+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Homotopy theory +--{: .hide} [[!include homotopy - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea An _interval object_ $I$ in a [[category]] $C$ is an [[object]] that behaves in $C$ roughly like the standard [[topological interval]] $I \coloneqq [0,1]$ with its two boundary point inclusions $$ {*}\amalg {*} \stackrel{[0, 1]}{\to} I $$ in the category [[Top]] of [[topological spaces]], where $[0,1]$ is the [[copairing]] of the [[global elements]] $0\colon {*} \to I$ and $1\colon {*} \to I$. A bare interval object may be nothing more than such a diagram. If $C$ admits sufficiently many [[limit]]s and [[colimit]]s, 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 $C$ supposed to be the right structure to ensure 1. that there is an object $I$ in $C$ such that for every object $X$ of $C$ the [[internal hom|internal hom object]] $[I,X]$ exists and behaves like a [[path space object]] for $X$; 2. that there is a notion of composition on these path objects which induces on $[I,X]$ a structure of a (higher) category internal to $C$: the [[fundamental category]] or [[fundamental groupoid]] of the object $X$, or rather its [[fundamental infinity-groupoid]]. For instance the choice $C =$ [[Top]] and $I = [0,1]$ should be an instance of a category with interval object, and the fundamental [[algebraic definition of higher category|algebraic]] [[n-groupoid]] $\Pi_n(X)$ obtained for any topological space $X$ from this data should be the fundamental $n$-groupoid as a [[Trimble n-category]]. We give two very similar definitions that differ only in some extra assumptions. * The first one was used by Berger and Moerdijk to generalize the Boardman&#8211;Vogt resolution of [[topological operad]]s to more general [[operad]]s. * The second is motivated from constructions appearing in the definitions of [[Trimble n-category]] and of [[generalized universal bundle]]. It includes the possibility that the interval is _not_ weakly equivalent to the point, in which case it may be used nontrivially to test for [[undirected object]]s and probe [[directed object]]s. ## Definitions ### Plain definition +-- {: .un_defn} ###### Definition (plain interval object) A **plain interval object** in a category $C$ is just a [[cospan]] diagram with equal feet $$ pt \stackrel{0}{\to} I \stackrel{1}{\leftarrow} pt $$ in $C$, with $I$ and $pt$ any two objects and $0$ and $1$ any two morphisms. =-- In categories with finite limits it is often required that $pt=*$ 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 $C$ is a [[homotopical category]], such as a [[model category]], there are natural further conditions to put on an interval object: #### Trimble interval object {#IntervalForTrimbledOmegaeCategories} The following definition is strongly related to the notion of [[Trimble omega-category]] where the interval object gives the internal hom $[I,X]$ the structure of an [[operad]] giving (by induction) the model of an $A_\infty$-[[A-infinity-category|category]] structure on $$ (X_0 := [pt,X]) \stackrel{s := [\sigma,X]}{\leftarrow} [I,X] \stackrel{t := [\tau, X]}{\to} (X_0 := [pt,X]) \,. $$ This internal $A_\infty$-category is denoted $$ \Pi_1(X) $$ This is in (a bit) more detail in [[Trimble omega-category]] and in [[fundamental (infinity,1)-category]] in the section "Fundamental algebraic $\infty$-categories". +-- {: .un_defn} ###### Definition (Trimble interval object) A **category with interval object** is * a symmetric [[closed monoidal homotopical category]] $V$; * with tensor unit being the [[terminal object]], which we write $pt$; * equipped with a [[bi-pointed object]] $$ \array{ && I \\ & {}^\sigma \nearrow && \nwarrow^{\tau} \\ pt &&&& pt } $$ in $V$, with $I$ called the **interval object**; such that * the [[pushout]] $$ \array{ && I^{\vee 2} := I \amalg_{pt} I \\ & \nearrow && \nwarrow \\ I &&&& I \\ & {}_{\tau}\nwarrow && \nearrow_{\sigma} \\ && pt } $$ exists in $V$, so that all compositions $$ \array{ && I^{\vee n} \\ & {}^\sigma \nearrow && \nwarrow^{\tau} \\ pt &&&& pt } $$ of $n \in \mathbb{N}$ copies of the [[co-span]] $I$ with itself by pushout over adjacent legs exist in $V$; * and for all $n$, the $V$-objects of morphisms ${}_{pt}[I, I^{\vee n}]_{pt}$ of cospans (as described at [[co-span]]) are weakly equivalent to the point $$ {}_{pt}[I, I^{\vee n}]_{pt} \,. $$ =-- #### Berger--Moerdijk segment- and interval object In [section 4](http://arxiv.org/PS_cache/math/pdf/0502/0502155v2.pdf#page=11) of * [[Clemens Berger]], [[Ieke Moerdijk]], _The Boardman-Vogt resolution of operads in monoidal model categories_ ([arXiv](http://arxiv.org/abs/math.AT/0502155)) the following definition is given: Let $V$ be a [[monoidal model category]] and write $pt$ for the tensor unit in $V$ (not necessarily the terminal object). A **[[segment object|segment (object)]]** $I$ in a [[monoidal model category]] $V$ is * a factorization $$ pt \amalg pt \stackrel{[0 , 1]}{\to} I \stackrel{\epsilon}{\to} pt $$ of the [[codiagonal morphism]] $$ pt \amalg pt \stackrel{[Id , Id]}{\to} pt $$ from the [[coproduct]] of $pt$ with itself that sends each component identically to $pt$. * together with an associative morphsim $$ \vee : I \otimes I \to I $$ which has 0 as its _neutral_ and 1 as its _absorbing_ element, and for which $\epsilon$ is a counit. If $V$ is equipped with the structure of a [[model category]] then a segment object is an **interval** in $V$ if $$ [0, 1]\colon pt \amalg pt \to I $$ is a cofibration and $\epsilon : I \to pt$ a weak equivalence. ### Interval type {#InHomotopyTypeTheory} 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]] $\Delta[1]$. * In a [[category of chain complexes]] the standard interval is the [[chain on a simplicial set|simplicial chain complex]] $C_\bullet(\Delta[1])$ 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]] $\{0\to 1\}$ (see [[co-span co-trace]]) * For $V = C =$ [[Top]] equipped with with the [[classical model structure on topological spaces]], the topological [[closed interval]] $I \colon [0,1]$ (with its [[Euclidean space|Euclidean]] [[metric topology]]) with $pt \stackrel{\sigma, \tau}{\to}I$ the maps to 0 and 1, respectively. This is the standard _[[topological interval]]_. This is the case described in detail at _[[Trimble n-category]]_. * For $V = \omega Cat$ the category of [[strict omega-category|strict omega-categories]] the first [[oriental]], the 1-[[globe]] $I = \{a \to b\}$ is an interval object. In this strict case in fact all hom objects are already equal to the point ${}_{pt}[I, I^{\vee n}]_{pt} = pt$ and $$ (X = [pt,X]) \stackrel{s := [\sigma,X]}{\leftarrow} [I,X] \stackrel{t := [\tau, X]}{\to} (X = [pt,X]) $$ is a strict co-category internal to $\omega$Cat. In this case, for $X$ any $\omega$-category the $A_\infty$-category $\Pi_1(X)$ is just an ordinary category, namely the 1-category obtained from truncation of $X$. Similarly, probably $\Pi_\omega(X) = X$ in this case. ### Standard intervals, cubes and simplices in $Top$ and $Diff$ Let $X = $ [[Top]] or $C = $ [[Diff]] be the category of [[topological space]]s or of [[manifold]]s. A standard choice of interval object in $C$ is $I = [0,1] \subset \mathbb{R}$ with the obvious two boundary inclusions $0,1 : {*} \to [0,1]$. But another possible choice is to let $I = \mathbb{R}$ be the whole real line, but still equipped with the two maps $0,1 : {*} \to \mathbb{R}$, that hit the $0 \in \mathbb{R}$ and $1 \in \mathbb{R}$, 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 $[0,1]$. While ${*} \stackrel{0}{\to}[0,1] \stackrel{1}{\leftarrow} {*}$ may seem like a more natural choice for a representative of the idea of the "standard interval", the choice ${*} \stackrel{0}{\to} \mathbb{R} \stackrel{1}{\leftarrow} {*}$ is actually more useful for many [[category theory|abstract nonsense]] constructions. But since it is hard to draw the full real line, in the following we depict the situation for the choice $I = [0,1]$. Then for low $n \in \mathbb{N}$ the above construction yields this * $n=0$ -- here $\Delta_I^0 = I^{\times 0} = {*}$ is the [[point]]. * $n=1$ -- here $\Delta_I^1 = I^{\times 1} = I$ is just the interval itself $$ \array{ (0) \to (1) } $$ The two face maps $\delta_1 {*} \to I$ and $\delta_0 : {*} \to I$ pick the boundary points in the obvious way. The unique degeneracy map $\sigma_0 : I \to {*}$ maps all points of the interval to the single point of the point. * $n=2$ -- here $\Delta_i^2 = I^{\times 2} = I \times I$ is the **standard square** $$ \array{ (0,1) &\to& (1,1) \\ \uparrow && \uparrow \\ (0,0) &\to& (1,0) } $$ But the three face maps $\delta_i : I \to I\times I$ of the cosimplicial object $\Delta_I$ constructed above don't regard the full square here, but just a triangle sitting inside it, in that pictorially they identify $(\Delta_I^1 = I)$-shaped boundaries in $I \times I$ as follows: $$ \array{ (0,1) &\to& (1,1) \\ \uparrow &^{= \delta_1(I)}\nearrow& \uparrow^{ = \delta_0(I)} \\ (0,0) &\stackrel{= \delta_2(I)}{\to}& (1,0) } $$ (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!) * $n=3$ -- here $\Delta_i^3 = I^{\times 3} = I \times I \times I$ is the **standard cube** +-- {: .un_example} ###### Exercise Insert the analog of the above discussion here and upload a nice graphics that shows the standard cube and how the cosimplicial object $\Delta_I$ picks a solid tetrahedron inside it. =-- As a start, we can illustrate how there are 6 3-simplices sitting inside each 3-cube. <img src="http://ncatlab.org/nlab/files/3simplex_in_3cube.jpg" width = "550"/> Once you see how the 3-simplices sit inside the 3-cube, the facemaps can be illustrated as follows: <img src="http://ncatlab.org/nlab/files/3simplex_facemaps.jpg" width = "550"/> Note that these face maps are to be thought of as maps into 3-simplices sitting inside a 3-cube. ### $\mathbb{A}^1$-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 $I$ is like picking a notion of _path_, in a category with interval object there is, under mild assumptions, for each object $X$ an [[infinity-category]] $\Pi_I(X)$ -- the fundamental $\infty$-category of $X$ with respect to $I$ -- whose [[k-morphism]]s are $k$-fold $I$-paths in $X$. This is described for two models for $(\infty,1)$-categories at [[fundamental (infinity,1)-category]] ## Homotopy localization induced from an interval Given a suitable interval obect in a [[site]] $C$, one may ask for [[∞-stack]]s on $C$ that are invariant under the notion of [[homotopy]] induced by $I$. These are obtained by [[homotopy localization]] of a full [[(∞,1)-category of (∞,1)-sheaves]] on $C$. ## Related concepts * [[interval]], [[interval type]] ## References * Clemens Berger, [[Ieke Moerdijk]], _The Boardman-Vogt resolution of operads in monoidal model categories_ ([arXiv](http://arxiv.org/abs/math.AT/0502155)), section 4, p.11 [[!redirects interval objects]] [[!redirects simplex in a lined topos]] [[!redirects contractible interval object]]