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Lurie define in HTT(Def. 1.1.1.6) a topological category as a enriched category over compactly generated (and weakly Hausdorff) topological spaces, but usually we define a topological category as lifting of sources (other definitions).

My question: it's equivalent the definition of topological category of Lurie with the usually?

Note: Lurie mention: "I take that definition for convenience" (Pg. 7), but he does not say anything about the connection with usual definition.

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    $\begingroup$ No, they are completely different concepts, as the second link you provide explains. $\endgroup$
    – Marc Hoyois
    Dec 13, 2017 at 1:33
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    $\begingroup$ There are actually at least three things that "topological category" can mean. In order of prevalence, they are: 1. category enriched in $Top$, 2. category internal to $Top$, 3. the notion defined in terms of sinks. (1) can be regarded as a special case of (2), but as Marc says, they are completely unrelated to (3). I think it's fair to say that most mathematicians outside of pure category theory are completely unaware of usage (3), whereas usage (1) and maybe (2) are well-known to many mathematicians in other areas (especially those in Lurie's audience who might use higher category theory.) $\endgroup$
    – Tim Campion ♦
    Dec 13, 2017 at 2:04

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