- added talk reference

highlighting combinatorialization property

]]>I started writing “framed-directed” instead of “framed” in a few places where the context isn’t clear, and I hope that improves the situation a bit. “Framed-directed” can then be abbreviated to “framed” if the context is clear. The following points were considered when the terminology was chosen:

- “directed” is a highly used term. in particular, the term “directed topological space” refers to a general idea and less so to a concrete notion. there are various ways of making that term concrete though, “framed-directed” topological spaces are one of them.
- the notion of “framed-directedness” and “ordinary framedness” are closely related (as explained to some extent at directed topological space). for instance, every stratum in a manifold diagrams is a framed manifold (in the ordinary sense). Moreover, a “framed-directed space” $X$, if $X$ is a manifold, is in particular a “ordinarily framed (namely, parallelizable) manifold $X$”. Generally, I’d argue that “framings” always refer to “choosing directions” in one way or the other.
- While “framed manifold” has an agreed-upon meaning, “framed (stratified) space” doesn’t yet.
- The terminology has been used with similar meaning in the context of stratified spaces in the work of Ayala-Francis-Tanaka-Rozenblyum: namely, they speak of “vari-framings”.

Given that the terminology “framed” here clashes with its ordinary use, and given that $\mathbb{R}^n$ equipped with such “framing” is then called a “directed space” anyways (here), why not speak of a “direction” or the like, instead of a “framing”?

]]>corrected terminology

]]>Added pointer to introductory n-category café post

]]>made definition explicit and self-contained, expanded article

]]>extended stub, including: - many motivational examples - a definitional sketch

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