In short, since both men and women are equally intelectually gifted, some people don't see the reasons for separating the competitions. I won't enter in the historical, social, cultural, physiological etc. details now, but if you are interested, you can read a very well argued blog post on this matter, written by Alexandra Kosteniuk, as a reply to the Wall Street Journal article.
I would only add one more thing: I have never come across to any reference about...hormonal construction. We all heard the terms 'testosterone', 'estrogen' or 'progesterone', hormones which influence our emotions, behaviours and even the outcome of our activities, phenomenon more obvious in the physical sports. And if we are speaking about a mind activity like chess, where good nerves and composure are highly required, the monthly cicle in a woman's body might have a word to say. It's clear that estrogen is closely linked with women's emotional well-being and to mood disruptions that occur only in women, that men cannot fully understand. What these effects mean in an individual woman is impossible to predict but they are there!
Ok. So we have separate events for various reasons. Than why should women benefit from both women and men prizes? Should they be allowed to compete in men's tournaments as well? If we admit that there are gender differences even in a brain's activity, why don't we apply the rules as in the Olympic Games? - these questions come from a professional chess player by the way, a man.
Alexandra is also touching the subject in her article: 'According to the IOC (International Olympic Committee) regulations women are not allowed to compete in men's events[...]Since chess wants to become a member of the olympic movement we should understand that soon we will need to deal with this issue and possibly women will not be allowed to compete in men's tournaments.'
I don't think we are in danger for this to happen soon: 'Chess is considered a "Mind Sport" and as part of its systematic review of the Sports Programme in 2002, the Olympic Programme Commission concluded that "Mind Sports" should not be eligible for admission to the Programme and there has been no change to these conclusions since then.' (you can read the IOC reaction in the New in Chess magazine, 2011#6, as a reply to why chess is not an Olympic Game)
Therefore, having women and men playing chess together is not an issue. In addition, there are two olympic sports where women are allowed to compete against men: sailing and equestrian. Most men's leagues are actually 'open' leagues in that women are allowed in them too. Example – all the professional leagues (NHL, MLB, NFL, etc.) allow women.
All men aren't better at all sports. However, most men have access to distinct biological advantages over women that, if utilized, can create a significant gap in competition that needs to be addressed by splitting the competition. It's not just height. It's not just weight. It's a combination of a variety of factors that affect what peak performance is for each gender. This is also what happens in chess: multiple variables must be taken into consideration.
Ignoring this, or not splitting the competition, is wholly unfair to a host of excellent female players who deserve their opportunity to display their talent without having their moves blocked by Anand or their achievements overshadowed by the results of Carlsen and Nakamura. I'm sure that, from an economic point of view, women are better off having their separate competitions.
Because women have historically faced athletic disadvantages, they should be able to play on all-female events if they choose. Women aren't being relegated to a sports ghetto by having a separate competition. It allows most of them just the opportunity to play the game. But they shouldn't be barred from playing on traditionally male tournaments either.
It's in our collective interest to create a playing structure that encourages men and women – at whatever level they can compete – to pass the pieces to one another. Professional chess shows us multiple ways to create compelling competition: rapid, blitz, knock out system etc. Or the recent pair-tournament, Botvinnik Memorial Rapid from Moscow, who wouldn't watch?
The road to coed play goes through many venues. Let's recognize that creating such opportunities is not only possible, but critical. Because sports – however much we may wish it were just play – carries wider social and political implications.