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The question is basically in the title. More generally, I would like to know if this, or any reasonable variant of it, is true. Or perhaps, to understand better the gap between 2-Segal spaces and Span-enriched $\infty$-categories.

Background

2-Segal spaces (see e.g. Dyckerhoff and Kapranov-s paper) are usually informally described as $\infty$-categories with multi-valued composition. This notion seems to be most naturally encoded by enrichment over the $\infty$-category of spans of spaces and this interpretation is mentioned both in the nLab entry and in this MO question by Tim Campion. Dyckerhoff and Kapranov prove something of this sort for 2-Segal sets, but for 2-Segal spaces there are instead some involved constructions with algebras in the $(\infty,2)$-category of BiSpans, which I am not sure how they relate to enrichment over Span.

It seems important to note that for 1-Segal spaces, which are "category objects" in spaces, we do need to impose completeness to get a model for usual $\infty$-categories. It therefore seems reasonable that general 2-Segal spaces encode "multi-compoisiontal category objects" and only after imposing some completeness condition, we might hope to get Span-enriched $\infty$-categories. In the above mentioned paper there is a definition of completeness for 2-Segal spaces (combine def 9.3.2 for completeness of (co)Segal fibrations and def 9.3.4 for the Hall (co)Segal fibration associated to a 2-Segal space). This might or might not be the relevant notion of completeness and I admit that I don't have a good understanding of it.

Another issue is the comparison of the morphisms on both sides. As pointed out by Tim Campion in his question, morphisms of 2-Segal spaces might need to correspond to Span-enriched functors which have actual maps (and not just spans) between the corresponding mapping spaces.

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  • $\begingroup$ There's some redundancy in the links -- I believe I wrote the relevant portions of the nlab page :). That's a very good point about the subtlety of completeness -- it's quite possible that I've jumped to conclusions and the story is more complicated than one might naively imagine... $\endgroup$
    – Tim Campion
    Nov 23, 2019 at 19:10

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Something like this is now a theorem of Walker Stern.

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