nLab
interval object

Contents

Idea

An interval object II in a category CC is an object that behaves in CC roughly like the unit interval I:=[0,1]I := [0,1] with its two boundary point inclusions

*⨿*[0,1]I {*}\amalg {*} \stackrel{[0, 1]}{\to} I

in the category Top of topological spaces, where [0,1][0,1] is the copairing of the global elements 0:*I0\colon {*} \to I and 1:*I1\colon {*} \to I.

A bare interval object may be nothing more than such a diagram. If CC admits sufficiently many limits and colimits, then from this alone a lot of structure derives. The precise definition of further structure and property imposed on an interval object varies with the intended context and applications.

Notably in a large class of applications the interval object in CC supposed to be the right structure to ensure

  1. that there is an object II in CC such that for every object XX of CC the internal hom object [I,X][I,X] exists and behaves like a path space object for XX;

  2. that there is a notion of composition on these path objects which induces on [I,X][I,X] a structure of a (higher) category internal to CC: the fundamental category or fundamental groupoid of the object XX, or rather its fundamental infinity-groupoid.

For instance the choice C=C = Top and I=[0,1]I = [0,1] should be an instance of a category with interval object, and the fundamental algebraic n-groupoid Π n(X)\Pi_n(X) obtained for any topological space XX from this data should be the fundamental nn-groupoid as a Trimble n-category.

We give two very similar definitions that differ only in some extra assumptions.

Definitions

Plain definition

Definition (plain interval object)

A plain interval object in a category CC is just a cospan diagram with equal feet

pt0I1pt pt \stackrel{0}{\to} I \stackrel{1}{\leftarrow} pt

in CC, with II and ptpt any two objects and 00 and 11 any two morphisms.

In categories with finite limits it is often required that pt=*pt=* is the terminal object and in this case the interval object is called cartesian interval object.

Examples for the use of this notion is at fundamental (infinity,1)-category in the section “fundamental geometric ∞-categories”.

In homotopical categories

If the ambient category CC is a homotopical category, such as a model category, there are natural further conditions to put on an interval object:

Trimble interval object

The following definition is strongly related to the notion of Trimble omega-category where the interval object gives the internal hom [I,X][I,X] the structure of an operad giving (by induction) the model of an A A_\infty-category structure on

(X 0:=[pt,X])s:=[σ,X][I,X]t:=[τ,X](X 0:=[pt,X]). (X_0 := [pt,X]) \stackrel{s := [\sigma,X]}{\leftarrow} [I,X] \stackrel{t := [\tau, X]}{\to} (X_0 := [pt,X]) \,.

This internal A A_\infty-category is denoted

Π 1(X) \Pi_1(X)

This is in (a bit) more detail in Trimble omega-category and in fundamental (infinity,1)-category in the section “Fundamental algebraic \infty-categories”.

Definition (Trimble interval object)

A category with interval object is

in VV, with II called the interval object;

such that

  • the pushout

    I 2:=I⨿ ptI I I τ σ pt \array{ && I^{\vee 2} := I \amalg_{pt} I \\ & \nearrow && \nwarrow \\ I &&&& I \\ & {}_{\tau}\nwarrow && \nearrow_{\sigma} \\ && pt }

    exists in VV, so that all compositions

    I n σ τ pt pt \array{ && I^{\vee n} \\ & {}^\sigma \nearrow && \nwarrow^{\tau} \\ pt &&&& pt }

    of nn \in \mathbb{N} copies of the co-span II with itself by pushout over adjacent legs exist in VV;

  • and for all nn, the VV-objects of morphisms pt[I,I n] pt{}_{pt}[I, I^{\vee n}]_{pt} of cospans (as described at co-span) are weakly equivalent to the point

    pt[I,I n] pt. {}_{pt}[I, I^{\vee n}]_{pt} \,.

Berger–Moerdijk segment- and interval object

In section 4 of

  • Clemens Berger, Ieke Moerdijk, The Boardman-Vogt resolution of operads in monoidal model categories (arXiv)

the following definition is given:

Let VV be a monoidal model category and write ptpt for the tensor unit in VV (not necessarily the terminal object).

A segment (object) II in a monoidal model category VV is

  • a factorization

    pt⨿pt[0,1]Iϵpt pt \amalg pt \stackrel{[0 , 1]}{\to} I \stackrel{\epsilon}{\to} pt

    of the codiagonal morphism

    pt⨿pt[Id,Id]pt pt \amalg pt \stackrel{[Id , Id]}{\to} pt

    from the coproduct of ptpt with itself that sends each component identically to ptpt.

  • together with an associative morphsim

    :III \vee : I \otimes I \to I

    which has 0 as its neutral and 1 as its absorbing element, and for which ϵ\epsilon is a counit.

If VV is equipped with the structure of a model category then a segment object is an interval in VV if

[0,1]:pt⨿ptI [0, 1]\colon pt \amalg pt \to I

is a cofibration and ϵ:Ipt\epsilon : I \to pt a weak equivalence.

Interval type

In homotopy type theory the cellular interval can be axiomatized as a higher inductive type. See interval type for more.

Examples

  • In sSet the standard interval object is the 1-simplex Δ[1]\Delta[1].

  • In a category of chain complexes the standard interval is the simplicial chain complex C (Δ[1])C_\bullet(\Delta[1]) on the 1-simplex, see at interval object in chain complexes.

  • The cube category is generated from a single interval object.

  • The standard interval object in Cat is the 1st oriental {01}\{0\to 1\} (see co-span co-trace)

  • For V=C=TopV = C = Top with its standard model structure the standard topological closed interval I:=[0,1]I := [0,1] with ptσ,τIpt \stackrel{\sigma, \tau}{\to}I the maps to 0 and 1, respectively. This is the case described in detail at Trimble n-category.

  • For V=ωCatV = \omega Cat the category of strict omega-categories the first oriental, the 1-globe I={ab}I = \{a \to b\} is an interval object. In this strict case in fact all hom objects are already equal to the point pt[I,I n] pt=pt{}_{pt}[I, I^{\vee n}]_{pt} = pt and

    (X=[pt,X])s:=[σ,X][I,X]t:=[τ,X](X=[pt,X]) (X = [pt,X]) \stackrel{s := [\sigma,X]}{\leftarrow} [I,X] \stackrel{t := [\tau, X]}{\to} (X = [pt,X])

    is a strict co-category internal to ω\omegaCat. In this case, for XX any ω\omega-category the A A_\infty-category Π 1(X)\Pi_1(X) is just an ordinary category, namely the 1-category obtained from truncation of XX. Similarly, probably Π ω(X)=X\Pi_\omega(X) = X in this case.

Standard intervals, cubes and simplices in TopTop and DiffDiff

Let X=X = Top or C=C = Diff be the category of topological spaces or of manifolds.

A standard choice of interval object in CC is I=[0,1]I = [0,1] \subset \mathbb{R} with the obvious two boundary inclusions 0,1:*[0,1]0,1 : {*} \to [0,1].

But another possible choice is to let I=I = \mathbb{R} be the whole real line, but still equipped with the two maps 0,1:*0,1 : {*} \to \mathbb{R}, that hit the 00 \in \mathbb{R} and 11 \in \mathbb{R}, respectively.

Either of these two examples will do in the following discussion. The second choice is to be thought of as obtained from the first choice by adding “infinitely wide collars” at both boundaries of [0,1][0,1]. While *0[0,1]1*{*} \stackrel{0}{\to}[0,1] \stackrel{1}{\leftarrow} {*} may seem like a more natural choice for a representative of the idea of the “standard interval”, the choice *01*{*} \stackrel{0}{\to} \mathbb{R} \stackrel{1}{\leftarrow} {*} is actually more useful for many abstract nonsense constructions.

But since it is hard to draw the full real line, in the following we depict the situation for the choice I=[0,1]I = [0,1].

Then for low nn \in \mathbb{N} the above construction yields this

  • n=0n=0 – here Δ I 0=I ×0=*\Delta_I^0 = I^{\times 0} = {*} is the point.

  • n=1n=1 – here Δ I 1=I ×1=I\Delta_I^1 = I^{\times 1} = I is just the interval itself

    (0)(1) \array{ (0) \to (1) }

    The two face maps δ 1*I\delta_1 {*} \to I and δ 0:*I\delta_0 : {*} \to I pick the boundary points in the obvious way. The unique degeneracy map σ 0:I*\sigma_0 : I \to {*} maps all points of the interval to the single point of the point.

  • n=2n=2 – here Δ i 2=I ×2=I×I\Delta_i^2 = I^{\times 2} = I \times I is the standard square

    (0,1) (1,1) (0,0) (1,0) \array{ (0,1) &\to& (1,1) \\ \uparrow && \uparrow \\ (0,0) &\to& (1,0) }

    But the three face maps δ i:II×I\delta_i : I \to I\times I of the cosimplicial object Δ I\Delta_I constructed above don’t regard the full square here, but just a triangle sitting inside it, in that pictorially they identify (Δ I 1=I)(\Delta_I^1 = I)-shaped boundaries in I×II \times I as follows:

    (0,1) (1,1) =δ 1(I) =δ 0(I) (0,0) =δ 2(I) (1,0) \array{ (0,1) &\to& (1,1) \\ \uparrow &^{= \delta_1(I)}\nearrow& \uparrow^{ = \delta_0(I)} \\ (0,0) &\stackrel{= \delta_2(I)}{\to}& (1,0) }

    (here the arrows do not depict morphisms, but the standard topological interval, i don’t know how to typeset just lines without arrow heads in this fashion!)

  • n=3n=3 – here Δ i 3=I ×3=I×I×I\Delta_i^3 = I^{\times 3} = I \times I \times I is the standard cube

    Exercise

    Insert the analog of the above discussion here and upload a nice graphics that shows the standard cube and how the cosimplicial object Δ I\Delta_I picks a solid tetrahedron inside it.

As a start, we can illustrate how there are 6 3-simplices sitting inside each 3-cube.

Once you see how the 3-simplices sit inside the 3-cube, the facemaps can be illustrated as follows:

Note that these face maps are to be thought of as maps into 3-simplices sitting inside a 3-cube.

𝔸 1\mathbb{A}^1-homotopy theory

See A1-homotopy theory.

Fundamental \infty-categories induced from intervals

The interest in interval objects is that various further structures of interest may be built up from them. In particular, since picking an interval object II is like picking a notion of path, in a category with interval object there is, under mild assumptions, for each object XX an infinity-category Π I(X)\Pi_I(X) – the fundamental \infty-category of XX with respect to II – whose k-morphisms are kk-fold II-paths in XX.

This is described for two models for (,1)(\infty,1)-categories at fundamental (infinity,1)-category

Homotopy localization induced from an interval

Given a suitable interval obect in a site CC, one may ask for ∞-stacks on CC that are invariant under the notion of homotopy induced by II. These are obtained by homotopy localization of a full (∞,1)-category of (∞,1)-sheaves on CC.

References

  • Clemens Berger, Ieke Moerdijk, The Boardman-Vogt resolution of operads in monoidal model categories (arXiv), section 4, p.11

Revised on November 8, 2012 17:24:06 by Stephan Alexander Spahn (79.227.175.40)