You can find Part One of this essay here and Part Two, here.
Like War and Peace, the lessons regarding Duty contained within Lysistrata touch upon ideas so ubiquitous within the Greek culture that they are rarely discussed in the abstract and must, instead, be inferred from the characters’ commentary on seemingly divergent topics. What makes Lysistrata stand out from the rest of her conspirators is not the abstention from sex but her willingness to first identify that a conflict has arisen between differing modes of duty and then reconcile them, thus bringing harmony not only to her own life but to all of Greece. Aristophanes places great emphasis throughout the play on women’s primary role in Greek culture as wives and keepers of the home . In the opening scene, as Lysistrata complains to her first arrival, Calonice, about the other women’s tardiness to hear her plan, her friend plainly lays out the many obligations that may have kept them from the appointment.
CALONICE: Don’t worry, dear, they’ll come. It’s not so easy for a wife to get out of the house, you know. They’ll all be rushing to and fro for their husbands, waking up the servants, putting the baby to bed or washing and feeding it—
LYSISTRATA: Damn it, there are more important things than that.
Note that Lysistrata does not say that those things are unimportant, only that there are concerns that go beyond them in dire need of resolution. What does this tell us? First, there are than more one set of duties that a woman might have an obligation to fulfill and that some of them are more important than others. Only the most banal would argue that women are born with no obligations to duty whatsoever. The fact that a woman is a human being, subject to the same laws of life and death that govern us all means that her first and most basic obligation is, in fact, to continue living as long as possible, just as it is for anyone else.
Unlike men, women carry the potential for creating more human life; a necessity, that must acknowledged as pertinent to some percentage of women of reproductive age or the species would have terminated aeons ago. So, looking at it from a strictly biological perspective, a woman’s most basic potential obligation is to, first, survive, and, second, potentially give birth and nurture that offspring until the child is capable of fending for her own survival. Through the course of human evolution, it became advantageous for a woman trying to accomplish goals number one and two to pair-bond with a capable male partner, creating a third sense of duty; expressed mostly simply as the relationship between husband and wife—though every culture and civilization develops an unique set of mores regarding this relationship (sometimes called marriage) reflective of its peculiar values and always subject to periodic re-examination.
These three obligations (to husband, to child, to self; listed in order of importance) can be thought of three-fourths of the Greek status quo for a woman’s duties as assumed by both Aristophanes and his audience. For the purpose of placing them on a more pragmatic continuum, I found it useful to think of these obligations as Internal at one extremity and progressing from self, to child, and then to husband (the opposite of Greek attitudes) towards another extreme that might be thought of as External. While the institution of Marriage (as observed by the Greeks) still represents an Internal obligation (as it is tied at least historically into both the preservation of self and nurturing of offspring), the final duty assumed in Lysistrata, a woman’s relations to the gods and goddesses represents the only socially acceptable foray into territory that is inarguably Exterior. As one might suspect, this piety is restricted to some extent to those goddesses specifically tied to home and the hearth. Athena, goddess of wisdom, war, and patroness to the city of Athens, despite her gender, is considered, to some extent, off limits to the women. As the women storm the Acropolis (Athena’s temple), the chorus of men sound their indignation.
Our women, if you please!
We’ve kept and fed within our doors
A pestilent disease!
They’ve seized our own Acropolis
With bars they’ve shut the gate!
They hold the statue of the Maid,
Protectress of our state!
Come on and let us hurry there
And put these logs around,
Smoke out the whole conspiracy
From Pallas’ sacred ground! 
The female chorus, a short time later responds, saying:
Some frail old men
Approach with limping gait,
And carry logs
Of an enormous weight.
Dire threats they make,
Our friends they hope to see
O Maid, this must not be!
No, may they save
All Greece from war insane
For that is why
They occupy thy fane. 
Later, as Lysistrata is able to make her case before the magistrate, another exchange is even more descriptive of this divine segregation inherent to the culture.
MAGISTRATE: Well, the first thing I want to know is—what in Zeus’ name do you mean by shutting and barring the gates of our own Acropolis against us?
LYSISTRATA: We want to keep the money safe and stop you from waging war.
MAGISTRATE: The war has nothing to do with the money—
LYSISTRATA: Hasn’t it? Why are Peisander and the other office-seekers always stirring things up? Isn’t it so they can take a few more dips in the public purse? Well, as far as we are concerned they can do what they like; only they’re not going to lay their hands on the money in there.
MAGISTRATE: Why, what are you going to do?
LYSISTRATA: Do? Why, we’ll be in charge of it.
MAGISTRATE: You in charge of our finances?
LYSISTRATA: Well, what’s so strange about that? We’ve been in charge of all your housekeeping finances for years.
MAGISTRATE: But that’s not the same thing.
LYSISTRATA: Why not?
MAGISTRATE: Because the money here is needed for the war!
LYSISTRATA: Ah, but the war itself isn’t necessary.
MAGISTRATE: Not necessary! How is the City going to be saved then?
LYSISTRATA: We’ll save it for you.
MAGISTRATE: This is intolerable!
LYSISTRATA: It may be, but it’s what’s going to happen.
MAGISTRATE: But Demeter!—I mean, it’s against nature. 
Parsing that passage, we see the magistrate open his attack with a double invocation of the patriarchal prerogative; first condemning her actions in the name of Zeus and then referring to the Acropolis as “our own”, meaning belonging rightly to the men. Sidestepping this condemnation to her authority, Lysistrata refutes his argument against it. He is left at the end only to invoke a “proper” female goddess, Demeter and labeling the entire spectacle as “against nature.”
Once Lysistrata moves beyond the conventions of her culture, she finds that there is very little that does not concern her more directly than she ever supposed, opening up a whole new sphere of duties that women might share alongside the men of Greece.
LYSISTRATA: In the last war we were too modest to object to anything you men did—and in any case you wouldn’t let us say a word. But don’t think we approved! We knew everything that was going on. Many times we’d hear at home about some major blunder of yours, and then when you came home, we’d be burning inside but we’d have to put on a smile and ask what it was you’d decided to inscribe on the pillar beneath the Peace Treaty.—And what did my husband always say?—“Shut up and mind your own business!” And I did.
MAGISTRATE: He’d have given you one if you hadn’t!
LYSISTRATA: Exactly—so I kept quiet. But, sure enough, next thing we knew you’d make an even sillier decision. And if I so much as said, “Darling, why are you still carrying on with this silly policy?” he would glare at me and say, “Back to your weaving, woman, or you’ll have a headache for a month. ‘Go and attend to your work; let war be the care of the menfolk.’”
MAGISTRATE: Quite right too, by Zeus.
LYSISTRATA: Right? That we should not be allowed to make the least little suggestion to you, no matter how much you mismanage the City’s affairs? And, now, look, every time two people meet in the street, what do they say? “Isn’t there a man in the country?” and the answer comes, “Not one.” That’s why we women got together and decided we were going to save Greece. What was the point of waiting any longer, we asked ourselves. Well now, we’ll make a deal. You listen to us—and we’ll talk sense, not like you used to—listen to us and keep quiet, as we’ve had to do up to now, and we’ll clear up the mess you’ve made. 
Now our continuum continues on its journey away from the traditional Greek sphere of women's obligation and into territory deemed by culture, convention and dogma to be exclusively male. This is Aristophanes’ radical leap as he begins to question what a woman’s obligations to her City, to Greece, and, indeed, to all humanity might very well be.