fundamental (infinity,1)-category


(,1)(\infty,1)-Category theory

Homotopy theory



In analogy to how a topological space has a fundamental ∞-groupoid, a directed space has a fundamental (∞,1)-category


Directed topological spaces


By a directed topological space we here mean specifically a pair (X,D)(X,D) consisting of

  • a topological space XX;

  • a subset DX [0,1]D \subset X^{[0,1]} of continuous maps γ:[0,1]X\gamma : [0,1] \to X that are are labeled as being directed ,

  • such that

    • for every orientation preserving homeomorphism ϕ:[0,1][0,1]\phi : [0,1] \to [0,1] and every directed map γ\gamma also γϕ\gamma \circ \phi is directed;

    • for any two directed paths γ 1,γ 2\gamma_1, \gamma_2 with γ 1(1)=γ 2(0)\gamma_1(1) = \gamma_2(0) also the concatenation

      [0,1]×2[0,2][0,1] *[0,1](γ 1,γ 2)X [0,1] \stackrel{\times 2}{\to} [0,2] \simeq [0,1] \coprod_{*} [0,1] \stackrel{(\gamma_1, \gamma_2)}{\to} X

      is a directed path.

A morphism (X 1,D 1)(X 2,D 2)(X_1,D_1) \to (X_2, D_2) of directed spaces is a continuous map X 1X 2X_1 \to X_2 that takes elements in D 1D_1 to elements in D 2D_2.

This defines the category DTopDTop of directed topological spaces.


(topological poset)

For XX a topological space equipped with the structure of a poset on its underlying set, say that γ:[0,1]X\gamma : [0,1] \to X is directed if for all xyx \leq y in [0,1][0,1] we have γ(x)γ(y)\gamma(x) \leq \gamma(y) in XX.


(directed geometric simplex)

For Δ Top k\Delta^k_{Top} the standard topological kk-simplex its standard directed paths are order-preserving maps into its 1-skeleton (the union of the 1-faces equipped with the evident poset-structure induced from that on the vertices), such that the endpoints land on vertices.


(directed geometric realization)

For CC a quasi-category its geometric realization C|C| becomes a directed topological space by taking the directed paths to be all maps that factor through its 1-skeleton:

[k]Δ 1Δ Top k×C k [k]ΔΔ Top k×C k=C \int^{[k] \in \Delta_{\leq 1}} \Delta^k_{Top} \times C_k \hookrightarrow \int^{[k] \in \Delta} \Delta^k_{Top} \times C_k = |C|

while preserving the canonical order and so that the endpoints land on vertices.

The previous example of the directed topological simplex is the special case of this for C=Δ[k]C = \Delta[k].

Fundamental (,1)(\infty,1)-category


The fundamental (,1)(\infty,1)-category of a directed topological space (X,D)(X,D) is given by the quasi-category Π(X,D)\Pi(X,D) whose k-morphisms are those continuous maps

σ:Δ Top kX \sigma : \Delta^k_{Top} \to X

that are morphisms of directed spaces with respect to the standard directed paths in Δ k\Delta^k.

Π(X,D) k:=Hom DTop(Δ Top k,(X,D)). \Pi(X,D)_k := Hom_{DTop}(\Delta^k_{Top}, (X,D)) \,.

This does indeed define a quasi-category


Use the standard retracts Δ Top kΛ i k\Delta^k_{Top} \to \Lambda^k_i of the topological horn inclusions Λ i kΔ k\Lambda^k_i \hookrightarrow \Delta^k to fill horns in Π(X,D)\Pi(X,D).

For k3k \geq 3 these retracts are the identity on 1-faces, hence trivially preserve the directed paths. For k=2k = 2 it is precisely the retract of the inner horn Λ 1 2\Lambda^2_1 that preserves the directed paths. This is sufficient to satisfy the inner horn filler condition of quasi-categories.



If XX is undirected in that all paths are labeled as directed – D=X [0,1]D = X^{[0,1]} – then Π(X,D)\Pi(X,D) coincides with the fundamental ∞-groupoid Π(X)\Pi(X) of XX.

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.

There are different ways to make this precise and realize it in detail. The main distinction is whether one uses


We describe an algebraic version in terms of Trimble omega-categories and then a geometric version in terms of cubical objects and simplicial objects.

Fundamental algebraic \infty-categories

The collection of objects { pt[I,I n] pt} n\{ {}_{pt}[I, I^{\vee n}]_{pt}\}_{n \in \mathbb{N}} in a category with interval object naturally comes equipped with the structure of an operad: this is the tautological co-endomorphism operad on the object II in the symmetric closed monoidal category of bi-pointed objects from ptpt to ptpt.

This induces in turn for all objects XVX \in V on the object [I,X][I,X] the structure of an operad, which is naturally interpreted as an internal 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)

and interpreted as the fundamental groupoid or rather, in general, the fundamental category of the object BB with respect to the interval object II – all internal to VV.

Moreover, by iterating this process as described at Trimble n-category one should obtain, if everything goes through, on XX the structure of a Trimble ω\omega-category and indeed a functor

Π ω:V 0TrimbleωCat. \Pi_\omega : V_0 \to Trimble \omega Cat \,.

(… to be continued …)

  • The condition that all pt[I,I n] pt{}_{pt}[I, I^{\vee n}]_{pt} are contractible is the coherence condition on all composition operations.

  • The above is not demanding that the interval object II itself is is weakly equivalent to the point. If it is, then Π 1(X)\Pi_1(X) is indeed a fundamental groupoid. If it is not, then Π 1(X)\Pi_1(X) may just be a fundamental category.

  • If X 0X_0 has a notion of path object one may consider imposing the condition that [I,X][I,X] is a path object of XX for all XX. Similarly, if V 0V_0 has a cylinder functor, one may consider imposing the condition that it is given by I-\otimes I.

Fundamental geometric \infty-categories

Let CC be a category with finite limits and (plain) interval object *0I1* {*} \stackrel{0}{\to} I \stackrel{1}{\leftarrow} {*}, where *{*} denotes the terminal object.

This may or may not come with further structures and properties as discussed in the definitions above. For the following however no more than that is neceesray.

Recall that the cube category is the initial strict monoidal category (M,,I)(M, \otimes, I) equipped with an object intint together with two maps i 0,i 1:Iinti_0, i_1: I \to int and a map p:intIp: int \to I such that pi 0=1 I=pi 1p i_0 = 1_I = p i_1.

In a tautological way, II induces a cocubical object in CC, a functor

I:C \Box_I : \Box \to C

from the cube category \Box to CC. This sends the abstract interval object intint that the cube category is freely generated from to the given II, and sends int nint^{\otimes n} to the nn-cuber I n:=I ×n\Box_I^n := I^{\times n}.

For every object XCX \in C homming cubes into XX thereby produces a cubical set

X I :int nC( I n,X). X^{\Box^\bullet_I} : int^n \mapsto C(\Box_I^n,X) \,.

One tends to want to regard this as the cubical incarnation of the fundamental \infty-category of XX with respect to the notion of path given by II.

However, while cubes are nice for many purposes, it is a sad fact of life that the homotopy theory for cubical structures (while certainly it does exist in full beauty in principle) is much less well developed to date (maybe that will change in the future) than that of simplicial structures. For many purposes in higher category theory, therefore, it will be useful to take a slightly different perspective on X X^{\Box}, without essentially changing it.

In fact, there is also naturally the structure of a cosimplicial object of the collection I ×I^\times \bullet of II-cubes. This differs from the cubical structure only in were precisely one injects the boundaries into an I ×nI^{\times n}

Definition (cosimplicial object induced from interval object)

Given a cartesian interval object ICI \in C, define a cosimplicial object

Δ I:ΔC \Delta_I : \Delta \to C

as follows:

  • the object in degree nn is the nn-fold product of II with itself:

    Δ I n:=I ×n \Delta_I^n := I^{\times n}
  • the degeneracy map σ i:Δ I n+1Δ I n\sigma_i : \Delta_I^{n+1} \to \Delta_I^n is given by projecting out the (i+1)(i+1)-factor:

    σ i:=p 1×p 2××p i×p i+2×p i+3××p n+1 \sigma_i := p_1 \times p_2 \times \cdots \times p_i \times p_{i+2} \times p_{i+3} \times \cdots \times p_{n+1}
  • the face map δ i:Δ I nΔ I n+1\delta_i : \Delta_I^{n} \to \Delta_I^{n+1} is given

    • for i=n+1i = n+1 by inserting 00 in the (n+1)(n+1)-factor:

      δ n+1:=p 1××p n×0 \delta_{n+1} := p_1 \times \cdots \times p_{n} \times 0
    • for i=0i = 0 by inserting 11 in the 00-factor:

      δ 0=1×p 1××p 2 \delta_0 = 1 \times p_1 \times \cdots \times p_2
    • for 0<i<n+10 \lt i \lt n+1 by duplicating the ii-factor:

      δ i:=p 1×× p i1×p i×p i×p i+1××p n \delta_i := p_1 \times \cdots \times_{p_{i-1}} \times p_i \times p_i \times p_{i+1} \times \cdots \times p_n

The maps defined this way indeed satisfy the simplicial identities.


This is straightforward to check, if a little tedious due to the many case distinctions.

Remark (unwrapping the definition)

It may be helpful to unpack the above definition a bit.

  • The two faces of Δ I 1\Delta_I^1 are just the “boundary points” of the interval itself.

    • (δ 0:*I)=(1:*I)(\delta_0 : {*} \to I) = (1 : {*} \to I)

    • (δ 1:*I)=(0:*I)(\delta_1 : {*} \to I) = (0 : {*} \to I)

  • The face maps of Δ I 2\Delta^2_I may be depicted as follows:

    δ 2:I × I Id 0 I × *,δ 1:I × I Id×Id I,δ 0:I × I 1 Id * × I \delta_2 : \array{ I &\times& I \\ \uparrow^{Id} && \uparrow^{0} \\ I &\times& {*} } \,, \;\;\;\;\; \delta_1 : \array{ I &\times& I \\ & \uparrow^{Id \times Id} \\ & I } \,, \;\;\;\;\; \delta_0 : \array{ I &\times& I \\ \uparrow^{1} && \uparrow^{Id} \\ {*} &\times& I }

    Therefore δ 1\delta_1, being a diagonal morphism (cartesian pairing of an identity with itself), literally identifies the diagonal in the “square” I×II \times I as the 1st 1-dimensional boundary.

Remark (collars)

This construction gives “collared simplices” in much the same sense as in collared cobordism? and in A1-homotopy:

there is no condition that the morphisms 0,1:*I0,1 : {*} \to I “hit a boundary point” – whatever that may mean in CC – of II. For instance in a lined topos II is canonically chosen to be the given line object and will hence typically “extended indefinitely” beyond these points.

An example of this in practice is the A1-homotopy theory of schemes – there is no exact analogue of the interval (with boundary points) in an algebraic setting, but the affine line A 1A^1 together with the canonical points 0 and 1 is an interval object.

So II need not “look” much like a 1-simplex, but the choice of boundary points δ 1=0:*I\delta_1 = 0 : {*} \to I and δ 0=1:*I\delta_0 = 1 : {*} \to I allows us to regard it as an interval for all practical purposes.

Similarly and more generally what the above construction manifestly defines are cubes I nI^n built from II. But then the simplicial choice of boundaries inside these cubes allows to think of them as just the simplices “sitting inside” these cubes.

All these statement become precise for specific typical choices of the ambient category CC, discussed in the examples below.

An important aspect is that once the cosimplicial object of collared simplices Δ I\Delta_I is used to form simplicial objects Π(X):=[Δ I ,X]\Pi(X) := [\Delta^\bullet_I,X] (discussed below) and when these are interpreted as models for ∞-groupoids, then the collars disappear: they are part of the model, but, roughly, don’t affect the equivalence class of the object that this model models.

For instance with *⨿*0,1{*}\amalg {*} \stackrel{0, 1}{\to}\mathbb{R} used as the interval in Top, a path in a topological space XX is an entire curve γ:X\gamma : \mathbb{R} \to X, but two such paths γ 1,γ 2\gamma_1,\gamma_2 are composable already when γ 1(1)=γ 2(0)\gamma_1(1) = \gamma_2(0), irrsepctive of how γ 1\gamma_1 extends >1\gt 1 and irrsepctive of how γ 2\gamma_2 extends <0\lt 0.

Moreover, the composite-up-to-homotopy of these two paths is an entire surface 2X\mathbb{R}^2 \to X in XX, but what only matters for this surface qualifying as a compositor of γ 1andγ 2\gamma_1 and \gamma_2 is that its δ 2\delta_2-segment {(x,y) 20x1,y=0}\{(x,y) \in \mathbb{R}^2 | 0 \leq x \leq 1, y = 0\} and its δ 0\delta_0segment {(x,y) 20y1,x=0}\{(x,y) \in \mathbb{R}^2 | 0 \leq y \leq 1, x = 0\} coincide with the corresponding segments in γ 1\gamma_1 and γ 2\gamma_2.

More on this in the following example section.

Example: 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


    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.

Fundamental little 1-cubes space induced from an interval

Urs Schreiber: something I am thinking about

The following is supposed to give an (∞,1)-operadic incarnation of the notion of fundamental \infty-groupoid induced from an interval object. It should resemble a geometric operadic version of the algebraic operadic version described further above.

Let Ω p\Omega^p be the category of planar trees, so that a presheaf on Ω p\Omega^p is a planar dendroidal set.

Given an interval object *0I1*{*} \stackrel{0}{\to} I \stackrel{1}{\leftarrow} {*} in a category CC, assume one isomorphism ϕ n[I,I n]\phi_n \in [I,I^{\vee n}] for each nn has been chosen.

Then there is a planar co-dendroidal object Ω C p:Ω pC\Omega^p_C : \Omega^p \to C in CC given by:

  • a tree TT with nn leaves is sent to I nI^{\vee n}

    (we think of the kk-th copy of II here as being associated to the kkth leaf of the planar tree);

  • every degeneracy map is sent to the corresponding identity morphism

    (this corresponds to the fact that the co-dendroidal object encodes no nontrivial unary (co)operations, only the (n>1)(n \gt 1)-ary operations encoded nontrivial infomation);

  • an outer face map on an nn-ary vertex is the identity on all copies of II corresponding to the unaffected leaves and is ϕ n\phi_n on the affected leaf;

  • an inner face map that contracts a k 1k_1-ary vertex with a k 2k_2-ary one is the identity on all unaffected leaves and is on the affected leaves the composition

    I (k 1+k 2)ϕ k 1+k 2 1Iϕ k 1I k 1(Id,,Id,ϕ k 2,Id,,Id)I (k 1+k 2). I^{\vee (k_1+k_2)} \stackrel{\phi_{k_1+k_2}^{-1}}{\to} I \stackrel{\phi_{k_1}}{\to} I^{\vee k_1} \stackrel{(Id, \cdots, Id,\phi_{k_2},Id, \cdots, Id )}{\to} I^{\vee (k_1+k_2)} \,.

Example. In Top with I=[0,1]I = [0,1] the standard interval, and I n=[0,n]I^{\vee n} = [0,n] let ϕ n:[0,1][0,n]\phi_n : [0,1] \to [0,n] be the map given by multiplication of real numbers by nn.

Then for the planar tree T 1T_1 given by

T 1=[ ] T_1 = \left[ \array{ \searrow && \swarrow \\ & \bullet \\ && \searrow && \swarrow \\ &&& \bullet \\ &&& \downarrow } \right]

the inclusion of the tree []\left[\downarrow\right] into TT given by identifying it with its root is sent to the map f:[0,1][0,3]f : [0,1] \to [0,3] that is the composite of the map ()2:[0,1][0,2](-)\cdot 2 : [0,1] \to [0,2] wth the map h:[0,2][0,3]h : [0,2] \to [0,3] that is multiplication by two on [0,1][0,1] and addition by 1 on [1,2][1,2].

On the other hand the inner face map from

T 2=[ ] T_2 = \left[ \array{ \searrow & \downarrow & \swarrow \\ & \bullet \\ & \downarrow } \right]

to T 1T_1 corresponds to the map [0,3][0,3][0,3] \to [0,3] that is the composite of the map ()/3:[0,3][0,1](-)/3 : [0,3] \to [0,1] with the above map [0,1][0,3][0,1] \to [0,3].

Now for XCX \in C any object, we obtain the planar dendroidal set

PathsX:THom C(Ω C p[T],X). Paths X : T \mapsto Hom_C( \Omega^p_C[T], X ) \,.

It assigns to any tree with nn leaves the hom-set Hom(I n,X)Hom(I^{\vee n},X). This we can think of as the set of standard parameterized paths of parameter length nn in XX. The action of tree morphisms T 1T 2T_1 \to T_2 on these sets is the reparameterization of these paths as encoded in the tree structure.

In particular, we have the dendroidal set Paths(I)Paths(I) of the interval object itself. This is something like the little 1-cubes operad as seen by II.

I think for every XX there is an evident morphism of dendroidal sets

PathsXPathsI. Paths X \to Paths I \,.

The component over the tree TT sends all of (PathsX)(T)=Hom C(I n,X)(Paths X)(T) = Hom_C(I^{\vee n},X) to ….

For XX a pointed object, there is the sub-dendroidal set ΩXPathsX\Omega X \subset Paths X of paths whose endpoints map to the basepoint.


The above definition, evident as it may be, does not seem to be in the literature. Articles that do discuss proposals for aspects of fundamental higher categories of directed spaces include

  • Marco Grandis, Directed homotopy theory, I. The fundamental category (arXiv)

  • Tim Porter, Enriched categories and models for spaces of evolving states, Theoretical Computer Science, 405, (2008), pp. 88–100.

  • Tim Porter, Enriched categories and models for spaces of dipaths. A discussion document and overview of some techniques (pdf)

The simplicially enriched categories obtained from a directed space according to the latter have as morphisms directed paths, and as 2-morphisms directed homotopies (paths of directed paths). This is in contrast to the above definition, where the 2-cells away from their boundary are unconstrained.

Accordingly it seems that the definitio by Porter produces indeed not a Kan-complex enriched category (a model for an (,1)(\infty,1)-category) but a quasi-category-enriched category, hence actually a model for an (∞,2)-categories, a candidate for the fundamental (,2)(\infty,2)-category of a space. Generally, in the fundamental (∞,n)-category the (kn)(k \leq n)-cells should be directed maps, and the k>nk \gt n-cells be otherwise unconstrained.

Revised on November 9, 2012 02:26:56 by Stephan Alexander Spahn (