A continuous map $p: E \to B$ is a Serre fibration if it has the homotopy lifting property for all CW complexes. Equivalently, it has the RLP with respect to $D^n \times \{0\} \hookrightarrow D^n \times [0,1]$ for all $n \geq 0$.

Examples:

• Fiber bundles are Serre fibrations.
• Covering maps are Serre fibrations.
• The projection $E \times F \to E$ is a (trivial) Serre fibration.
• The path fibration $PX \to X$ (evaluating a path at its endpoint) is a Serre fibration with fiber $\Omega X$.

A map $p: X \to Y$ of simplicial sets is a Kan fibration if it has the RLP with respect to all horn inclusions $\Lambda^n_k \hookrightarrow \Delta[n]$ for $n \geq 1$, $0 \leq k \leq n$.

When $Y = \Delta[0]$, a Kan fibration $X \to \Delta[0]$ means $X$ is a Kan complex. In general, Kan fibrations are "families of Kan complexes parametrized by $Y$" (the fibers are Kan complexes).

The geometric realization of a Kan fibration (with countable base) is a Serre fibration.

In $\mathbf{Ch}_{\geq 0}(R)$ with the projective model structure, a chain map $f: C \to D$ is a fibration if $f_n: C_n \to D_n$ is surjective for all $n \geq 1$.

The fibration condition does not require surjectivity in degree $0$. A trivial fibration is a surjective quasi-isomorphism (surjective in all degrees, including $0$, and inducing isomorphisms on homology).

In the Quillen model structure on $\mathbf{Top}$, cofibrations are retracts of relative CW inclusions: maps $i: A \to X$ where $X$ is obtained from $A$ by attaching cells.

Cofibrant objects are retracts of CW complexes. Since every space is fibrant, the bifibrant objects are exactly the CW complexes (up to retract).

The cofibration $S^{n-1} \hookrightarrow D^n$ (including the boundary sphere into the disk) is a generating cofibration: all cofibrations are built from these by pushout, transfinite composition, and retract.

In the Kan--Quillen model structure on $\mathbf{sSet}$, cofibrations are monomorphisms (levelwise injective maps). Since every simplicial set is a colimit of $\Delta[n]$'s and the boundary inclusions $\partial\Delta[n] \hookrightarrow \Delta[n]$ generate all monomorphisms, these are the generating cofibrations.

Every simplicial set is cofibrant (the map $\emptyset \hookrightarrow X$ is always a monomorphism). This is one of the main technical advantages of working in $\mathbf{sSet}$.

In $\mathbf{Ch}_{\geq 0}(R)$ with the projective model structure, cofibrations are monomorphisms with levelwise projective cokernel. The generating cofibrations are the maps $0 \to D^n$ (the disk chain complex: $R$ in degrees $n$ and $n-1$, with identity differential) and $S^{n-1} \hookrightarrow D^n$ (the sphere to disk inclusion).

Cofibrant chain complexes are complexes of projective modules.

The Kan--Quillen model structure on $\mathbf{sSet}$ is cofibrantly generated:

• $I = \{\partial\Delta[n] \hookrightarrow \Delta[n] : n \geq 0\}$ (generating cofibrations).
• $J = \{\Lambda^n_k \hookrightarrow \Delta[n] : n \geq 1, 0 \leq k \leq n\}$ (generating trivial cofibrations).

The fibrations are maps with RLP against $J$ (Kan fibrations), and the trivial fibrations are maps with RLP against $I$ (maps surjective on all simplices).

The Quillen model structure on $\mathbf{Top}$ is cofibrantly generated:

• $I = \{S^{n-1} \hookrightarrow D^n : n \geq 0\}$ (boundary sphere inclusions).
• $J = \{D^n \times \{0\} \hookrightarrow D^n \times [0,1] : n \geq 0\}$ (cylinder inclusions).

Cofibrations in any model category are closed under:

• Composition: if $i: A \to B$ and $j: B \to C$ are cofibrations, so is $j \circ i$.
• Pushout: if $i: A \to B$ is a cofibration and $f: A \to C$ is any map, then $C \to B \sqcup_A C$ is a cofibration.
• Transfinite composition: a transfinite composition of cofibrations is a cofibration.
• Retract: a retract of a cofibration is a cofibration.
• Coproduct: a coproduct of cofibrations is a cofibration.

The same closure properties hold for fibrations (with pullback replacing pushout, and limits replacing colimits).

In $\mathbf{sSet}$, the pushout of $\partial\Delta[n] \hookrightarrow \Delta[n]$ along a map $\partial\Delta[n] \to X$ produces $X$ with an $n$-cell attached:

$X \hookrightarrow X \sqcup_{\partial\Delta[n]} \Delta[n]$

This is a cofibration (since cofibrations are closed under pushout). Cell attachment is the basic building block for constructing CW-type objects in any cofibrantly generated model category.

If $p: E \to B$ is a Kan fibration and $f: B' \to B$ is any map, the pullback $f^*(E) = E \times_B B' \to B'$ is again a Kan fibration. The fiber over a point $b' \in B'$ is the same as the fiber of $p$ over $f(b')$.

This is the simplicial analogue of pulling back a fiber bundle along a map.