"Models" in homotopy theory 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?
 A: It is not related to model categories. It's related to mathematical modeling. The point is that we have many possible choices for what we mean by "space." You might mean "topological space" but someone else might mean "simplicial set" or "Delta generated space." It's important to specify which definition we are using for the concept we care about. Different choices have different pros and cons. For example, if by "space" you mean "topological space" and you prove something about spaces, then it holds more generally than if by "space" you meant "CW complex." On the other hand, if by "space" you mean "simplicial set" then you have more tools at your disposal. Your category of spaces is locally presentable, for example.
Same for $(\infty,1)$-categories. There are many choices for what you could mean by that. For example, you might mean "quasicategory" and someone else might mean "relative category" or "simplicial category." All concepts are equivalent on the homotopy category level, but it's still important to specify which mathematical model you are using for the phenomenon you wish to study.
A: 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...).
A: Maybe a "lower" analogue would be helpful.
An ordered pair is an object that contains two pieces of data, the first component and the second component.
Suppose we want to make this precise in the language of ZFC, where everything is a set.
Then there are several ways we could "implement" the construction of ordered pairs.
For example, we could define
$$
(a, b)_1 = \{\{a\}, \{a, b\}\}
$$
or
$$
(a, b)_2 = \{\{\{a\}, \emptyset\}, \{b\}\}.
$$
We could refer to these as different "models" for the ordered pair $(a, b)$ and, if $X$ and $Y$ are sets, we could also refer to the sets
$$
X \times_1 Y = \{(x, y)_1 \mid x \in X, y \in Y\}
$$
and
$$
X \times_2 Y = \{(x, y)_2 \mid x \in X, y \in Y\}
$$
as different "model sets" for the cartesian product $X \times Y$.
These sets $X \times_1 Y$ and $X \times_2 Y$ do not contain the same elements, so we should regard them as distinct.
In practice the existence of these different models is irrelevant for most ordinary mathematics because:

*

*Most mathematicians don't actually care about reducing their arguments to formal proofs in the specific system ZFC, so we might as well just imagine (and do in practice) that the notion of an ordered pair is primitive and behaves in the ways we expect.

*Most mathematicians are already accustomed to treating two sets that are isomorphic in an obvious way (like $X \times_1 Y$ and $X \times_2 Y$) as "the same", even if they are not literally made up of the same elements.


Now when it comes to homotopy theory and other higher structures, the corresponding views are not so mainstream.
Spaces are usually not taken as a primitive notion,
but instead something we can present using many different kinds of "models", for example using topological spaces, or simplicial sets, etc.
We would like to think of two equivalent spaces as "the same",
but if we can only access spaces through, say, topological spaces,
we certainly don't think of two topological spaces as "the same" when they have the same weak homotopy type.
Moreover the different "model categories" of topological spaces, simplicial sets, etc. are certainly not "the same" category, or even equivalent categories, either.
So in short, the purpose of "models" is to define higher structures in terms of ordinary set-level ones.
(Homotopy type theory is a system that explicitly embraces these notions 1 and 2 in the higher setting: "space" is a primitive concept, and two equivalent spaces are "the same" (the univalence axiom).
But it is still open whether homotopy type theory or any similar system can encode all of mathematics (e.g., the theory of $\infty$-categories) with this perspective.)
