Like many modernist texts, Ulysses ransacks mythology to provide a fractured modern world with some underlying order.This is certainly a very familiar idea, almost a cliche. One tires of hearing how Joyce shows how modern life reveals deep mythic structures, how the Homeric narrative silently supports the 'surface story' behind the backs of the characters themselves.
The only problem is that this idea doesn't really bear scrutiny. There are few one to one and systematic Homeric parallels in Ulysses. It is not as if incidents in the immediate story are systemically translatable back into their Homeric ‘equivalents'. Take for example the ‘no man’ of the Cyclops episode – this epithet could refer to the narrator but also to Bloom, whose polyoptical view of things (he can always 'see the othe fellow's point of view'), various names (Bloom/ Flower/ Virag) and fleeting connection with Everyman make him well fitted to this. Odysseus’ 20 year exile is glimpsed, arguably, in the 20-1 odds on Throwaway, like a tiny ironic splinter of the original story. The blinding of Polyphemus is not simply turned into Bloom’s metaphorical blinding of the monofocal Citizen, but is glimpsed in the sweep’s brush that nearly has the anonymous narrator’s eye out. In such trivial incidents are glimpsed the refracted light of the dead mythic star. The Homeric content is shattered and re-distributed, a single element appearing in Joyce’s Dublin as several splinters. Molly is not only Penelope but at one point Circe too, so that ‘Penelope’ and ‘Circe’ slide and reattach. New resemblances and significances are produced from the ruins of a distant mythic substrate.
We might think about this in terms of what Georg Lakoff (discussing metaphor) terms ‘cross-domain’ mapping. For example, when one says ‘she was really cold to me’, emotion is understood – or ‘mapped’ – in terms of temperature; when we say ‘the past is behind me’, time is mapped in terms of space. One ‘domain’ (the target domain) is understood in terms of the other (the source domain). So it is that Ulysses is understood as a kind of cross-domain mapping with 1904 Dublin as the target domain and Homeric myth as source domain. It's this which seems to me only half-true. What happens rather is that the ‘source domain’ (the Homeric) is broken up and disseminated through the target domain; and instead of being a domain of stable meaning which spontaneously 'reads' early 20th C Dublin, it is used for jokes, puns, semantic hyperlinks and so on. The Homeric narrative is used as a grammar which can generate very different sentences; in short, a compositional device.
If we see this from the point of view of novelistic construction, it is clear that a mythic element, such as the ‘no man’ of the Cyclops episode, is not approached in terms of ‘what is the contemporary equivalent of this? (ie who is the modern signifier of this signified)’ but ‘what different meanings can this signifier generate?’ You then put the signifier to work in the text, like a little programme, throwing up various puns, correspondences etc.
The 'unity' or 'order' imposed by the Homeric in no way points to the underlying order of the apparently chaotic world; it is instead used in ludic or ironic ways. It has been dismantled and turned into pliable material surrendered into the hands of the inventor (JJ).