In a model category $\mathcal{M}$, the homotopy limit $\operatorname{holim}_I F$ of a diagram $F: I \to \mathcal{M}$ is computed by:

1. Replace $F$ by a fibrant diagram (a pointwise fibrant replacement).
2. Apply a derived limit construction (e.g., the Bousfield--Kan formula).

The Bousfield--Kan formula gives:

$\operatorname{holim}_I F = \operatorname{eq}\left(\prod_{i \in I} F(i)^{N(I/i)} \rightrightarrows \prod_{(i \to j) \in I} F(j)^{N(I/i)}\right)$

In the $\infty$-categorical setting, homotopy limits ARE limits: there is no need for fibrant replacement or derived constructions because the quasi-category already incorporates the homotopy theory.

The homotopy pullback of $X \xrightarrow{f} Z \xleftarrow{g} Y$ is the $\infty$-categorical limit of the diagram $\Lambda^2_1 \to \mathcal{C}$. Concretely:

$X \times^h_Z Y \simeq X \times_Z Z^{\Delta[1]} \times_Z Y$

This is the space of triples $(x, \gamma, y)$ where $x \in X$, $y \in Y$, and $\gamma$ is a path from $f(x)$ to $g(y)$ in $Z$.

For the diagram $* \to BG \leftarrow *$ (in spaces), the homotopy pullback is $G$ (the group as a space). The ordinary pullback would be just a point.

Homotopy products (limits over discrete indexing categories) agree with ordinary products when the factors are fibrant. For spaces: $\prod^h X_i = \prod X_i$ (ordinary product with the product topology).

For chain complexes: the homotopy product is the ordinary product of chain complexes (products of chain complexes are exact, so no derived correction is needed).

The homotopy pushout of $Y \xleftarrow{f} X \xrightarrow{g} Z$ is:

$Y \sqcup^h_X Z \simeq Y \sqcup_X (X \times \Delta[1]) \sqcup_X Z$

This is the double mapping cylinder: glue $Y$ and $Z$ together via a cylinder on $X$.

For the diagram $* \leftarrow S^{n-1} \to D^n$ (in spaces), the homotopy pushout is $S^n$ (the $n$-sphere). The ordinary pushout $D^n / S^{n-1}$ gives the same answer since the maps are cofibrations.

Homotopy coproducts over discrete indexing categories agree with ordinary coproducts: $\coprod^h X_i = \coprod X_i$. For simplicial sets, disjoint union is already homotopically correct.

A sequential homotopy colimit $\operatorname{hocolim}(X_0 \to X_1 \to X_2 \to \cdots)$ is the telescope construction:

$\operatorname{Tel}(X_\bullet) = \operatorname{hocolim}_n X_n \simeq \operatorname{colim}\left(X_0 \xrightarrow{f_0} X_1 \xrightarrow{f_1} X_2 \to \cdots\right)$

when the maps are cofibrations. In general, one replaces the maps by cofibrations first (using the mapping cylinder).

Example: the sequential colimit $S^0 \hookrightarrow S^1 \hookrightarrow S^2 \hookrightarrow \cdots$ (equatorial inclusions) has homotopy colimit $S^\infty \simeq *$ (contractible).

The $\infty$-category $\mathcal{S}$ of spaces has all limits and colimits:

• Products are Cartesian products.
• Pullbacks are homotopy pullbacks (fiber products with path data).
• Coproducts are disjoint unions.
• Pushouts are homotopy pushouts (double mapping cylinders).
• Totalizations (limits over $\Delta$) compute spaces of sections.

The homotopy groups of a limit diagram are related by the Milnor exact sequence and the Bousfield--Kan spectral sequence.

In the derived $\infty$-category $D(R)$:

• Products are products of chain complexes.
• Pullbacks are derived fiber products (mapping cones shifted).
• Coproducts are direct sums.
• Pushouts involve mapping cones: $\operatorname{cofib}(A \to B) = B/A$ (the cofiber).

The distinguished triangles of the triangulated homotopy category are shadows of pushout/pullback squares in $D(R)$.

The $\infty$-category $\operatorname{Cat}_\infty$ has all limits and colimits:

• Products $\mathcal{C} \times \mathcal{D}$ are Cartesian products of quasi-categories.
• Pullbacks are homotopy pullbacks.
• Functor categories $\operatorname{Fun}(\mathcal{C}, \mathcal{D})$ are exponential objects.
• Localizations $\mathcal{C}[W^{-1}]$ are colimits in $\operatorname{Cat}_\infty$.

In a quasi-category $\mathcal{C}$, the limit $\lim F$ of a diagram $F: K \to \mathcal{C}$ satisfies:

$\operatorname{Map}_{\mathcal{C}}(c, \lim F) \simeq \operatorname{Map}_{\operatorname{Fun}(K, \mathcal{C})}(\underline{c}, F)$

where $\underline{c}: K \to \mathcal{C}$ is the constant diagram at $c$. This is the mapping space version of the classical universal property.

Dually, $\operatorname{Map}_{\mathcal{C}}(\operatorname{colim} F, c) \simeq \operatorname{Map}_{\operatorname{Fun}(K, \mathcal{C})}(F, \underline{c})$.

These are equivalences of spaces (Kan complexes), not just bijections of sets. This is the key difference from ordinary category theory.