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In Tammo tom Dieck's 'Algebraic Topology' (corrected, 2nd printing from 2010) on page 24, it says

The passage from TOP to h-TOP may be interpreted as a passage from "continuous mathematics" to "discrete mathematics".

and I have discovered sentences like these a couple of times in a similar context. I'm not completely sure if I'm understanding this correctly, though. The core of this statement seems to be the more general notion, that we gain an Algebraic viewpoint during this passage.

But in what sense is an Algebraic viewpoint really 'discrete'? Is it the notion that we are looking at structures coming from finitary operations instead of structures together with which we are considering limits? Can somebody elaborate on this a little bit (preferably not by using too much higher category theory, I only know some basic concepts and not all of them...)?

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I'm also looking forward to any answers to this question, but no one answered so far. So I'm trying to provide some viewpoints.


The functor $\pi_0$

Let's look at a simpler example: the functor $\pi_0:\mathbf{Top}\to \mathbf{Set}$ defined in Chapter 2.1.

The functor $\pi_0$ sends a topological space $X$ to the set of its path components, and a continuous function to the induced set map. If we consider New Zealand as a topological space $X$, then $\pi_0(X)$ would become a two-element set $\{\textit{North Island, South Island}\}$ (ignoring other smaller islands).

After applying $\pi_0$, every pair of points in $X$ that is connected by a path are shrunk to a same element in $\pi_0(X)$:

Since all paths are glued onto two points (precisely: elements), the notion of path has vanished in the RHS above and discrete information of the original space (the number of path components) has shown.

In $\mathbf{Top}$ we first relate points/mappings/paths by a topological concept, then glue the related objects together, and finally get a new (often discrete) structure where the topological concept we initially introduced vanishes.

It is just like: in algebra we quotient a group $G$ along its commutator subgroup $C_G$, and get an Abelian group $G/C_G$, in which all commutators vanish (become identity).


The functor $\pi_1$

In advance, let the annulus in the following picture be a pointed topological space $X$. The fundamental group functor $\pi_1:\mathbf{Top^0}\to\mathbf{Grp}$ makes every class of paths that is related by homotopies in $X$ become a single element in $\pi_1(X)$:

The paths which are related by a topological concept (homotopy) are shrunk to be a single element in $\pi_1(X)$. Therefore, roughly speaking, in $\pi_1(X)$ no two distinct elements are related by a homotopy now (a topological concept vanishes).

Eliminating such topological concept in a space yields a discrete structure $\pi_1(X)\cong (\mathbb{Z}, +)$. Operations defined for classes of paths (eg. concatenation) may look like manipulating discrete elements in $\pi_1(X)$.


From $\mathbf{Top}$ to $\textbf{h-Top}$

Likewise, in the category $\mathbf{Top}$, there are continuous maps that can be associated by homotopies (in some sense, associated by structures of continuum):

In $\textbf{h-}\mathbf{Top}$, these maps are quotiented together:

Thus, (maybe more than) a continuum of morphisms are packed into one. We take a "bunch" of morphisms at once, and study interactions between "bunches". These bunches again considered to be discrete, since no pairs of "big" morphisms are homotopic anymore.

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    $\begingroup$ Great answer! :D I guess this combines very well with what I have written in my question about finitary operations, since you have given some very nice exposition on what sets these operations are defined on and how they relate to the continuous structures they come from :) $\endgroup$ – polynomial_donut Jun 18 '17 at 18:18

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