what is the exact borders of Bococa?

I think this is fairly standard, but everyone has their own definnition of the boundaries:

Carroll Gardens: Douglass to Huntington Streets and Columbia to Smith Streets. Boerum Hill: Smith Street to Third Avenue and Atlantic Avenue to Fifth Street. Cobble Hill: Atlantic Avenue to Douglass Street and Columbia to Smith Streets.
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It's hard to use streets for borders, since for the most part the boundaries may not be defined by them.

Carroll Gardens: East - Canal, South - Hamilton Pkwy, West - BQE, North - Degraw. Douglass street is wrong, don't know who came up with that one. You may say: but, but, but east of smith is gowanus hollow or some other nonsense, not until recently when some brokers got $$$ in their eyes. The boundaries have been in effect since the 60s, before which all three areas were simply South Brooklyn-along with a few others.

Cobble Hill: East - Court St, West - BQE, North - Atlantic, South - Degraw

Boerum Hill - East - 3rd Ave, West - Boerum Place, North - Atlantic/State, South - Degraw

The State St thing is a debate. I personally think it's Downtown brooklyn if you are east of Court and Brooklyn heights if you are west of court. but people who live on State Street east of court claim it's boerum hill.

The Southern Boundary of Boerum hill is a bit debatable, but I think it's roughly Degraw or Warren. The Southern boundary of Cobble Hill is definitely Degraw.