$\operatorname{Ho}(\mathbf{Top})$ has objects all topological spaces and morphisms are homotopy classes of maps between CW replacements:

$\operatorname{Hom}_{\operatorname{Ho}(\mathbf{Top})}(X, Y) = [QX, Y]$

where $QX$ is a CW approximation of $X$. For CW complexes $X$ and $Y$, this is simply $[X, Y]$, the set of homotopy classes of continuous maps.

Since all spaces are fibrant in $\mathbf{Top}$, we only need cofibrant replacement.

$\operatorname{Ho}(\mathbf{sSet})$ has morphisms computed as $[X, RY]$ where $RY$ is a fibrant replacement (Kan complex approximation) of $Y$. Since all simplicial sets are cofibrant, we only need fibrant replacement.

For Kan complexes $X$ and $Y$: $\operatorname{Hom}_{\operatorname{Ho}(\mathbf{sSet})}(X, Y) = \pi_0(\operatorname{Map}(X, Y))$, the connected components of the mapping space (simplicial function complex).

$\operatorname{Ho}(\mathbf{Ch}(R))$ is the derived category $D(R)$. Morphisms are computed via projective or injective resolutions:

$\operatorname{Hom}_{D(R)}(C, D) = [PC, D] = [C, ID]$

where $PC$ is a projective resolution and $ID$ is an injective resolution.

This recovers the classical derived category construction via chain homotopy classes of maps between resolutions.

For a pointed model category $\mathcal{M}_*$ (with a zero object), $\operatorname{Ho}(\mathcal{M}_*)$ is a pointed category with suspension and loop functors:

$\Sigma X = X \otimes S^1 = \operatorname{cofib}(X \to *), \quad \Omega Y = Y^{S^1} = \operatorname{fib}(* \to Y)$

These satisfy $[\Sigma X, Y] \cong [X, \Omega Y]$ (the loop-suspension adjunction in the homotopy category).

For a ring $R$, the tensor product $- \otimes_R M: \mathbf{Ch}(R) \to \mathbf{Ch}(R)$ does not preserve quasi-isomorphisms in general. Its left derived functor is:

$\mathbf{L}(- \otimes_R M)(C) = PC \otimes_R M$

where $PC$ is a projective resolution. The homology $H_n(PC \otimes_R M) = \operatorname{Tor}_n^R(C, M)$ recovers the classical Tor functors.

The internal Hom $\operatorname{Hom}_R(-, M)$ has right derived functor:

$\mathbf{R}\operatorname{Hom}_R(C, M) = \operatorname{Hom}_R(C, IM)$

where $IM$ is an injective resolution. Its cohomology $H^n(\operatorname{Hom}_R(C, IM)) = \operatorname{Ext}^n_R(C, M)$ recovers the classical Ext functors.

For a scheme $X$ and a sheaf $\mathcal{F}$, the global sections functor $\Gamma(X, -)$ has right derived functor $\mathbf{R}\Gamma(X, \mathcal{F})$, computed via injective resolution. The cohomology $H^n(X, \mathcal{F}) = R^n\Gamma(X, \mathcal{F})$ is sheaf cohomology.

The homotopy category $\operatorname{Ho}(\mathcal{M})$ loses crucial higher homotopical information:

1. Mapping spaces: $\operatorname{Ho}(\mathcal{M})$ only records $\pi_0$ of the mapping spaces, not the full mapping space. The composition $[X, Y] \times [Y, Z] \to [X, Z]$ forgets the homotopy-coherent composition.

2. Homotopy limits/colimits: The homotopy category does not have good limits and colimits. Homotopy pullbacks in $\operatorname{Ho}(\mathcal{M})$ do not satisfy the universal property of pullbacks.

3. Obstruction theory: Higher obstructions to lifting/extending maps require mapping space data, not just homotopy classes.

This is the fundamental motivation for $\infty$-categories: they capture the full mapping spaces, not just $\pi_0$.

The homotopy category $\operatorname{Ho}(\mathbf{Top}_*)$ of pointed spaces is not abelian and not even additive in general. However, for stable homotopy theory ($\operatorname{Ho}(\mathbf{Sp})$ for spectra), the homotopy category is triangulated. This triangulated structure is a remnant of the richer stable $\infty$-categorical structure.

Consider $X = S^2$ and $Y = S^2$ in $\mathbf{Top}$. The set $[S^2, S^2] = \mathbb{Z}$ (classified by degree). But the full mapping space $\operatorname{Map}(S^2, S^2)$ has infinitely many connected components (one for each degree), and the component of degree-$n$ maps has nontrivial homotopy groups related to the homotopy groups of $S^2$.

The homotopy category sees only $\pi_0 = \mathbb{Z}$, losing all this higher information. An $\infty$-category retains the full mapping space.

The localization functor $\gamma: \mathcal{M} \to \operatorname{Ho}(\mathcal{M})$ is the identity on objects and sends each morphism to its homotopy class (after fibrant-cofibrant replacement). It is universal among functors inverting weak equivalences:

For any functor $F: \mathcal{M} \to \mathcal{D}$ sending weak equivalences to isomorphisms, there exists a unique functor $\overline{F}: \operatorname{Ho}(\mathcal{M}) \to \mathcal{D}$ with $F = \overline{F} \circ \gamma$.

This universal property justifies the homotopy category as the "best ordinary category approximation" to the homotopy theory of $\mathcal{M}$.