nerve in nLab

Contents

Contents

The nerve is the right adjoint of a pair of adjoint functors that exists in many situations. For the general abstract theory behind this see

• nerve and realization.

Idea

As soon as any locally small category CC comes equipped with a cosimplicial object

Δ C:Δ→C \Delta_C : \Delta \to C

that we may think of as determining a realization of the standard nn-simplex in CC, we make every object of CC probeable by simplices in that there is now a way to find the set

N(A) n:=Hom C(Δ C[n],A) N(A)_n := Hom_C(\Delta_C[n],A)

of ways to map the nn-simplex into a given object AA.

These collections of sets evidently organize into a simplicial set

N(A):Δ op→Set. N(A) : \Delta^{op} \to Set \,.

This simplicial set is called the nerve of AA (with respect to the chosen realization of the standard simplices in CC). Typically the nerve defines a functor N:C→Set Δ opN \colon C \to Set^{\Delta^op} that has a left adjoint |⋅|:Set Δ op→C|\cdot| \colon Set^{\Delta^op} \to C called realization.

There are many generalizations of this procedure, some of which are described below.

Definition

(notice that for the moment the following gives just one particular case of the more general notion of nerve)

Let SS be one of the categories of geometric shapes for higher structures, such as the globe category GG, the simplex category Δ\Delta, the cube category □\Box, the cycle category Λ\Lambda of Connes, or certain category Ω\Omega related to trees whose corresponding presheaves are dendroidal sets.

If CC is any locally small category or, more generally, a VV-enriched category equipped with a functor

i:S→C i : S \to C

we obtain a functor

N:C→V S op N : C \to V^{S^{op}}

from CC to globular sets or simplicial sets or cubical sets, respectively, (or the corresponding VV-objects) given on an object c∈Cc \in C by the restricted Yoneda embedding

N i(c):S op→iC op→C(−,c)V. N_i(c) : S^{op} \stackrel{i}\to C^{op} \stackrel{C(-,c)}{\to} V \,.

This N i(c)N_i(c) is the nerve of cc with respect to the chosen i:S→Ci : S \to C. In other words, N=i *∘YN = i^* \circ Y where Y:C→[C op,V]Y: C \to [C^{op}, V] is the curried Hom functor; if V=SetsV=\mathsf{Sets} then YY is the Yoneda embedding.

Typically, one wants that ii is dense functor, i.e. that every object cc of CC is canonically a colimit of a diagram of objects in MM, more precisely,

colim((i/c)⟶pr SS→iC)=c, \mathrm{colim}((i/c)\stackrel{\mathrm{pr}_S}{\longrightarrow} S \stackrel{i}{\to} C) = c,

which is equivalent to the requirement that the corresponding nerve functor is fully faithful (in other words, if ii is inclusion then SS is a left adequate subcategory of CC in terminology of Isbell 60). The nerve functor may be viewed as a singular functor? of the functor ii.

Examples

Nerve of a 1-category

For fixing notation, recall that the source and target maps of a small category form a span in the category Span(Set)Span(Set) where composition is given by a pullback (fiber product). The pairs of composable morphisms of a category are then obtained composing its source/target span with itself.

Definition

More abstractly, this construction is described as follows. Recall that

So the set N(𝒞) nN(\mathcal{C})_n of nn-simplices of the nerve is the set of functors {0→1→⋯→n}→𝒞\{0 \to 1 \to \cdots \to n\} \to \mathcal{C}. This is clearly the same as the set of sequences of composable morphisms in 𝒞\mathcal{C} of length nn obtained by iterated fiber product (as above for pairs of composables):

N(𝒞) n=Mor(𝒞)× Obj(𝒞)Mor(𝒞)× Obj(𝒞)⋯× Obj(𝒞)Mor(𝒞)⏟ nfactors N(\mathcal{C})_n = \underbrace{ Mor(\mathcal{C}) \times_{Obj(\mathcal{C})} Mor(\mathcal{C}) \times_{Obj(\mathcal{C})} \cdots \times_{Obj(\mathcal{C})} Mor(\mathcal{C}) }_{n \medspace factors}

The collection of all functors between linear orders

{0→1→⋯→n}→{0→1→⋯→m} \{ 0 \to 1 \to \cdots \to n \} \to \{ 0 \to 1 \to \cdots \to m \}

is generated from those that map almost all generating morphisms k→k+1k \to k+1 to another generating morphism, except at one position, where they

• map a single generating morphism to the composite of two generating morphisms

  δ i n:[n−1]→[n] \delta^n_i : [n-1] \to [n]

  δ i n:((i−1)→i)↦((i−1)→i→(i+1)) \delta^n_i : ((i-1) \to i) \mapsto ((i-1) \to i \to (i+1))

• map one generating morphism to an identity morphism

  σ i n:[n+1]→[n] \sigma^n_i : [n+1] \to [n]

  σ i n:(i→i+1)↦Id i \sigma^n_i : (i \to i+1) \mapsto Id_i

It follows that, for instance

• for (d 0→f 1d 1,d 1→f 2d 2,d 2→f 3d 3)∈N(D) 3(d_0 \stackrel{f_1}{\to} d_1, d_1 \stackrel{f_2}{\to} d_2, d_2 \stackrel{f_3}{\to} d_3) \in N(D)_3 the image under d 1:=N(𝒞)(δ 1):N(𝒞) 3→N(𝒞) 2d_1 := N(\mathcal{C})(\delta_1) : N(\mathcal{C})_3 \to N(\mathcal{C})_2 is obtained by composing the first two morphisms in this sequence: (d 0→f 2∘f 1d 2,d 2→f 3d 3)∈N(𝒞) 2(d_0 \stackrel{f_2 \circ f_1}{\to} d_2, d_2 \stackrel{f_3}{\to} d_3) \in N(\mathcal{C})_2

• for (d 0→f 1d 1)∈N(𝒞) 1(d_0 \stackrel{f_1}{\to} d_1) \in N(\mathcal{C})_1 the image under s 1:=N(𝒞)(σ 1):N(𝒞) 1→N(𝒞) 2s_1 := N(\mathcal{C})(\sigma_1) : N(\mathcal{C})_1 \to N(\mathcal{C})_2 is obtained by inserting an identity morphism: (d 0→f 1d 1,d 1→Id d 1d 1)∈N(𝒞) 2(d_0 \stackrel{f_1}{\to} d_1, d_1 \stackrel{Id_{d_1}}{\to} d_1) \in N(\mathcal{C})_2.

In this way, generally the face and degeneracy maps of the nerve of a category come from composition of morphisms and from inserting identity morphisms.

In particular in light of their generalization to nerves of higher categories, discussed below, the cells in the nerve N(𝒞)N(\mathcal{C}) have the following interpretation:

• N(𝒞) 0={d|d∈Obj(𝒞)}N(\mathcal{C})_0 = \{d | d \in Obj(\mathcal{C})\} is the collection of objects of 𝒞\mathcal{C};

• N(𝒞) 1=Mor(𝒞)={d→fd′|f∈Mor(D)}N(\mathcal{C})_1 = Mor(\mathcal{C}) = \{d \stackrel{f}{\to} d' | f \in Mor(D)\} is the collection of morphisms of DD;

• N(𝒞) 2={ d 1 f 1↗ ⇓ ∃! ↘ f 2 d 0 →f 2∘f 1 d 2|(f 1,f 2)∈Mor(D) t× sMor(D)}N(\mathcal{C})_2 = \left\{ \left. \array{ && d_1 \\ & {}^{f_1}\nearrow &\Downarrow^{\exists !}& \searrow^{f_2} \\ d_0 &&\stackrel{f_2 \circ f_1}{\to}&& d_2 } \right| (f_1, f_2) \in Mor(D) {}_t \times_s Mor(D) \right\} is the collection of composable morphisms in 𝒞\mathcal{C} as in the diagram The 2-cell itself is to be read as the composition operation, which is unique for an ordinary category (there is just one way to compose two morphisms);

• N(𝒞) 3={d 1 →f 2 d 2 f 1↑ f 2∘f 1↗ ↓ f 3 d 0 →f 3∘(f 2∘f 1) d 3⇒∃!d 1 →f 2 d 2 f 1↑ ↘ f 3∘f 2 ↓ f 3 d 0 →(f 3∘f 2)∘f 1 d 3|(f 3,f 2,f 1)∈Mor(D) t× sMor(D) t× sMor(D)}N(\mathcal{C})_3 = \left\{ \left. \array{ d_1 &\stackrel{f_2}{\to}& d_2 \\ {}^{f_1}\uparrow & {}^{f_2 \circ f_1}\nearrow & \downarrow^{f_3} \\ d_0 &\stackrel{f_3\circ (f_2\circ f_1)}{\to}& d_3 } \;\;\;\;\;\stackrel{\exists !}{\Rightarrow} \;\;\;\;\; \array{ d_1 &\stackrel{f_2}{\to}& d_2 \\ {}^{f_1}\uparrow & \searrow^{f_3\circ f_2} & \downarrow^{f_3} \\ d_0 &\stackrel{(f_3\circ f_2) \circ f_1}{\to}& d_3 } \right| (f_3,f_2, f_1) \in Mor(D) {}_t \times_s Mor(D) {}_t \times_s Mor(D) \right\} is the collection of triples of composable morphisms as in the diagram to be read as the unique associators that relate one way to compose three morphisms using the above 2-cells to the other way.

Examples

Properties

The following lists some characteristic properties of simplicial sets that are nerves of categories.

See at Segal condition for more on this.

(e.g. Kerodon, Prop. 1.2.3.1; see also at inner fibration.)

(cf. e.g. Kerodon, Prop. 1.2.4.2)

Here the point as compared to the previous statements is that in particular all the outer horns have fillers for n>3n \gt 3.

The existence of inverse morphisms in CC corresponds to the fact that in the Kan complex N(C)N(C) the “outer” horns

d 0 ↘ f d 1 →Id d 1 d 1and d 1 f↗ d 0 →Id d 0 d 1 \array{ && d_0 \\ & && \searrow^{f} \\ d_1 &&\stackrel{Id_{d_1}}{\to} && d_1 } \;\;\; \;\;\; and \;\;\; \;\;\; \array{ && d_1 \\ & {}^f\nearrow && \\ d_0 &&\stackrel{Id_{d_0}}{\to} && d_1 }

have fillers

d 0 f −1↗ ↘ f d 1 →Id d 1 d 1and d 1 f↗ ↘ f −1 d 0 →Id d 0 d 0 \array{ && d_0 \\ & {}^{f^{-1}}\nearrow&& \searrow^{f} \\ d_1 &&\stackrel{Id_{d_1}}{\to} && d_1 } \;\;\; \;\;\; and \;\;\; \;\;\; \array{ && d_1 \\ & {}^f\nearrow && \searrow^{f^{-1}} \\ d_0 &&\stackrel{Id_{d_0}}{\to} && d_0 }

(even unique fillers, due to the above).

It suggests the sense that a Kan complex models an ∞-groupoid. The possible lack of uniqueness of fillers in general gives the ‘weakness’ needed, whilst the lack of a coskeletal property requirement means that the homotopy type it represents has enough generality, not being constrained to be a 1-type.

(e.g Kerodon, Prop. 1.2.2.1; Rezk 2022, Prop. 4.10)

So functors between locally small categories are in bijection with morphisms of simplicial sets between their nerves.

Unwrapping this definition inductively in (n+m)(n+m), this says that a simplicial set is the nerve of a category if and only if all its cells in degree ≥2\geq 2 are unique compositors, associators, pentagonators, etc of composition of 1-morphisms. No non-trivial such structure cells appear and no further higher cells appear.

This characterization of categories in terms of nerves directly leads to the model of (∞,1)-category in terms of complete Segal spaces by replacing in the above discussion sets by topological spaces (or something similar, like Kan complexes) and pullbacks by homotopy pullbacks.

(e.g. Duskin 1975, §0.18(b), Joyal 2008, Cor. 1.2)

Hence in the nerve of a category, all horn inclusions Λ[n] i↪Δ[n]\Lambda[n]_i \hookrightarrow \Delta[n] have unique fillers for n>3n \gt 3, and all boundary inclusions ∂Δ[n]↪Δ[n]\partial \Delta[n] \hookrightarrow \Delta[n] have unique fillers for n≥3n \geq 3.

In summary:

Nerve of a 2-category

For 2-categories modeled as bicategories the nerve operation is called the Duskin nerve.

This is theorem 8.6 in (Duskin)

For a 2-category, regarded as a Cat-internal category one can apply the nerve operation for categories in stages, to obtain the double nerve.

Nerve of a 3-category

One also has a nerve operation for 3-categories modeled as tricategories: the Street nerve.

This is the main result of (Carrasco, 2014).

Nerve of an ω\omega-category

• For strict omega-categories there is a nerve induced by the orientals; see omega-nerve.

Nerve of chain complexes

Let Ch +Ch_+ be the category of chain complexes of abelian groups, then there is a cosimplicial chain complex

C •:Δ→Ch + C_\bullet : \Delta \to Ch_+

given by sending the standard nn-simplex Δ[n]\Delta[n] first to the free simplicial group F(Δ[n])F(\Delta[n]) over it and then that to the normalized Moore complex. This is a small version of the ordinary homology chain complex of the standard nn-simplex.

The nerve induced by this cosimplicial object was first considered in

• D. Kan, Functors involving c.s.s complexes, Transactions of the American Mathematical Society, Vol. 87, No. 2 (Mar., 1958), pp. 330–346 (jstor)

The nerve/realization adjunction induced from this is the Dold-Kan correspondence. See there for more details.

Geometric realization

Often the operation of taking the nerve of a (higher) category is followed by forming the geometric realization of the corresponding cellular set.

Nerves and higher categories

For many purposes it is convenient to conceive categories and especially ∞-categories entirely in terms of their nerves: those simplicial sets that arise as certain nerves are usually characterized by certain properties. So one can turn this around and define an ∞-category as a simplicial set with certain properties. This is the strategy of a geometric definition of higher category. Examples for this are complicial sets, Kan complexes, quasi-categories, simplicial T-complexes,…

Internal nerve

A variant of the nerve construction can also be applied internally within a category, to any internal category, see the discussion at internal category.

Direct categories versus finite simplicial sets

If a direct category has finitely many objects then its nerve is a finite simplicial set. Conversely, if a finite simplicial set is the nerve of a category then the category is a direct category with finitely many objects.

Properties

(Non-)Preservation of colimits

While the nerve operation is a right adjoint (this Prop.) and hence preserves all limits, the nerve operation does not preserve all colimits (Exp. ), hence is not a left adjoint.

However, it does preserve some colimits (Exp. ); rather special ones, but of central importance in the theory of classifying spaces constructed via geometric realization of simplicial topological spaces (Exp. ).

(In the following Exp. we use “card” instead of the more common notation “|−|{\vert - \vert}” for cardinality (of underlying sets) in order not to clash with the notation for geometric realization, even if the latter is not directly involved in the following examples.)

The principle behind Exp. is readily seen to be, more generally, the following:

• Barratt nerve

• Duskin nerve

• Street nerve

• ∞-nerve

• homotopy coherent nerve

• dg-nerve

• monad with arities

References

For covers

The notion of the nerve of a cover (in modern parlance: of its Cech groupoid) appears in:

• Paul Alexandroff, Section 9 of: Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung, Mathematische Annalen 98 (1928), 617–635 (doi:10.1007/BF01451612).

For categories

The notion of the nerve of a general category already appears in

• Alexander Grothendieck, above Proposition 4.1 of: Techniques de construction et théorèmes d’existence en géométrie algébrique III : préschémas quotients, Séminaire Bourbaki : années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, (numdam:SB_1960-1961__6__99_0, pdf)

Another early appearance in print is:

• Graeme Segal, Section 2 of: Classifying spaces and spectral sequences, Publications Mathématiques de l’IHÉS, Volume 34 (1968), p. 105-112 (numdam:PMIHES_1968__34__105_0)

  (in the context of constructing classifying spaces for principal bundles in algebraic topology)

Review and exposition:

• Saunders MacLane, §XII.2 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

• Kerodon, The Nerve of a Category (002L)

• Kerodon, The Nerve of a Groupoid (0035)

• Charles Rezk, Part 1 of: Introduction to quasicategories (2022) [pdf, pdf]

See also:

• Tom Leinster, p. 117 onwards in: Higher operads, higher categories , London Mathematical Society Lecture Note Series, 298. Cambridge Univ. Press 2004. xiv+433 pp. ISBN: 0-521-53215-9 (arXiv:math.CT/0305049)

• Tom Leinster, How I learned to love the nerve construction, nn-Category Café, January 6, 2008.

  (an explanation of how the simplex category and the nerve construction arise canonically from the free category monad)

See also:

• John Isbell, Adequate subcategories, Illinois J. Math. 4, 541–552 (1960) (doi:10.1215/ijm/1255456274)

• W. G. Dwyer, D. M. Kan, Singular functors and realization functors, Nederl. Akad. Wetensch. Indag. Math. 46 (1984), no. 2, 147–153. pdf

For higher categories

For strict omega-categories:

• Ross Street, The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987) 283-335; MR89a:18019 (pdf, doi:10.1016/0022-4049(87)90137-X).

For 2-categories and bicategories:

• Manuel Bullejos, Antonio M. Cegarra, On the geometry of 2-categories and their classifying spaces, K-Theory 29 3 (2003) 211-229 [doi:10.1023/B:KTHE.0000006921.50151.00, pdf]

• Stephen Lack, Simona Paoli, 2-nerves for bicategories, K-Theory 38 (2008) [arXiv:math/0607271, doi:10.1007/s10977-007-9013-2]

• Paul Bressler, Alexander Gorokhovsky, Ryszard Nest, Boris Tsygan, Formality for algebroids I: Nerves of two-groupoids [arxiv:1211.6603v3]

For 3-categories:

• Pilar Carrasco, Nerves of Trigroupoids as Duskin-Glenn’s 33-Hypergroupoids, Applied Categorical Structures 23.5 (2015): 673-707 (doi:10.1007/s10485-014-9374-7)