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Tim Porter
Tim Porter
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My answer would be that his insight was firstly that it pays to take what Grothendieck said in his various long manuscripts, extremely seriously and then to devote a very large amount of thought, time and effort. Many of the methods in HTT have been available from the 1980s and the importance of quasi-categories as a way to boost higher dimensional category theory was obvious to Boardman and Vogt even earlier. Lurie then put in an immense amount of work writing down in detail what the insights from that period looked like from a modern viewpoint.It worked as the progress since that time had provided tools ripe for making rapid progress on several linked problems. His work since HTT continues the momentum that he has built up.

As far as `abstracter than thou' goes, I believe that Groothendieck'sGrothendieck's ideas in Pursuing Stacks were not particularly abstract and Lurie's continuation of that trend is not either. Once you see that there are some good CONCRETE models for $\infty$-categories the geometry involved gets quite concrete as well. Simplicial sets are not particularly abstract things, although they can be a bit scary when you meet them first. Quasi-categories are then a simple-ish generalisation of categories, but where you can use both categorical insights and homotopy insights. That builds a good intuition about infinity categories... now bring in modern algebraic topology with spectra, etc becoming available.

My answer would be that his insight was firstly that it pays to take what Grothendieck said in his various long manuscripts, extremely seriously and then to devote a very large amount of thought, time and effort. Many of the methods in HTT have been available from the 1980s and the importance of quasi-categories as a way to boost higher dimensional category theory was obvious to Boardman and Vogt even earlier. Lurie then put in an immense amount of work writing down in detail what the insights from that period looked like from a modern viewpoint.It worked as the progress since that time had provided tools ripe for making rapid progress on several linked problems. His work since HTT continues the momentum that he has built up.

As far as `abstracter than thou' goes, I believe that Groothendieck's ideas in Pursuing Stacks were not particularly abstract and Lurie's continuation of that trend is not either. Once you see that there are some good CONCRETE models for $\infty$-categories the geometry involved gets quite concrete as well. Simplicial sets are not particularly abstract things, although they can be a bit scary when you meet them first. Quasi-categories are then a simple-ish generalisation of categories, but where you can use both categorical insights and homotopy insights. That builds a good intuition about infinity categories... now bring in modern algebraic topology with spectra, etc becoming available.

My answer would be that his insight was firstly that it pays to take what Grothendieck said in his various long manuscripts, extremely seriously and then to devote a very large amount of thought, time and effort. Many of the methods in HTT have been available from the 1980s and the importance of quasi-categories as a way to boost higher dimensional category theory was obvious to Boardman and Vogt even earlier. Lurie then put in an immense amount of work writing down in detail what the insights from that period looked like from a modern viewpoint.It worked as the progress since that time had provided tools ripe for making rapid progress on several linked problems. His work since HTT continues the momentum that he has built up.

As far as `abstracter than thou' goes, I believe that Grothendieck's ideas in Pursuing Stacks were not particularly abstract and Lurie's continuation of that trend is not either. Once you see that there are some good CONCRETE models for $\infty$-categories the geometry involved gets quite concrete as well. Simplicial sets are not particularly abstract things, although they can be a bit scary when you meet them first. Quasi-categories are then a simple-ish generalisation of categories, but where you can use both categorical insights and homotopy insights. That builds a good intuition about infinity categories... now bring in modern algebraic topology with spectra, etc becoming available.

Source Link
Tim Porter
Tim Porter
  • 9.6k
  • 1
  • 27
  • 41

My answer would be that his insight was firstly that it pays to take what Grothendieck said in his various long manuscripts, extremely seriously and then to devote a very large amount of thought, time and effort. Many of the methods in HTT have been available from the 1980s and the importance of quasi-categories as a way to boost higher dimensional category theory was obvious to Boardman and Vogt even earlier. Lurie then put in an immense amount of work writing down in detail what the insights from that period looked like from a modern viewpoint.It worked as the progress since that time had provided tools ripe for making rapid progress on several linked problems. His work since HTT continues the momentum that he has built up.

As far as `abstracter than thou' goes, I believe that Groothendieck's ideas in Pursuing Stacks were not particularly abstract and Lurie's continuation of that trend is not either. Once you see that there are some good CONCRETE models for $\infty$-categories the geometry involved gets quite concrete as well. Simplicial sets are not particularly abstract things, although they can be a bit scary when you meet them first. Quasi-categories are then a simple-ish generalisation of categories, but where you can use both categorical insights and homotopy insights. That builds a good intuition about infinity categories... now bring in modern algebraic topology with spectra, etc becoming available.