I heard that there is an $\infty$-category $\mathbf{Top}\_\infty$ whose objects are topological spaces, whose 1-morphisms are continuous maps, whose 2-morphisms are homotopies, whose 3-morphisms are homotopies between homotopies, and so on.


*Question 1:* What is a homotopy between homotopies? What is a homotopy between *such* homotopies? (I can't find a definition using Google.)


*Question 2:* Is there any definition of $\mathbf{Top}\_\infty$ in the literature if we model $\infty$-categories as *quasicategories*?


This $\infty$-category should have at least the property that its homotopy category $\mathrm{h}\mathbf{Top}\_\infty$ is ["the naive homotopy category"](https://en.wikipedia.org/wiki/Homotopy_category#The_naive_homotopy_category).


My motivation is the following: If we have defined $\mathbf{Top}\_\infty$, then we can consider the subcategory $\mathbf{Type}\_\infty\subseteq \mathbf{Top}\_\infty$ of all CW complexes. We need that category in order to formulate Grothendieck's homotopy hypothesis:



> 
> There is an equivalence of $\infty$-categories $\mathbf{Type}\_\infty \to \infty\mathbf{Grp}\_\infty$.
> 
> 
> 


Note that the $\infty$-category $\infty\mathbf{Grp}\_\infty$ of $\infty$-groupoids has already been defined in the literature. It is discussed in Chapter 3 of Lurie's *Higher Topos Theory* (consider the subcategory of the $\infty$-category of all $\infty$-categories consisting of Kan complexes).


*Question 3:* Can the homotopy hypothesis be proved in this setting?



Precise definition of the $\infty$-category of spaces, continuous maps, homotopies, homotopies between homotopies, and so on
Here's an answer for question 1. (This bothered me for a long time too, I also could never find a formal definition in the literature!)




---


If one uses 'nice' topological spaces (so that Top has an [internal hom](https://mathoverflow.net/questions/403714)), then one can define a **homotopy** from a map $f: X\to Y$ to another map $g:X\to Y$ to be a continuous map
$$H:[0,1]\to\mathbf{Map}(X,Y)$$
with $H(0)=f$ and $H(1)=g$.




---


Then, given two such homotopies $H\_1,H\_2:[0,1]\to\mathbf{Map}(X,Y)$, we can define a **homotopy of homotopies** or a **$2$-homotopy** $\eta: H\_1\to H\_2$ to be a continuous map
$$\eta:[0,1]\to\mathbf{Map}([0,1],\mathbf{Map}(X,Y))$$
with $\eta(0)=H\_1$ and $\eta(1)=H\_2$.




---


One can keep doing this, defining a notion of **$n$-homotopy** using that of an **$(n-1)$-homotopy**.




---


To get the usual definition (which works for arbitrary topological spaces!), one uses that the functors $(-)\times X$ and $\mathbf{Map}(X,-)$ are adjoint.




---


So a **homotopy**
$$H:[0,1]\to\mathbf{Map}(X,Y)$$
is the same as a continuous map
$$H^\dagger:X\times[0,1]\to Y,$$
where the endpoint conditions are now given by
\begin{align\*}
 H^\dagger(x,0) &= f(x),\\
 H^\dagger(x,1) &= g(x)
\end{align\*}
for all $x\in X$.




---


Similarly, a **homotopy between homotopies**
$$\eta:[0,1]\to\mathbf{Map}([0,1],\mathbf{Map}(X,Y))$$
is the same as a map
$$\eta^\dagger:[0,1]\times[0,1]\to\mathbf{Map}(X,Y)$$
such that
\begin{align\*}
 \eta^\dagger(0,t) &= H\_1(t),\\
 \eta^\dagger(1,t) &= H\_2(t)
\end{align\*}
for all $t\in[0,1]$, which is the same as a map
$$\eta^\ddagger:[0,1]\times[0,1]\times X\to Y$$
such that
\begin{align\*}
 \eta^\ddagger(0,t,x) &= H^\dagger\_1(t,x),\\
 \eta^\ddagger(1,t,x) &= H^\dagger\_2(t,x)
\end{align\*}
for all $t\in[0,1]$ and all $x\in X$.




---


One can keep going like this for $n$-homotopies, applying adjointness $0$ to $n-1$ times, leading to $n$ equivalent definitions of an $n$-homotopy!


