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7.6 Examples of Limits and Colimits

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. In §7.1, we introduced the notion of limit and colimit for an arbitrary morphism of simplicial sets $\sigma : K \rightarrow \operatorname{\mathcal{C}}$. Our goal in this section is to make the general theory more explicit for some special classes of diagrams which arise frequently in practice.

We begin in §7.6.1 by considering the case where $K$ is a discrete simplicial set. In this case, specifying a functor $\sigma : K \rightarrow \operatorname{\mathcal{C}}$ is equivalent to specifying a collection of objects $\{ Y_ k \in \operatorname{\mathcal{C}}\} _{k \in K}$, indexed by the collection of vertices of $K$. We say that an object of $\operatorname{\mathcal{C}}$ is a product of the collection $\{ Y_ k \} _{k \in K}$ if it is a limit of the diagram $\sigma $, and a coproduct of the collection $\{ Y_ k \} _{k \in K}$ if it is a colimit of the diagram $\sigma $. These conditions can be formulated purely in terms of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, provided that we regard $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as enriched over the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$ (see Remark 7.6.1.5). In particular, the forgetful functor from $\operatorname{\mathcal{C}}$ to (the nerve of) its homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ preserves products and coproducts (Warning 7.6.1.2).

In §7.6.2, we allow $K$ to be an arbitrary simplicial set, but require $\sigma : K \rightarrow \operatorname{\mathcal{C}}$ to be a constant diagram taking some value $Y \in \operatorname{\mathcal{C}}$. In this case, we will denote a limit of $\sigma $ (if it exists) by $Y^{K}$ and a colimit of $\sigma $ (if it exists) by $K \otimes Y$ (Notation 7.6.2.5). We refer to $Y^{K}$ as a power of $Y$ by $K$, and $K \otimes Y$ as a tensor product of $Y$ by $K$. These notions can again be formulated purely at the level of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, regarded as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched category (Definition 7.6.2.1 and Remark 7.6.2.6).

In §7.6.3, we study limit and colimit diagrams indexed by the simplicial set $K = \Delta ^1 \times \Delta ^1$. Let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories, which we depict as a diagram

\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d]^-{f_0} \\ X_1 \ar [r]^-{f_1} & X. } \]

We say that $\sigma $ is a pullback square if it is a limit diagram, and a pushout square if it is a colimit diagram (Definition 7.6.3.1). Beware that these conditions cannot be formulated at the level of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, even if its $\mathrm{h} \mathit{\operatorname{Kan}}$-enrichment is accounted for: see Warning 7.6.3.3 Example 7.6.3.4.

It follows from Proposition 7.5.4.13 that a (strictly commutative) diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_1 \ar [r] & X } \]

determines a pullback square in the $\infty $-category $\operatorname{\mathcal{S}}$ if and only if it is a homotopy pullback square. However, not every pullback square in the $\infty $-category $\operatorname{\mathcal{S}}$ arises in this way. In §7.6.4, we give a detailed classification of all pullback squares in the $\infty $-category $\operatorname{\mathcal{S}}$ (Corollary 7.6.4.10). In particular, for every pair of morphisms of Kan complexes $f_0: X_0 \rightarrow X$ and $f_1: X_1 \rightarrow X$, we construct a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ X_0 \times ^{\mathrm{h}}_{ X} X_1 \ar [r] \ar [d] & X_0 \ar [d]^-{f_0} \\ X_1 \ar [r]^-{f_1} & X } \]

in the $\infty $-category $\operatorname{\mathcal{S}}$ (Example 7.6.4.11); beware that this diagram usually does not commute in the ordinary category of simplicial sets. Our analysis can be applied more generally to any $\infty $-category which arises as the homotopy coherent nerve of a locally Kan simplicial category (Corollary 7.6.4.12); in particular, it can be applied to the $\infty $-category $\operatorname{\mathcal{C}}= \operatorname{\mathcal{QC}}$ of small $\infty $-categories (see Proposition 7.6.4.8 and Corollary 7.6.4.9).

Let $(\bullet \rightrightarrows \bullet )$ denote the simplicial set given by the coproduct $\Delta ^1 \coprod _{ \operatorname{\partial \Delta }^1} \Delta ^1$ (Notation 7.6.5.1). In §7.6.5, we study limits and colimits of diagrams indexed by $( \bullet \rightrightarrows \bullet )$. For any $\infty $-category $\operatorname{\mathcal{C}}$, functors $\sigma : ( \bullet \rightrightarrows \bullet ) \rightarrow \operatorname{\mathcal{C}}$ can be identified with pairs $f_0, f_1: Y \rightarrow X$ of morphisms in $\operatorname{\mathcal{C}}$ having the same source and target. In this case, we denote a limit of $\sigma $ (if it exists) by $\operatorname{Eq}(f_0, f_1)$, and a colimit of $\sigma $ (if it exists) by $\operatorname{Coeq}(f_0, f_1)$ (Notation 7.6.5.5). We refer to $\operatorname{Eq}(f_0, f_1)$ as an equalizer of the pair $(f_0, f_1)$, and to $\operatorname{Coeq}( f_0, f_1 )$ as a coequalizer of $(f_0, f_1)$ (Definition 7.6.5.4). Beware that, as with pullbacks and pushouts, the notions of equalizer and coequalizer cannot be formulated purely in terms of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$; in particular, the forgetful functor from $\operatorname{\mathcal{C}}$ to (the nerve of) its homotopy category need not preserve equalizers and coequalizers.

Let $\operatorname{\mathbf{Z}}_{\geq 0}$ denote the set of nonnegative integers, endowed with its usual linear ordering. In §7.6.6, we study colimits of diagram $X: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{\mathcal{C}}$, which we represent informally as

\[ X(0) \xrightarrow {f_0} X(1) \xrightarrow {f_1} X(2) \xrightarrow {f_2} X(3) \xrightarrow {f_3} X(4) \rightarrow \cdots \]

In the special case where $\operatorname{\mathcal{C}}= \operatorname{\mathcal{S}}$ is the $\infty $-category of spaces, we show that the colimit $\varinjlim X(n)$ (formed in the ordinary category of simplicial sets, using the transition morphisms $f_ i$) is also a colimit in the $\infty $-category $\operatorname{\mathcal{S}}$ (Variant 7.6.6.9). Similarly, if $Y: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{S}}$ is a diagram which we depict informally as

\[ \cdots \rightarrow Y(4) \xrightarrow {g_3} Y(3) \xrightarrow {g_2} Y(2) \xrightarrow { g_1} Y(1) \xrightarrow {g_0} Y(0), \]

then the usual inverse limit $\varprojlim Y(n)$ (formed in the category of simplicial sets, using the transition morphisms $g_ n$) is also a limit in the $\infty $-category $\operatorname{\mathcal{S}}$, provided that each of the maps $g_ n$ is a Kan fibration (Variant 7.6.6.11). These assertions have counterparts for sequential limits and colimits in the $\infty $-category $\operatorname{\mathcal{QC}}$: see Examples 7.6.6.8 and 7.6.6.10.

Though the classes of diagrams we study in this section are of a very restricted type, they are nonetheless useful for analyzing limits and colimits in general. If $K$ is a complicated simplicial set which can be decomposed into simpler constituents, then we can often use Proposition 7.5.8.12 to reduce questions about $K$-indexed (co)limits to questions about (co)limits indexed by those constituents. We will consider several variants on this theme:

  • If a simplicial set $K$ decomposes as a disjoint union $\coprod _{j \in J} K_ j$, then we can often rewrite $K$-indexed limits as products; see Proposition 7.6.1.17.

  • If a simplicial set $K$ fits into a categorical pushout diagram

    \[ \xymatrix@R =50pt@C=50pt{ K_{01} \ar [r] \ar [d] & K_0 \ar [d] \\ K_1 \ar [r] & K, } \]

    then we can often rewrite $K$-indexed limits as pullbacks; see Proposition 7.6.3.17.

  • If $\operatorname{\mathcal{C}}$ is an $\infty $-category which admits finite products, then an equalizer of a pair of morphisms $f_0, f_1: Y \rightarrow X$ (if it exists) is characterized by the existence of a pullback diagram

    \[ \xymatrix@R =50pt@C=50pt{ \operatorname{Eq}(f_0, f_1) \ar [r] \ar [d] & Y \ar [d]^-{ (f_0, f_1) } \ar [d] \\ X \ar [r]^-{ \delta _ X } & X \times X; } \]

    see Proposition 7.6.5.22.

  • If $\operatorname{\mathcal{C}}$ is an $\infty $-category which admits finite products, then a pullback of a diagram $X_0 \xrightarrow {f_0} X \xleftarrow {f_1} X_1$ can be rewritten as the equalizer of a diagram $X_0 \times X_1 \rightrightarrows X$ (Proposition 7.6.5.23).

  • If $\operatorname{\mathcal{C}}$ is an $\infty $-category which admits countable products, then the limit of a tower

    \[ \cdots \rightarrow X(3) \xrightarrow {f_2} X(2) \xrightarrow {f_1} X(1) \xrightarrow {f_0} X(0) \]

    can be rewritten as an equalizer $\operatorname{Eq}( f, \operatorname{id}_ X )$, where $X$ is the product $\prod _{n \geq 0} X(n)$ and $f: X \rightarrow X$ is the endomorphism of $X$ determined by the sequence $\{ f_ n \} _{n \geq 0}$; see Proposition 7.6.6.16.

  • If $K$ is a simplicial set which can be written as the colimit of a sequence

    \[ K(0) \rightarrow K(1) \rightarrow K(2) \rightarrow K(3) \rightarrow \cdots , \]

    then we can often rewrite $K$-indexed limits as sequential limits (Corollary 7.6.6.14).

By applying these observations iteratively, one can build arbitrarily complicated limits (and colimits) out of the constructions studied in this section. For example, we show that an $\infty $-category $\operatorname{\mathcal{C}}$ admits finite limits if and only if it admits pullbacks and has a final object (Corollary 7.6.3.18).

Structure

  • Subsection 7.6.1: Products and Coproducts
  • Subsection 7.6.2: Powers and Tensors
  • Subsection 7.6.3: Pullbacks and Pushouts
  • Subsection 7.6.4: Examples of Pullback and Pushout Squares
  • Subsection 7.6.5: Equalizers and Coequalizers
  • Subsection 7.6.6: Sequential Limits and Colimits
  • Subsection 7.6.7: Small Limits