The cat offers a vivid, appealing, and highly appropriate metaphor for the fog in T. S. Eliot's “The Love Song of J. Alfred Prufrock.” The job of a metaphor is to describe something unknown in terms of something familiar. Most people are much more familiar with the behavior of cats than they are with a city fog. It is easy to picture a cat rubbing against something and sliding around corners, flicking its tongue and lounging comfortably, leaping with ease and gliding about silently, and then curling up and falling asleep. The fog does the same, according to Eliot. It, too, rubs and slides, licks and lounges, leaps and glides, and then curls up around the house and falls asleep. By describing the fog as a cat, Eliot has created an intense picture of this yellow fog and its behavior.
The repetition of the words “yellow” and “window-panes” add to the strangeness of the scene and emphasize the separation of the speaker from others. Fog and smoke are not typically yellow in most people's experiences, yet they are in this city. They are yellow from the chemicals in the air, chemicals that stain the fog and the buildings and the streets. This yellow smoke and fog encase the city in grime and pollution, and their strange color makes the scene eerie and rather revolting.
Further, the repetition of the compound “window-panes” emphasizes the separation of the external and the internal worlds. The speaker is watching the fog from outside. The windowpanes block the fog from coming in, but they also keep out the speaker, emphasizing his alienation from the people within the buildings and symbolizing his alienation from the world and even from himself. Windowpanes are a barrier, and the speaker encounters plenty of other barriers as he attempts to interact with others and to discover who he is and what he is supposed to do.
The description of the fog as a cat is the use of personification. Eliot gives the fog the qualities of a living creature in order to give it more importance. The fog underlines the feeling of loneliness established in the first section of the poem, also suggested by the "one night cheap hotels" and the "sawdust restaurants". The description mirrors the loneliness of the Prufrock, who is searching for some meaningful human connection. The sleepy, cat-like fog is something tangible that Prufrock identifies with in his own search. He says there will be "time" for the fog, just as there will be "time" for him to "meet the faces that you meet."
Personifying the fog makes it easier to show Prufrock's identification with it - one living creature to another. The description of the fog, combined with the repetition of certain words and the rhyme scheme, heighten the song-like qualities of the poem. Eliot labels this a "love song" and uses these elements to the romantic introspection of the speaker, Prufrock. This is not a love song in the traditional sense, of course. ...
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This poem is one of the first modern poems, and Eliot mixes these traditional romantic elements with the modern images of urban loneliness and social anxiety to challenge the traditional. Prufrock is an anti-hero, an awkward and unsureprotagonist in the story of his own life.
The main purpose would be to simply be poetic, and use an interesting metaphor for the fog. Cats are known to be quiet, stealthy, adept, and able to scale and inhabit places that other creatures can't. So, comparing the fog to a cat is an apt description because fog too is silent, stealthy, and creeps into places that others can't. Also, in this beginning part of the poem, Eliot is describing a run-down part of the city; he mentions the "half-deserted streets", the "one-night cheap hotels," the "sawdust restaurants with oyster shells." He emphasizes the run-down atmosphere by using the word "yellow" repeatedly; this is no beautiful fog that creeps beautifully before a glowing moon. Describing it as yellow emphasizes the industrial, polluted nature of the fog. Most likely, in the city, the pollution from industrialization is so dense that it turns the fog yellow. That this yellow, dirty color brushes against the windowpanes indicates that it is right at their door, and hard to escape. Eliot was a modernist writer, and modernists tended to bemoan the dehumanization and pollution of the industrial age, so Eliot's mentioning of the "yellow" fog plays into that.
Also, the part of the city that he is describing is rather decrepit, and with that image comes the fact that there are probably many stray cats that are running around in the alleys. So, as the fog creeps through this part of the city, alley-cats are probably fairly common and a part of that run-down atmosphere. So, rather than comparing the fog to say, a snake, Eliot picks an animal that is regional and logical for that part of town.
I hope that those thoughts help a bit; good luck!