,.$$

The bicategorical nerve is the nerve induced from that. So for $C$ a bicategory we have

$$N(C) : [k] \mapsto BiCat_{NLax}(\Delta[k], C) \,.$$

There are also an oplax version and two non-normalized versions.

Properties

• The simplicial sets in the image of the geometric nerve are 3-coskeletal.

• The geometric nerve identifies precisely bigroupoids with 2-hypergroupoids: those Kan complexes for which the horn fillers in dimension $\geq 3$ are unique . (Theorem 8.6 in (Duskin 2002))

(This shows in particular that bigroupoids model all homotopy 2-types.)

• The nerve is a full and faithful functor $BiCat_{NLax}\to sSet$.

Example

Any strict 2-category determines both a ‘bicategory’ in the above sense (since a ‘strict’ thing is also a ‘weak’ one) and a simplicially enriched category. The latter is found by taking the nerve of each ‘hom-category’. The Duskin nerve of a 2-category is the same as the homotopy coherent nerve of the corresponding $sSet$-category. This can also be applied to 2-groupoids and, thus, results in a classifying space construction for crossed modules.

Picturing the Duskin nerve

Following (Johnson–Yau, Section 5.4), one may picture the Duskin nerve $N(\mathcal{C})$ of a bicategory $(\mathcal{C},1^{\mathcal{C}},\circ_{\mathcal{C}},\alpha^{\mathcal{C}},\lambda^{\mathcal{C}},\rho^{\mathcal{C}})$ as follows:

1. The $0$-simplices of $N(\mathcal{C})$ are the objects of $\mathcal{C}$;

2. The $1$-simplices of $N(\mathcal{C})$ are the $1$-morphisms of $\mathcal{C}$;

3. The $2$-simplices of $N(\mathcal{C})$ are quadruples $(i,j,k,\theta)$ as in the diagram

   

   where $A,B,C\in\mathrm{Obj}(\mathcal{C})$, $i,j,k\in\mathrm{Mor}_1(\mathcal{C})$ and $\theta\colon j\circ i\Rightarrow k$ is a $2$-morphism of $\mathcal{C}$;

4. The $3$-simplices of $N(\mathcal{C})$ are $14$-tuples

   $$(A_{0},A_{1},A_{2},A_{3},f_{01},f_{02},f_{03},f_{12},f_{13},f_{23},\theta_{012},\theta_{013},\theta_{023},\theta_{123})$$

   as in the diagram such that we have an equality of pasting diagrams in $\mathcal{C}$;

5. The $n$-simplices of $N(\mathcal{C})$ consist of

   • A collection $\{A_{i}\}_{0\leq i\leq n}$ of objects of $\mathcal{C}$,
   • A collection $\{f_{ij}\colon A_{i}\longrightarrow A_{j}\}_{0\leq i\lt j\leq n}$ of $1$-morphisms of $\mathcal{C}$, and
   • A collection $\{\theta_{ijk}\colon f_{jk}\circ f_{ij}\Rightarrow f_{ik}\}_{0\leq i\lt j\lt k\leq n}$ of $2$-morphisms of $\mathcal{C}$

   such that, for each $i,j,k\in\mathbb{N}$ with $0\leq i\lt j\lt k\leq n$, we have an equality of pasting diagrams in $\mathcal{C}$;

6. The degeneracy map

   $$\mathrm{s}^{0}_{0}\colon N_{0}(\mathcal{C})\longrightarrow N_{1}(\mathcal{C})$$

   of $N(\mathcal{C})$ in degree $0$ is the map sending a $0$-simplex $A$ of $N(\mathcal{C})$ (i.e. an object $A$ of $\mathcal{C}$) to the $1$-simplex $\mathrm{id}_{A}\colon A\to A$.

7. The degeneracy maps

   $$\mathrm{s}^{1}_{0} \colon N_{1}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),$$
   $$\mathrm{s}^{1}_{1} \colon N_{1}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),$$

   of $N(\mathcal{C})$ in degree $1$ are the maps described as follows: given a $1$-simplex $\sigma=(A\xrightarrow{f}B)$ of $N(\mathcal{C})$, we have

8. The degeneracy maps in degree $2$

   $$\mathrm{s}^{2}_{0} \colon N_{2}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),$$
   $$\mathrm{s}^{2}_{1} \colon N_{2}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),$$
   $$\mathrm{s}^{2}_{2} \colon N_{2}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),$$

   of $N(\mathcal{C})$ in degree $2$ are the maps described as follows: given a $2$-simplex of $N(\mathcal{C})$, we have

9. The face maps

   $$\mathrm{d}^{1}_{0} \colon N_{1}(\mathcal{C})\longrightarrow N_{0}(\mathcal{C}),$$
   $$\mathrm{d}^{1}_{1} \colon N_{1}(\mathcal{C})\longrightarrow N_{0}(\mathcal{C}),$$

   of $N(\mathcal{C})$ in degree $1$ are given by

   $$\mathrm{d}^{1}_{0}(A\xrightarrow{f}B)=B$$
   $$\mathrm{d}^{1}_{1}(A\xrightarrow{f}B)=A$$

10. The face maps

    $$\mathrm{d}^{2}_{0} \colon N_{2}(\mathcal{C})\longrightarrow N_{1}(\mathcal{C}),$$
    $$\mathrm{d}^{2}_{1} \colon N_{2}(\mathcal{C})\longrightarrow N_{1}(\mathcal{C}),$$
    $$\mathrm{d}^{2}_{1} \colon N_{2}(\mathcal{C})\longrightarrow N_{1}(\mathcal{C}),$$

    of $N(\mathcal{C})$ in degree $2$ are described as follows: given a $2$-simplex

of $N(\mathcal{C})$, we have 11. The face maps

$$\mathrm{d}^{3}_{0} \colon N_{3}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),$$

$$\mathrm{d}^{3}_{1} \colon N_{3}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),$$

$$\mathrm{d}^{3}_{2} \colon N_{3}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),$$

$$\mathrm{d}^{3}_{3} \colon N_{3}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),$$

of $N(\mathcal{C})$ in degree $3$ are described as follows: given a $3$-simplex of $N(\mathcal{C})$, we have 12. The face maps

$$\mathrm{d}^{4}_{0} \colon N_{4}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),$$

$$\mathrm{d}^{4}_{1} \colon N_{4}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),$$

$$\mathrm{d}^{4}_{2} \colon N_{4}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),$$

$$\mathrm{d}^{4}_{3} \colon N_{4}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),$$

$$\mathrm{d}^{4}_{4} \colon N_{4}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),$$

of $N(\mathcal{C})$ in degree $4$ are described as follows: given a $4$-simplex $\sigma$ of $N(\mathcal{C})$ as in the diagram we have

Related concepts

• nerve

• geometric nerve of a bicategory

• geometric nerve of a tricategory

• ∞-nerve

• homotopy coherent nerve

References

• Ross Street, The algebra of oriented simplexes, Journal of Pure and Applied Algebra, Volume 49, Issue 3, December 1987, Pages 283–335

• John Duskin, Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories, Theory and Applications of Categories 9 10 (2002) 198–308 [tac:9-10]

• V. Blanco, M. Bullejos, E. Faro, A Full and faithful Nerve for 2-categories, Applied Categorical Structures, Vol 13-3, 223-233, 2005. (See also arxiv).

• Niles Johnson, Donald Yau, 2-Dimensional Categories (arXiv:2002.06055).