Precise definition of the $\infty$-category of spaces, continuous maps, homotopies, homotopies between homotopies, and so on

Question

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".

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?

Answer 1

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!)

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If one uses 'nice' topological spaces (so that Top has an internal hom), 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 $H(1)=g$.

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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 $\eta(1)=H_2$.

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

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

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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 $x\in X$.

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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 $x\in X$.

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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!

Answer 2

Zhen Lin gives the recipe precisely for an answer to question 2.

The notion of homotopy between homotopies, etc. can be approached in several different ways, and I must admit is not obvious to do it simplicially. Papers on homotopy coherent nerve (e.g. J.-M. Cordier and T. Porter, Vogt’s Theorem on Categories of Homotopy Coherent Diagrams, Math. Proc. Camb. Phil. Soc. 100 (1986), 65–90.) may help. With Heiner Kamps, I wrote a textbook 'Abstract Homotopy and Simple Homotopy Theory, 1997, World Scientific.) in which we tried to discuss and where possible to answer a lot of related questions. The point is to build up the intuition as well as the technical details.

I wrote some notes (a long time ago) that might help (https://ncatlab.org/nlab/files/S-cat-notes.pdf) especially section 3.4. There is also another set of notes (with some overlap) here.

For the so called Grothendieck homotopy hypothesis, Grothendieck discussed this in his 'letter to Quillen' which is the first few pages of Pursuing Stacks (which is available, retyped and beautiful). It is very worth while going back to that source. The point about Kan complexes is in a letter from me to Grothendieck (16/6/1983), which should shortly be available (retyped). (See the Zulip thread on Grothendieck.) Of course, there has been an enormous amount of work on this since, and you probably know Joyal's and Lurie's theory of $\infty$-categories as quasi-categories, better than I do, but if you define $\infty$-groupoids to be Kan complexes, and use classical simplicial homotopy theory, you can put together a coherent picture of why you can define things not only to work, but to make a lot of geometric sense.

Edit: In answer to the comment of user997814, I am adding in something which is too big to put in a comment.

If $H, H':X\times I\to Y$ are homotopies between $f$ and $g$, the `obvious' thing to try for a homotopy from $H$ to $H'$ is a map $K: X\times I\times I\to Y$, so thinking of the homotopies just as maps. That gives one approach and leads to a cubical approach to homotopy theory and that is explored in the book with Kamps. That view has a square with H on the bottom H' on the top and constant homotopies down the sides, on f and g respectively.

Alternatively one can choose to define homotopies between homotopies with the sides possibly non-contant, and you will find that situation fits well with a simplicial viewpoint. This occurs when looking at what it might mean to have a homotopy coherent diagram in Top in the form of a 3-simplex. You would have spaces at the vertices, maps between them, for each face of the tetrahedron you would have homotopies between the composite of two sides and the remaining edge . (See diagram in example 1, on p. 14 of the S-cat notes). These homotopies in the 4 different faces compose and one wants a homotopy linking these composite homotopies and the sensible thing to do (due, I think, to Boardman and Vogt) is to use a square (see example 3 p.14 of those notes) and that does not have constant homotopies down the sides.

If you try to workout what a 3rd level homotopy is .... clearly one could try $X\times I^3\to Y$ and the same sort of thing works. This occurs if one tries to specify a homotopy coherent diagram (corresponding to a 4-simplex). This is discussed in various places including Kamps and Porter round about p. 312-315 and in Chapter 11 of the Crossed Menagerie, but can be worked out from the previous case (except for a twist in the story that is mentioned in the discussions).

Answer 3

In a nice setting where you have function spaces, if you have two points $x, y \in X$, a "homotopy from $x$ to $y$" is a path $I \to X$ whose endpoints are $x$ and $y$.

But you don't merely have a set of "homotopies from $x$ to $y$", you have a space of them, namely the subspace of $X^I$ consisting of those points that correspond to paths from $x$ to $y$. I will call this space $Map(x, y)$.

Well, $Map(x,y)$ is a space. So if you have two points $f, g \in Map(x,y)$... a "homotopy from $f$ to $g$ is a path $I \to Map(x,y)$ whose endpoints are $f$ and $g$. So that's what a "homotopy between $f$ and $g$ which are themselves homotopies from $x$ to $y$" is.

Now, $Map(f,g)$ is a subspace of $Map(x,y)^I$ which in turn is a subspace of $X^{I \times I}$, and you can interpret points of $Map(f,g)$ as maps $I \times I \to X$ as described in the other posts. But I find this finer description to be more illuminating: in short:

A "homotopy between homotopies" is meant to be thought of as a path in the "space of homotopies from $x$ to $y$".

Although be warned, sometimes people are actually interested in unbased homotopies between homotopies; that is, asking for paths in $X^I$ rather than paths in $Map(x,y)$. But people will usually use different language in that case.