> There is a category called woman, it’s defined by something that’s identify related.

But that’s not how it’s defined. People have been using that word in every language humans ever invented for thousands of years to mean biological female. If you want to argue that there is something else that isn’t biological sex and you want to invent a word for it, go nuts, but “woman” is already defined. Words can and do change definitions over time, of course. If it’s your contention that the definition by consensus has already changed, say so, but there are billions of people on this earth who haven’t got the message, which seems odd for something determined by consensus of the people who use language.

Putting that aside, since sports are about physicality and accomplishing things in the real world, it makes no sense to base them on “identity” - something that cannot be detected or defined by anyone but the self identifier - rather they should be based on physical aspects of reality.

This implies that males who identify as women but do not undergo HRT are not women in the context of sports (and their gender in other contexts remains ill defined, especially in the absence of perverse incentive). This is a form of misgendering, which is what we were trying to avoid in the first place.

    This is a position that one could take up, but it comes
    at a steep cost. It holds the societal acceptance of
    transgenderism hostage to a biological account of
    sex-gender. This is problematic for several reasons.

    Moreover, it is worth highlighting the problems with
    suggesting that sex, as biologically based, determines
    the gender with which one psychologically identifies
    [...] Second, whatever criterion is offered to ground
    this similarity would inevitably disqualify many women,
    for not all women share the same hormone levels,
    reproductive capacity, gonadal structure, genital
    makeup, and so on. (Tuvel 2017)