nLab directed topological space




topology (point-set topology, point-free topology)

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A directed topological space is a topological space XX in which there is some ‘sense of direction’. This can happen in various different ways and the level of the ‘directedness’ can be different in different situations, so naturally there are several ‘competing’ ideas, but the beginning of a consensus on what the overarching idea is.

If one bases homotopy theory on the idea of a singular simplex or more generally a singular cell of any shape, then there is no way in which a ‘sense of direction’ can be encode. If we have a path in a space we can go along it (traverse it) in either direction, from 0 to 1 or from 1 to 0. From this perspective a directed space is one in which not every singular cell Δ nX\Delta^n \to X (for Δ n\Delta^n the standard topological simplex) is supposed to be traversable in all directions, in some sense: instead these kk-dimensional paths may have a direction .

As an example one can base the ‘sense of direction’ on a closed preorder or partial order, (that is a pospace),so that the paths from the directed interval [0,1][0,1] with the usual order to the space XX, can only be ‘traversed’ in one direction. Another example which does not fit into this first type would be the directed circle.


In other words, a circle with direction determined by the anticlockwise sense. Again it is easy to see that there are certain paths that respect the direction, ‘directed paths’ whilst others do not.

So far there exists a well-developed theory for a notion of directed spaces XX where 1-dimensional paths given by maps [0,1]X[0,1] \to X from the interval into the space are equipped with a direction. See in particular the book by Marco Grandis on Directed Algebraic Topology listed below. Another suggested notion for modelling directed spaces is that of framed spaces, which is tailored towards certain higher categorical applications.

Note that a directed space is like a generalised space; not every directed space need be a space in the traditional sense, in accordance with the red herring principle. As an instance of this, note that Marco Grandis in his book Directed Algebraic Topology handles the directed homotopy of small categories, and of cubical complexes, since this is useful for comparison an interpretation of directed homotopy ‘invariants’.

Directed spaces are studied in directed homotopy theory, a relatively young topic. In generalization of how a topological space has a fundamental groupoid, a directed space has a fundamental category.

Homotopy-theoretic perspective

From a homotopy theoretic perspective one would wish that notions of directed spaces might serve to generalize the homotopy hypothesis – which identifies ordinary (undirected) topological spaces with ∞-groupoids, i.e., with (∞,0)-categories – to a more general context where (∞,0)-categories are generalized to (∞,r)-categories with r>0r \gt 0:

An (∞,r)-category in this context might correspond to a rr-directed topological space , one that comes equipped with a notion of orientation of its kk-cells for 0kr0 \leq k \leq r, but was impartial on direction above that dimension.

If such a definition exists, it may need to use filtered topological spaces instead of bare topological spaces.

Even in the absence of a homotopy-theoretic definition of rr-directed space in this sense, from the perspective of homotopy theory one might take the standpoint of the homotopy hypothesis and define a (nice) rr-directed space to be an (∞,r)-category, just as it makes good sense and is nowadays common practice in algebraic topology to define a nice topological space to be an ∞-groupoid.

See (n,r)-category for more on that.



A directed topological space or d-space is pair (X,dX)(X, d X) consisting of a topological space XX and a subset dXC(I,X)d X \subset C(I,X) of continuous maps from the interval I=[0,1]I = [0,1] into XX – called directed paths or d-paths – satisfying the following conditions:

  1. (constant paths) every constant map IXI\to X is directed,

  2. (reparametrisation) dXdX is closed under composition with increasing maps III\to I,

  3. (concatenation) dXdX is closed under path-concatenation: if the d-paths a,ba, b are consecutive in XX (a(1)=b(0))(a(1) = b(0)), then their ordinary concatenation a+ba+b is also a d-path

(a+b)(t)=a(2t),if0t12,(a+b)(t) = a(2t),\,\text{if}\, 0\le t\le \frac{1}{2},
(a+b)(t)=b(2t1),if12t1.(a+b)(t) = b(2t-1),\,\text{if}\, \frac{1}{2}\le t\le 1.

A morphism of directed topological spaces f:(X,dX)(Y,dY)f : (X, d X)\to (Y , d Y) is a morphism of topological spaces f:XYf: X \to Y which preserves directed paths in that for every γ:IX\gamma: I \to X in dXd X the path f *γ:IγXfYf_* \gamma : I \stackrel{\gamma}{\to} X \stackrel{f}{\to} Y is in dYd Y.


Basic examples of dd-spaces include:

  • The standard directed interval is I d=([0,1],dI)I_d = ([0,1], d I) with dId I the set of all monotonic continuous maps [0,1][0,1][0,1] \to [0,1] is a d-space

  • Any pospace XX gives rise to a d-space by taking the directed paths to be, well, directed paths, i.e. continuous order-preserving maps from I dI_d to XX.

Many other example can be found in the references.


A different definition comes from Sanjeevi Krishnan, A Convenient Category of Locally Preordered Spaces, Applied Categorical Structures, 2009, vol. 17, no 5, p. 445-466 (arxiv):

Definition A stream is a tuple X, X, \leq_{-}, where \leq_{-} assigns to each open subset UXU \subset X a preorder U\leq_U, such that:

iU i= i U i\leq_{\bigcup_i U_i} = \bigvee_i \leq_{U_i}

Here, U i,iIU_i, i \in I is a collection of open sets, and \bigvee is pointwise or of relations.


  • Morally, a point in a stream is less or equal to another point if it is less or equal in any open set. The relation X\leq_X for the whole space does not hold much information.
  • In the typical example of the clockwise oriented circle, each point is less or equal to every other one in the relation S 1\leq_{S^1}; but for a contractible subset US 1U \subset S^1, a point xx is less or equal than yy if yy can be reached from xx in a monotonous clockwise path.
  • The defining equation of a stream can be thought of as a (co)sheaf condition, and there is indeed a cosheafification result.
  • Every d-space gives rise to a stream.
  • The category of streams has good properties. In particular, there is an further notion of compactly flowing streams_ extending the notion of compactly generated Hausdorff spaces, and indeed the forgetful functor creates limits and colimits.
  • The category of compactly flowing streams is Cartesian closed.

Framed spaces


Another way to endow spaces with directions is via framings, i.e. choices of a “basis of the vector space of tangential directions at each point”. Somewhat abstracting this idea, frames may also be thought about in terms of projections, as the following remark motivates.


Let VV be an nn-dimensional vector space with an inner product gg. The following structures on VV are equivalent.

  • An orthonormal frame of VV, i.e. an ordered sequence of vectors v iv_i, 1in1 \leq i \leq n, such that g(v i,v j)=δ ijg(v_i,v_j)=\delta_{ij}.
  • A sequence of surjections V iV i1V_i \to V_{i-1}, 1in1 \leq i \leq n, where V iV_i is an oriented ii-dimensional vector space (and V n=VV_n = V).

The two structures are related by setting v iv_i to be the unit vector spanning ker(V iV i1)\ker(V_i \to V_{i-1}) such that V i1v iV_{i-1} \oplus v_i recovers the orientation of V iV_i (note that all V iV_i canonically embed in VV as the orthogonal complement of the kernel of the composite map VV iV \to V_i).

In the absence of inner products, one cannot speak of orthonormal frames any longer. However, sequences of projections can still be defined, and may be regarded as playing the role of “metric-free orthonormal” frames. (A vaguely analogous line of thinking is that a Morse function MM \to \mathbb{R} provides useful “direction” information on MM, e.g. for the construction of handlebodies, that is ultimately independent from any chosen metric on MM.) Moreover, this approach offers room for generalization by varying the length of the projection sequence and the vector space dimensions.


The standard orthonormal frame of nn-dimensional euclidean space n\mathbb{R}^n consists for the ordered sequence of vectors e 1=(1,0,...,0)e_1 = (1,0,...,0), e 2=(0,1,0,...,0)e_2 = (0,1,0,...,0), …, e n=(0,...,0,1)e_n = (0,...,0,1). By the previous remark, this orthonormal frame is equivalently described by the sequence of projections π i: i= i1× i1\pi_i : \mathbb{R}^i = \mathbb{R}^{i-1} \times \mathbb{R} \to \mathbb{R}^{i-1} (each i\mathbb{R}^i being endowed with standard orientation).

When forgetting basepoints, then the previous remark and example equally apply to affine spaces, now endowing each point in the space with a basis of frames. Using affine standard framed n\mathbb{R}^n (or rather compact contractible patches of it, that interact nicely with the projections, see below) as our “local models” for framed spaces one may define global framed spaces and their maps.



Inductively in nn \in \mathbb{N}, an nn-framed patch U nU \subset \mathbb{R}^n is a non-empty subspace of n\mathbb{R}^n with the property that its projection π n(U)\pi_n(U) is an (n1)(n-1)-framed patch, such that π n:Uπ n(U)\pi_n : U \to \pi_n(U) has fibers π n 1(x)\pi^{-1}_n(x) of the form [γ (u),γ +(u)][\gamma_-(u),\gamma_+(u)] for two continuous sections γ ±:π n(U)π n(U)×\gamma_\pm : \pi_n(U) \to \pi_n(U) \times \mathbb{R}. Given two nn-framed patches UU and VV, a (partial) nn-framed patch map F:UVF : U \to V is a (partial) continuous map which descends along π n\pi_n to a (partial) (n1)(n-1)-framed patch map F n1:π n(U)π n(V)F_{n-1} : \pi_n (U) \to \pi_n(V) such that mappings of fibers F:π n 1(x)π n 1(F n1(x))F : \pi^{-1}_n(x) \to \pi^{-1}_n(F_{n-1}(x)) are monotone.

Note that nn-framed patches are compact and contractible spaces.


The standard example of an nn-framed patch is the closed nn-cube I n=[1,1] n n\mathbf{I}^n = [-1,1]^n \subset \mathbb{R}^n. However, in general nn-framed patches need not be “nn-dimensional”: for instance, the 00th slice {0}×I n1\{0\} \times \mathbf{I}^{n-1} of the nn-cube is itself an nn-framed patch, and so are the “kk-directed intervals” I k:={0} k1×I×{0} nk\mathbf{I}_k := \{0\}^{k-1} \times \mathbf{I} \times \{0\}^{n-k}.


Let XX be a topological space. Fix nn \in \mathbb{N}.

  • An nn-framed chart (U,γ)(U,\gamma) in XX is an embedding γ:U n\gamma : U \hookrightarrow \mathbb{R}^n of a subspace UXU \subset X whose image im(γ)\im(\gamma) is an nn-framed patch.

  • Two nn-framed charts (U,γ)(U,\gamma), (V,ρ)(V,\rho) in XX are compatible if ργ 1\rho \circ \gamma^{-1} is a partial map of nn-framed patches.

An nn-framed space is a space XX endowed with an nn-framing structure 𝒜\mathcal{A}, which is an “atlas” of compatible nn-framed charts {(U i,γ i)}\{(U_i,\gamma_i)\} such that U iU_i are a locally finite cover of XX.


The condition for covers to be locally finite is convenient as it describes the situation of locally finite cell complexes. However, the condition could be replaced, and the definition generalized, in several ways. The above version of the definition should be considered a first approximation to a potentially more general notion of framed spaces.

Maps of framed spaces could be defined along the following lines.


Given spaces with nn-framing structure (X,𝒜)(Y,)(X,\mathcal{A}) \to (Y,\mathcal{B}) then a framed map F:XYF : X \to Y is a map such that for any charts (U,γ)𝒜(U,\gamma) \in \mathcal{A} and (V,ρ)(V,\rho) \in \mathcal{B}, FF yields a partial map ρFγ 1:im(γ)im(ρ)\rho \circ F \circ \gamma^{-1} : \mathrm{im}(\gamma) \to \mathrm{im}(\rho) of nn-framed patches.

Framed spaces and framed maps are, in a sense, “very rigid” variants of directed spaces. Nonetheless, they are interesting to study as they turn out to have a rich combinatorial theory associated to them. This combinatorial counterpart is particular useful when translating between the topology of directed spaces and the combinatorics of higher categories: an example of this is the definition of manifold diagrams in the language of framed spaces.


Realizations of geometric computads (resp. their functors) have the structure of framed spaces (resp. of framed maps), see Dorn23.

Comparison to dd-spaces


One may want to define directed paths in an nn-framed space as maps from “the framed interval” into that space: but, in fact, there are now nn different framed intervals, corresponding to the nn possible directions of nn-framings. These are precisely modelled by the nn-patches I k\mathbf{I}_k mentioned in a previous example. Note, there is a framed bijection I kI j\mathbf{I}_k \to \mathbf{I}_j if and only if kjk \leq j (in particular, directions aren’t interchangeable at all: analogously, note that in an n-category a (nk)(n-k)-morphisms can be (possibly degenerate) (nj)(n-j)-morphisms only if kjk \leq j). Given a framed space (X,𝒜)(X,\mathcal{A}), and defining d kX=Map fr(I k,(X,𝒜))d_k X = \mathrm{Map}_{\mathrm{fr}}(\mathbf{I}_k, (X,\mathcal{A})), we thus obtain a filtration of directed paths:

d nXd n1X...d 1XMap(I,X) d_n X \subset d_{n-1} X \subset ... \subset d_1 X \subset \Map(\mathbf{I},X)

instead of a single path subset dXMap(I,X)dX \subset \Map(\mathbf{I},X), as required in the definition of dd-spaces.


  • If we can equip directed spaces with an internal hom, then a directed space with at least one directed path should be a strictly directed object in the category of directed spaces, with respect to the standard directed interval as the interval object, while an ordinary topological space regarded as a directed space should be an undirected object.


The above definition of dd-spaces is from

This has now developed into a book

A discussion of reparameterization of directed paths in directed topological spaces is in

  • Ulrich Fahrenberg and Martin Raussen, Reparametrizations of Continuous Paths (arXiv, (blog))

Further references are given in directed homotopy theory.

Some ‘local combinatorial’ aspects of framed spaces are discussed in:

A global (higher-categorical) perspective on directed spaces is taken in:

Last revised on May 13, 2023 at 12:30:45. See the history of this page for a list of all contributions to it.