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edited Apr 13, 2017 at 12:58

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You might want to take a look at the responses to How to think about model categories?How to think about model categories?

Even if you only cared about presentable (∞,1)-categories, you still might prefer the general (∞,1)-categorical machinery to that of model categories, if only to speak of the (∞,1)-category of all presentable (∞,1)-categories. I don't know of any corresponding "model 2-category" of model categories (though one can see traces of what it ought to be in the work of Dugger).

However, model categories do have certain advantages. There are some model categories arising from algebra for which I don't know any alternative description of the associated (∞,1)-category, such as the model category of chain complexes of comodules over a Hopf algebra. Even when this is not the case, model categories are often more convenient for computation. For instance, I can write down a (∞,1)-categorical description of some group cohomology, but to compute it I'll probably write down a resolution of something, which is the prescription coming from model categories.

You might want to take a look at the responses to How to think about model categories?

Even if you only cared about presentable (∞,1)-categories, you still might prefer the general (∞,1)-categorical machinery to that of model categories, if only to speak of the (∞,1)-category of all presentable (∞,1)-categories. I don't know of any corresponding "model 2-category" of model categories (though one can see traces of what it ought to be in the work of Dugger).

However, model categories do have certain advantages. There are some model categories arising from algebra for which I don't know any alternative description of the associated (∞,1)-category, such as the model category of chain complexes of comodules over a Hopf algebra. Even when this is not the case, model categories are often more convenient for computation. For instance, I can write down a (∞,1)-categorical description of some group cohomology, but to compute it I'll probably write down a resolution of something, which is the prescription coming from model categories.

You might want to take a look at the responses to How to think about model categories?

Even if you only cared about presentable (∞,1)-categories, you still might prefer the general (∞,1)-categorical machinery to that of model categories, if only to speak of the (∞,1)-category of all presentable (∞,1)-categories. I don't know of any corresponding "model 2-category" of model categories (though one can see traces of what it ought to be in the work of Dugger).

However, model categories do have certain advantages. There are some model categories arising from algebra for which I don't know any alternative description of the associated (∞,1)-category, such as the model category of chain complexes of comodules over a Hopf algebra. Even when this is not the case, model categories are often more convenient for computation. For instance, I can write down a (∞,1)-categorical description of some group cohomology, but to compute it I'll probably write down a resolution of something, which is the prescription coming from model categories.

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created Dec 12, 2009 at 17:32

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You might want to take a look at the responses to How to think about model categories?

Even if you only cared about presentable (∞,1)-categories, you still might prefer the general (∞,1)-categorical machinery to that of model categories, if only to speak of the (∞,1)-category of all presentable (∞,1)-categories. I don't know of any corresponding "model 2-category" of model categories (though one can see traces of what it ought to be in the work of Dugger).

However, model categories do have certain advantages. There are some model categories arising from algebra for which I don't know any alternative description of the associated (∞,1)-category, such as the model category of chain complexes of comodules over a Hopf algebra. Even when this is not the case, model categories are often more convenient for computation. For instance, I can write down a (∞,1)-categorical description of some group cohomology, but to compute it I'll probably write down a resolution of something, which is the prescription coming from model categories.