I am not really familiar with homotopical category theory, so please forgive me if I make rude mistakes. I know quite a bit of common category theory, as well as familiar with algebraic topology.
How can one construct exact sequences abstractly in the framework of homotopical categories? Consider any model you like (topologically or simplicially enriched, or $(\infty,1)$-categories). nLab teaches us that to construct exact sequences in topology one needs to consider fiber sequences and homotopy limits/colimits. A fiber sequence $F\to E \to B$ is a homotopy pullback
\begin{array}{ccc} F & \rightarrow & E\\\
\downarrow & & \downarrow\\\
\ast & \rightarrow & B\\\
\end{array}
The loop space $\Omega X$ of $X$ is defined as a homotopy pullback
\begin{array}{ccc} \Omega X & \rightarrow & \ast \\\
\downarrow & & \downarrow\\\
\ast & \rightarrow & X\\\
\end{array}
Dually, one can define cofiber sequences and suspensions $\Sigma X$. We get a standard adjunction $\Sigma \vdash \Omega$. Given a fiber sequence, one can have a long fiber sequence $\dots\to\Omega E \to \Omega B \to F \to E \to B$. To begin with, consider a homotopy pullback rectangle
\begin{array}{ccc} \Omega B & \rightarrow & \ast \\\
\downarrow & & \downarrow\\\
F & \rightarrow & E\\\
\downarrow & & \downarrow\\\
\ast & \rightarrow & B\\\
\end{array}
Standard diagram chasing and (homotopic) universality arguments show that we get a fiber sequence $\Omega \to F \to E$ etc. Now if we apply the functor $\pi_0 \simeq \pi(\ast,-)$, we get a sequence $$\dots \to \pi_0(\Omega E) \to \pi_0(\Omega B) \to \pi_0(F) \to \dots$$
Using the fact that $\pi_0$ preserves fiber sequences (because it is representable) and $\pi_0(\Omega^i X) \simeq \pi_i (X)$ we get a classical long exact sequence of homotopy groups for a fibration.
Now things get trickier if we consider homotopy pushouts instead of pullbacks. If we have a cover $(U,V)$ of space $X$, then we have a homotopy pushout diagram
\begin{array}{ccc} U\cap V & \rightarrow & U \\\
\downarrow & & \downarrow\\\
V & \rightarrow & X\\\
\end{array}
Standard arguments allow to replace this diagram with cofiber sequence $U \vee V \to X \to \Sigma (U\cap V)$. In theory it should give Seifert-van Kampen theorem and Mayer-Vietoris sequences in homology and cohomology in a similar way. In practice, I don't understand how can this happen. To get Seifert-van Kampen theorem, one should apply $\pi_1\simeq \pi_0 \circ \Omega$, but $\Omega$ fails to preserve cofiber sequences. Dold-Thom theorem tells us that $H_i = \pi_i \circ S$, where $S$ is the topological abelianization functor. We're fine with $S$, because it's a left adjoint to forgetful functor, but we have the same problem with $\pi_i$. It doesn't preserve cofiber sequences. Even if it did, we would have terms like $\pi_i (S \Sigma X)=H_i (\Sigma X) = H_{i-1}(X)$ for $i>0$, and that is nothing like an ascending sequence of homology groups. Even cohomology isn't fine: we can take the space of maps to $K(\mathbb{Z},1)$ and it turns cofiber sequences into fiber sequences, but we still have a problem with grading, like in case of homology.
So my question is: how can we deal with these difficulties and get the mentioned theorems? I didn't have any success till now in finding references on homotopy limits/colimits technique, although many sources mention them and deal with technical properties, like their existence in model categories. Any references would be greatly appreciated!
Best Answer
I don't think the Seifert-van Kampen theorem follows from these kinds of considerations. Rather, it is the statement that the fundamental groupoid functor $\tau_{\leq 1}$ preserves homotopy colimits. That's because it is left adjoint to the inclusion of 1-groupoids in $\infty$-groupoids (a generalization to any $\infty$-category is in Higher Topos Theory, Prop. 5.5.6.18).
[Removed the part on cohomology because it got me confused!]
Added: I attempted to explain the long exact sequences in cohomology without using spectra, but that was wrong. Here's a correct explanation. The reduced cohomology of a pointed space $X$ depends only on its stabilization $\Sigma^\infty X$: it is given by $H^n(X;A)=[\Sigma^\infty X,\Sigma^n HA]$ where $HA$ is an infinite delooping of $K(A,0)$. The functor $\Sigma^\infty$ preserves cofiber sequences (being left adjoint). Now if you have a cofiber sequence in a stable category, you get long exact sequences of abelian groups when you apply functors like $[E,-]$ or $[-,E]$.
Added later: My original answer was correct, but like I said I got confused… Here it is again. Let $A\to B\to C$ be a cofiber sequence of pointed spaces. As you say in your question, you get a fiber sequence of mapping spaces
$Map(C,X)\to Map(B,X)\to Map(A,X)$
for any $X$, because $Map$ transforms homotopy colimits in its first variable into homotopy limits. In its second variable it preserves homotopy limits, so $\Omega Map(A,X)=Map(A,\Omega X)$. Applying to $X=K(G,n)$ gives you the usual long exact sequence in cohomology, but only from $H^0$ to $H^n$.