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5.4 Size Conditions on $\infty $-Categories

Recall that a small category $\operatorname{\mathcal{C}}$ consists of the following data:

  • A set $\operatorname{Ob}(\operatorname{\mathcal{C}})$, whose elements are referred to as objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, a set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$, whose elements are referred to as morphisms from $X$ to $Y$.

  • For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, a composition law

    \[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]

    which is required to be unital and associative.

This definition treats categories as algebraic objects akin to groups (though somewhat more general), which is perfectly adequate for many purposes. However, it is often useful to apply the theory to categories which are not small, such as the category of sets $\operatorname{\mathcal{C}}= \operatorname{Set}$. In this case, $\operatorname{Ob}(\operatorname{\mathcal{C}})$ is the collection of all sets, and must be treated with a bit of care to avoid paradoxes.

Example 5.4.0.1. When speaking informally, it is common to say that the category $\operatorname{Set}$ has all limits and colimits. A more precise statement is that the category $\operatorname{Set}$ has all small limits and colimits; that is, every diagram $F: \operatorname{\mathcal{J}}\rightarrow \operatorname{Set}$ indexed by a small category $\operatorname{\mathcal{J}}$ has a limit and colimit. Here the size restriction on $\operatorname{\mathcal{J}}$ cannot be omitted. For example, if $\{ S_ j \} _{j \in J}$ is a collection of sets indexed by another set $J$, then it is permissible to form the coproduct $\coprod _{j \in J} S_ j$. However, it is not permissible to form the coproduct $\coprod _{S \in \operatorname{Ob}(\operatorname{Set})} S$ of all sets.

In the setting of higher category theory, one encounters similar issues. In §1.3, we defined an $\infty $-category to be a simplicial set $\operatorname{\mathcal{C}}_{\bullet }$ which satisfies a filling condition for inner horns (Definition 1.3.0.1). By analogy with the discussion above, we might be better to refer to such objects as small $\infty $-categories. However, we will often want to apply the ideas developed in this book to $\infty $-categories $\operatorname{\mathcal{C}}_{\bullet }$ which are not small, because the collections $n$-simplices $\operatorname{\mathcal{C}}_{n}$ are “too big” to be sets (this situation arises, for example, if $\operatorname{\mathcal{C}}_{\bullet }$ is the nerve of a large category). For the most part, we will ignore the set-theoretic issues which are raised by allowing such objects into our discourse. However, this is not always possible: as Example 5.4.0.1 illustrates, it is sometimes important to track the distinction between “large” and “small.”

The first goal of this section is to introduce some language for quantifying the sizes of category-theoretic objects. Let $\kappa $ be an infinite cardinal. We will say that a set is $\kappa $-small if its cardinality is strictly smaller than $\kappa $ (Definition 5.4.3.1). We will say that a simplicial set $S$ is $\kappa $-small if the collection of nondegenerate simplices of $S$ is $\kappa $-small (Definition 5.4.4.1). We summarize the basic properties of $\kappa $-small sets and simplicial sets in §5.4.3 and §5.4.4, respectively. Beware that $\kappa $-smallness is not a homotopy invariant condition: that is, it is possible for a $\kappa $-small $\infty $-category to be equivalent to an $\infty $-category which is not $\kappa $-small. In §5.4.5, we address this point by introducing the notion of essential smallness. If $\kappa $ is an uncountable cardinal, we say that an $\infty $-category $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small if it is equivalent to a $\kappa $-small $\infty $-category (Definition 5.4.5.1). One can formulate this condition also in the case $\kappa = \aleph _0$, but it is poorly behaved: it is very rare for finite simplicial sets to be $\infty $-categories (see Warning 5.4.5.6).

The second goal of this section is to provide a concrete criterion which can be used to test if an $\infty $-category is essentially $\kappa $-small. For simplicity, let us assume that $\kappa $ is an (uncountable) regular cardinal. We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is locally $\kappa $-small if, for every pair of objects $C,D \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ is essentially $\kappa $-small (Definition 5.4.8.1). In §5.4.8, we show that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small if and only if it locally $\kappa $-small and the set of isomorphism classes $\pi _0( \operatorname{\mathcal{C}}^{\simeq } )$ is $\kappa $-small (Proposition 5.4.8.8). We are therefore reduced to the problem of testing essential $\kappa $-smallness of Kan complexes. In §5.4.7, we address this problem by showing that a Kan complex $X$ is essentially $\kappa $-small if and only if the set $\pi _0(X)$ is $\kappa $-small and the homotopy groups $\{ \pi _{n}(X,x) \} _{n > 0}$ are $\kappa $-small for every vertex $x \in X$ (Proposition 5.4.7.1).

The proofs of Propositions 5.4.8.8 and 5.4.7.1 will use a common strategy. In both cases, the hard part is to show that if $\operatorname{\mathcal{C}}$ is an $\infty $-category for which certain homotopy-invariant quantities are bounded in size, then $\operatorname{\mathcal{C}}$ is equivalent to an $\infty $-category $\operatorname{\mathcal{C}}_0$ for which the collection of simplices is bounded in size. We will prove this using the theory of minimal models. We say that an $\infty $-category $\operatorname{\mathcal{C}}_0$ is minimal if the datum of a simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ is determined by its homotopy class relative to the boundary $\operatorname{\partial \Delta }^ n$ (see Definition 5.4.6.1). In §5.4.6, we will prove the following:

  • For every $\infty $-category $\operatorname{\mathcal{C}}$, there exists an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{C}}_0$ is minimal (Proposition 5.4.6.12). Moreover, $\operatorname{\mathcal{C}}_0$ is uniquely determined up to isomorphism (Corollary 5.4.6.11).

  • If $\operatorname{\mathcal{C}}_0$ is a minimal $\infty $-category, then every equivalence of $\infty $-categories $\operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ is a monomorphism of simplicial sets (Lemma 5.4.6.8). Consequently, $\operatorname{\mathcal{C}}_0$ is essentially $\kappa $-small if and only if it is $\kappa $-small (Corollary 5.4.6.9).

Remark 5.4.0.2. Throughout this section, we will need some elementary properties of cardinals and cardinal arithmetic. For the reader's convenience, we briefly review the set-theoretic prerequisites in §5.4.1 and §5.4.2.

Remark 5.4.0.3. The notion of minimal $\infty $-category was introduced by Joyal in [joyalnotes]. In the setting of Kan complexes, the theory of minimal models is much older (see [Barratt1959OnSF]).

Remark 5.4.0.4. Let $\kappa $ be an uncountable regular cardinal. We will see later that the essentially $\kappa $-small $\infty $-categories admit a more intrinsic characterization: they are precisely the $\kappa $-compact objects of the $\infty $-category $\operatorname{\mathcal{QC}}$ of $\infty $-categories (see Proposition ).

Remark 5.4.0.5. There are several means of handling the technical difficulties inherent in working with “large” mathematical objects (in either classical or higher category theory):

$(1)$

One can employ some set-theoretic device that enables one to distinguish between large and small. For example, one can assume the existence of a strongly inaccessible cardinal $\kappa $ (Definition 5.4.3.19), and define an $\infty $-category $\operatorname{\mathcal{C}}$ to be small (essentially small, locally small) if it is $\kappa $-small (essentially $\kappa $-small, locally $\kappa $-small).

$(2)$

One can work exclusively with small $\infty $-categories, and mirror the distinction between large and small by keeping careful track of relative sizes (using the language developed in this section).

$(3)$

One can simply ignore the set-theoretic difficulties inherent in considering large $\infty $-categories, provided that one avoids doing anything which is obviously illegal (see Example 5.4.0.1).

In practice, we will often switch between these approaches, favoring clarity of exposition over mathematical rigor.

Structure

  • Subsection 5.4.1: Ordinals and Well-Orderings
  • Subsection 5.4.2: Cardinals and Cardinality
  • Subsection 5.4.3: Smallness of Sets
  • Subsection 5.4.4: Small Simplicial Sets
  • Subsection 5.4.5: Essential Smallness
  • Subsection 5.4.6: Minimal $\infty $-Categories
  • Subsection 5.4.7: Small Kan Complexes
  • Subsection 5.4.8: Local Smallness