My first and ongoing reaction to New York City is: I have seen all this in the movies. I already knew that the "Bronx is up and the Battery’s down" - important because it is the reverse of British usage where you go "up" to London.
This is the view if you climb out on to the fire escape of our hotel.

In fact there are a lot of dogs in New York and most of their owners seem responsible

These fire escapes look like they were afterthoughts but they were probably part of the original design

Possibly a sewer vent

Outside Schatzie Prime Meats on Broadway just up from 101st Street

Looking up Broadway from the corner of 102nd Street. Our hotel is at the centre of this photo

Cropped out of the previous image. The Marrakech Hotel at 2588 Broadway on the corner of 103rd street. I assume the hotel occupies all four of the upper floors of this building and there is no elevator! Fortunately we were on what the Americans call the second floor.

Just south of the cathedral: the Peace_Fountain by Greg Wyatt in the West 111th Street People's Garden

New York delights in superlatives. This claims to be the biggest Anglican cathedral in the world.

The choir parapet of notable figures from each century since the birth of Christ here we see, from right to left: 14xx Christopher Columbus, 15xx Galileo, 16xx Shakespeare, 17xx George Washington and 18xx Abraham Lincoln. The parapet was installed in 1922 and they left the last niche empty until the century had ended.

When that happened, they could not agree on a single person so they gave us this interesting quartet: Martin Luther King Jr., Albert Einstein, Susan B. Anthony (American social reformer and women's rights activist born into a Quaker family) and Mohandas Gandhi.

The building that New Yorkers call the Port Authority is in fact a bus terminal. Nearby it and parodying the name Lesley spotted the Beer Authority. Which is a hopeful sign that the USA is not a total lager desert. The place has sixty taps serving this wide range of beers.

The Beer Authority is entered from West 40th Street. this is the view from its window on to 8th Avenue

Harry Beck designed a diagrammatic map of the London Underground back in 1931. But the Metropolitan Transportation Authority sticks doggedly to this approximately geographical format. Points to note: it has been rotated about 30° anti-clockwise from north-up so Manhattan is straight up and down and Central Park looks unnaturally wide because of stretching needed to show the detail in Midtown.
I also found this even more complicated map dated 2010 June in their archive.

Getting about NYC is generally quite easy with their boring grid system of streets and avenues. But one trap to catch the unwary is that you can have separate subway stations with the same name. There are three stations called 103rd Street in Manhattan at least a kilometre apart from each other and another in Queens. This is 103rd Street served by the train.
Those parts of the subway which are actually underground were mainly built by cut-and-cover. Here we are looking up at the sky. Up top these are just metal grilles in the cetral reservation of Broadway.

The guard rides in a tiny cubicle midway down the train. These monitors, I think at 103rd Street for the train, enable them to check the state of the closing doors

Monster vacuum cleaner! Do we have these on the London Underground?

Automated display on the train. Lights on to show where it has yet to stop - important because the <> train does not stop at certain stations.

South Ferry station is the downtown terminus of the train. It is built on a very tight curve - you can only get out from the first few cars and they have these extending platforms to cover the gap. These three imaages show the platform extended - note two yellow strips and …

here is the platform retracted.

An hallmark of the New York subway is the lines of pillars close to the platform edge arising from they way they do cut-and-cover. This is Fulton Street station on the and lines.

An exception to the cut-and-cover construction had to be used at Washington Heights which include the highest natural point in Manhattan at 81 metres (265 feet) above sea level. So the platforms here at 181st Street station on the tunnelling construction was used. The line is reached by "three massive elevators" and and we have a wide space free of pillars.

Clearly the subway builders were not familiar with deep tube building techniques. There was a fatal accident during construction in 1903 and in 2009 an eight metre long section of the brick lining collapsed.

And even here, if we look in the opposite direction along the platform, they have reverted to their beloved pillars.

But by the time we get to Dyckman Street the subway has become overground.