There are several variants of the definition.





Framed spaces

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

\begin{rmk} Let $V$ be an $n$-dimensional [[vector space]] with an [[inner product]] $g$. The following structures on $V$ are equivalent. \begin{enumerate} \item An orthonormal frame of $V$, i.e. an ordered sequence of vectors $v_i$, $1 \leq i \leq n$, such that $g(v_i,v_j)=\delta_ij$. \item A sequence of surjections $V_i \to V_{i-1}$, $1 \leq i \leq n$, where $V_i$ is an oriented $i$-dimensional vector space (and $V_n = V$). \end{enumerate} The two structures are related by setting $v_i$ to be the unit vector spanning $\ker(V_i \to V_{i-1})$ such that $V_{i-1} \oplus v_i$ recovers the orientation of $V_i$ (note that all $V_i$ canonically embed in $V$ as the orthogonal complement of the kernel of the composite map $V \to V_i$).

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]] $M \to \mathbb{R}$ provides useful “direction” information on $M$, e.g. for the construction of handlebodies, that is ultimately independent from any chosen metric on $M$.) Moreover, this approach offers room for generalization by varying the length of the projection sequence and the vector space dimensions. \end{rmk}

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

When forgetting basepoints, then the previous remark and example equally apply to [[affine space affine spaces]], now endowing each point in the space with a basis of frames. Using affine standard framed $\mathbb{R}^n$ as our local model for framed spaces, one may define global framed (and thereby, directed) spaces and their maps as follows.

\begin{defn}[Framed space] \label{defn:framed-space} Let $X$ be a ({convenient}) topological space. Fix $n \in \lN$ and denote by $\pi_{> i}$ the projection $\mathbb{R}^i \times \mathbb{R}^{n-i} \to \mathbb{R}^i$. \begin{enumerate} \item An $n$-framed chart $(U,\gamma)$ in $X$ is an embedding $\gamma : U \into \mathbb{R}^n$ of a subspace $U \subset X$. If $U’ \subset U$, then $\gamma$ restricts to the restricted chart $(U’,\gamma)$. \item Given two $n$-framed charts $(U,\gamma)$, $(V,\rho)$ in spaces $X$ resp. $Y$, a map $F : X \to Y$ is said to be framed on the given charts, if it restricts to a map $F : U \to V$ such that, for all $i$, there is a continuous map $F_i : \pi_{> i}(U) \to \pi_{> i}(V)$ making $\pi_{> i} \circ \rho \circ F = F_i \circ \pi_{> i} \circ \gamma$ commute. \item Two $n$-framed charts $(U,\gamma)$, $(V,\gamma’)$ in $X$ are framed compatible if $\id : X = X$ is a framed map on the restricted charts $(U \mathcal{A}p V, \gamma)$ resp. $(U \mathcal{A}p V,\gamma’)$. \end{enumerate} An $n$-framing structure on $X$ is an “atlas” $\mathcal{A}$ of framed compatible charts ${(U_i,\gamma_i)}$ such that the subspaces $U_i$ are a locally finite compact cover of $X$. An $n$-framed space is a space endowed with an $n$-framing structure. \end{defn}

\begin{defn} Given spaces with $n$-framing structure $(X,\mathcal{A}) \to (Y,\mathcal{B})$ then a map $F : X \to Y$ is said to be a framed map if for each point $x \in X$, denoting by $(U_i,\gamma_i) \in \mathcal{A}$ the charts containing $x$ and by $(V_j,\rho_j) \in \mathcal{B}$ the charts containing $F(x)$, the map $F : (U_i \mathcal{A}p F\inv(V_j), \gamma_i) \to (V_j,\rho_j)$ is framed (for all $i,j$). \end{defn}

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 [[framed combinatorial topology 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 diagram manifold diagrams]] in the language of framed spaces.


For framed spaces see:

  • {#DornDouglas21} Christoph Dorn and Christopher Douglas, Framed combinatorial topology, 2021 (pdfs)

  • {#DornDouglas22B} Christoph Dorn and Christopher Douglas, A brief introduction to framed combinatorial topology, 2022 (pdfs)