What do people in homotopy theory mean when they say "... is a model for spaces / $(\infty,1)$-categories"? And why does one need models?


Is this related to the notion of a model category?



"Models" in homotopy theory
Yes, this is absolutely related to model categories (although the concept of models for an homotopy theory is much more general): the notion of model category was introduced by Quillen exactly to express the idea of models for homotopy types. I quote Quillen's *Homotopical Algebra* (Chapter I, page 0.3):



> 
> The term "model category" is short for "a category of models for a
> homotopy theory" where the homotopy theory associated to a model
> category $\underline C$ is the homotopy category $Ho \, \underline C$
> [...]. The same homotopy theory may have different models e.g.
> ordinary homotopy theory with a basepoint is the homotopy theory of
> the following model categories: 0-connected topological spaces,
> reduced simplicial sets, and simplicial groups.
> 
> 
> 


In fact, Quillen introduced model structures to give sufficient conditions for two model categories to induce equivalent homotopy theories in a way which is compatible with possible extra structures of interest (e.g. suspension, cofiber sequences, etc): the notion of Quillen equivalence.


We need models because we need to define a written formal language to speak of mathematical objects, including $\infty$-categories or $\infty$-groupoids (=spaces or anima...). And a formal language is made of letters, the concatenation of which is strictly associative, and basic induction rules will follow the principle of modus ponens which is a strictly associative process as well. In particular, with so much strictly associative processes to begin with, the formal language we choose will define a $1$-category. Now, there is no unicity of language. For instance, if we could decide to interpret $\infty$-groupoids as CW-complexes (morphisms in $\infty$-groupoids being paths), or as Kan complexes (morphisms in $\infty$-groupoids being $1$-dimensional simplices) we get two interpretations. It is a theorem of Milnor that the homotopy theory of CW-complexes up to homotopy is equivalent to the homotopy theory of Kan complexes up to simplicial homotopy. This can be promoted to a Quillen equivalence between suitable model category structures à la Quillen on topological spaces and on simplicial sets. Having a Quillen equivalence implies that we have in fact an equivalence of $\infty$-categories, out of which one can define all the extra structures that Quillen was thinking about when he wrote his monograph on homotopical algebra. This why, nowadays, we define homotopy theories as $\infty$-categories.


Now, there are infinitely many different models for the homotopy theory of $\infty$-groupoids - and similarly for $\infty$-categories (in fact for any homotopy theory coming from an $\infty$-category...).


