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homotopy coherent diagram

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Higher category theory

higher category theory

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Extra properties and structure

1-categorical presentations

Homotopy theory

Contents

Idea

A homotopy coherent diagram is a diagram of objects in a homotopical category, where commutativity is replaced by explicit homotopies, those homotopies are to then be coherently linked by higher homotopies … and so on.

It is a model for an (∞,1)-functor.

The idea perhaps intuitively makes sense but the management of the interactions between the various levels of homotopy requires care. The ideas were handled in various ways , but we will concentrate on approaches linked to the initial work of Michael Boardman and Rainer Vogt and then developed further by Jean-Marc Cordier and Tim Porter.

We will often use h.c. as an abbreviation for ‘’homotopy coherent’’.

Definition

In components

The original definition of Vogt, 1973 is essentially the following.

Suppose now that we have the h.c. diagram F:S(𝔸)F : S(\mathbb{A}) \to \mathcal{B}. This is specified by assignments:

  • to each object aa of 𝔸\mathbb{A}, it assigns an object F(a)F(a) of \mathcal{B};

  • for each string of composable morphisms in 𝔸\mathbb{A},

σ=(f 0,,f n)\sigma = (f_0, \ldots, f_n)

starting at aa and ending at bb, a simplicial map

F(σ):S(𝔸)(0,n+1)(F(a),F(b)),F(\sigma) : S(\mathbb{A})(0,n+1) \to \mathcal{B}(F(a), F(b)),

that is, a higher homotopy

F(σ):Δ[1] n(F(a),F(b)),F(\sigma) : \Delta[1]^n \to \mathcal{B}(F(a), F(b)),

such that

(i) if f 0=idf_0 = id, F(σ)=F( 0σ)(proj×Δ[1] n1)F(\sigma) = F(\partial_0\sigma)(proj \times \Delta[1]^{n-1})

(ii) if f i=idf_i = id, 0<i<n0\lt i \lt n

F(σ)=F( iσ(.(I i×m×I ni),F(\sigma) = F(\partial_i\sigma(.(I^i \times m \times I^{n-i}),

where m:I 2Im : I^2 \to I is the multiplicative structure on I=Δ[1]I = \Delta[1] by the ‘max’ function on {0,1}\{0,1\};

(iii) if f n=idf_n = id, F(σ)=F( nσ)F(\sigma) = F(\partial_n \sigma);

(iv)i_{i} F(σ)|(I i1×{0}×I ni)=F( iσ),1in1F(\sigma)|(I^{i-1}\times \{0\} \times I^{n-i}) = F(\partial_i\sigma), 1 \leq i \leq n-1;

(v)i_{i} F(σ)|(I i1×{1}×I ni)=F(σ i ).F(σ i)F(\sigma)|( I^{i-1}\times \{1\} \times I^{n-i}) = F(\sigma^\prime_i) . F(\sigma_i), where σ i=(f 0,,f i1\sigma_i = (f_0, \ldots, f_{i-1} and σ =(f i,,f n)\sigma^\prime = (f_i, \ldots, f_n). We have used i\partial_i for the face operators in the nerve of 𝔸\mathbb{A}.

This original form can be very useful for checking (bare hands!) within an application that a diagram is h.c., although the SSetSSet-functor approach is for many uses more compact and maniable and allows functorial constructions more easily. The link with the bar construction and comonadic resolution? approaches give suggestive links to interpretation of cohomology classes.

As algebras over an operad.

For \mathcal{E} a symmetric monoidal category, and CC a small \mathcal{E}-enriched category, there is an operad Diag CDiag_C whose algebras over an operad are \mathcal{E}-enriched functors

F:C F : C \to \mathcal{E}

hence CC-diagrams in \mathcal{E}.

If \mathcal{E} is also a monoidal model category with an interval object HH in a sufficiently nice way, then there exists the Boardman-Vogt resolution

HoCoDiag C:=W(H,Diag C). HoCoDiag_C := W(H, Diag_C) \,.

The algebras over this operad are then precisely homotopy coherent diagrams over CC in \mathcal{E}. For =\mathcal{E} = Top regarded with the standard model structure on topological spaces and H=[0,1]H = [0,1] the standard interval, this reproduces the ordinary notion of homotopy coherent diagrams (BergerMoerdijk)

Properties

Equivalence

(i) If X:AX : \mathbf{A}\to Top is a commutative diagram and we replace some of the X(a)X(a) by homotopy equivalent Y(a)Y(a) with specified homotopy equivalence data:

f(a):X(a)Y(a),g(a):Y(a)X(a)f(a) : X(a) \to Y(a), \quad g(a) : Y(a) \to X(a)
H(a):g(a)f(a)Id,K(a):f(a)g(a)Id,H(a) : g(a)f(a) \simeq Id, \quad K(a) : f(a)g(a) \simeq Id,

then we can combine these data into the construction of a h. c. diagram YY based on the objects Y(a)Y(a) and homotopy coherent maps

f:XY,g:YX,etc.,f : X\to Y, \quad g : Y \to X, etc.,

making XX and YY homotopy equivalent as h.c. diagrams.

(This applied to a GG-space, XX, shows that if we replace XX by a homotopy equivalent YY, then YY will be a h. c. version of a GG-space, i.e. a h. c. diagram of shape BGBG, the corresponding one object groupoid to GG.)

Homotopy coherent nerve

Theorem

Cordier (1980)

Given 𝔸{\mathbb{A}}, a small category, the sSet-enriched category S(𝔸){S(\mathbb{A})} called the homotopy coherent nerve is such that a h.c. diagram of shape 𝔸{\mathbb{A}} in Top is given precisely by an sSet-enriched functor

F:S(𝔸)Top F : {S(\mathbb{A})} \to Top

This suggested the extension of h.c. diagrams to other contexts such as a general locally Kan SSetSSet-category, \mathcal{B} and suggests the definition of homotopy coherent diagram in a 𝒮\mathcal{S}-category and thus a homotopy coherent nerve of an SSetSSet-category.

To understand simplical h.c. diagrams and thus the h.c. simplicial nerve N()N(\mathcal{B}), we unpack the definition of homotopy coherence, for conveneince, repeating some points made in homotopy coherent nerve.

The first thing to note is that for any nn and 0i<jn0\leq i\lt j\leq n, S[n](i,j)Δ[1] ji1S[n](i,j) \cong \Delta[1]^{j-i-1}, the (ji1)(j-i-1)-cube given by the product of ji1j-i-1 copies of Δ[1]\Delta[1]. Thus we can reduce the higher homotopy data to being just that, maps from higher dimensional cubes.

Next some notation:

Given simplicial maps

f 1:K 1(x,y),f_1: K_1 \to \mathcal{B}(x,y),
f 2:K 2(y,z),f_2: K_2 \to \mathcal{B}(y,z),

we will denote the composite

K 1×K 2(x,y)×(y,z)c(x,z)K_1 \times K_2 \to \mathcal{B}(x,y)\times \mathcal{B}(y,z) \stackrel{c}{\to} \mathcal{B}(x,z)

just by f 2.f 1f_2.f_1 or f 2f 1f_2f_1. (We already have seen this in the h.c. diagram above for 𝔸=[3]\mathbb{A} = [3]. X(123)X(01)X(123)X(01) is actually X(123)(I×X(01))X(123)(I \times X(01) ), whilst X(23)X(012)X(23)X(012) is exactly what it states.)

Rectification

Every homotopy coherent diagram is weakly equivalent to a strict diagram, a phenomenon known as rectification.

Vogt’s theorem

Theorem

(Vogt)

If A\mathbf{A} is a small category, there is a category Coh(A,Top)\mathbf{Coh(A,Top)} of h.c. diagrams and homotopy classes of h. c. maps between them. Moreover there is an equivalence of categories

Coh(A,Top)Ho(Top A).\mathbf{Coh(A,Top)} \stackrel{\simeq}{\to} \mathbf{Ho(Top^A)}.

Berger-Moerdijk theorem

If we think of hc diagrams as algebras over an operad, then this rectification is a special case of the general rectification theorem for such algebras. See model structure on algebras over an operad for details.

Examples

h.c. diagrams in a category with cylinder functor, denoted ×I-\times I

  1. A diagram indexed by the small category, [2][2].

is h.c. if there is specified a homotopy

X(012):X(0)×IX(2),X(012) : X(0)\times I \to X(2),
X(012):X(02)X(12)X(01).X(012) : X(02) \simeq X(12)X(01).
  1. For a diagram indexed by [3][3]: Draw a 3-simplex, marking the vertices X(0)X(0), \ldots, X(3)X(3), the edges X(ij)X(ij), etc., the faces X(ijk)X(ijk), etc. The homotopies X(ijk)X(ijk) fit together to make the sides of a square
X(13)X(01) X(123)X(01) X(23)X(12)X(01) X(013) X(23)X(012) X(03) X(023) X(23)X(02) \begin{matrix} X(1 3)X(0 1)&\xrightarrow{X(1 2 3)X(0 1)}&X(2 3)X(1 2)X(0 1)\\ \mathllap{X(0 1 3)}\left\uparrow\space{30}{20}{0}\right.& &\left.\space{30}{20}{0}\right\uparrow\mathrlap{X(2 3) X(0 1 2)}\\ X(0 3)&\xrightarrow[\qquad X(0 2 3)\qquad]{\quad}&X(2 3)X(0 2) \end{matrix}

and the diagram is made h.c. by specifying a second level homotopy

X(0123):X(0)×I 2X(3)X(0123) : X(0)\times I^2\to X(3)

filling this square, in the sense that restricting to each side of the square in the double homotopy gives the correspondingly labelled homotopy from the diagram.

These can be continued for larger [n][n], and the results glued together to make larger h.c. diagrams. Of course, this is not how it is done, but it provides some understanding of the basic idea.

References

For Vogt’s theorem, the original reference is

  • Rainer Vogt, Homotopy limits and colimits, Math. Z., 134 (1973) 11-52.

A generalisation of his theorem using simplicially enriched categories and the homotopy coherent nerve of such a thing, is to be found in

  • J.-M. Cordier and T. Porter, Vogt’s Theorem on Categories of Homotopy Coherent Diagrams, Math. Proc. Camb. Phil. Soc. 100 (1986) pp. 65-90.

A neat application to changing objects in diagrams within a homotopy type can be found in

  • J.-M. Cordier and T. Porter, Maps between homotopy coherent diagrams, Top. and its Appls., 28, (1988), 255 – 275.

A summary of homotopy coherence can be found in Chapter 5 of

and in chapter 10 of

The operad-theoretic description of homotopy-coherent diagrams is in

See model structure on algebras over an operad for more on this.

Revised on April 9, 2014 06:12:03 by Tim Porter (2.26.27.237)