To crawl or not to crawl

Crawl  

The question here is: is it proper to crawl at a, or would that be the route to defeat?

Black's sente  

Black's move is sente against the life of the group. turns it into a living shape, the J+1 group. Black has thus blocked the left side in sente.

White's big move  

Hence, prevents Black's sente block. Moreover, next White can get ahead at a, with a hane at the head of two making much of Black's influence going to waste.

Next crawling question  

So, Black will usually respond, to keep his thick position on the third line.^[1] After all, White is crawling. White faces a new decision, to crawl or not to crawl again?

No sente  

This time, Black's move is not sente against the life of the corner.

White's loss  

Therefore, White's crawling once more does nothing more than crawling, which is the route to defeat. Black happily expands his influence from the third line.

Joseki  

Note: There is a reason that White plays before in the joseki. After the exchange, - , Black is already strong, and is less likely to reply to with .