用户: Scavenger/Infinity Category and Higher Algebra/Accessible and Presentable Categories

Setting 0.1. Throughout these notes, we fix an inaccessible cardinal $\varkappa$ as our size parameter. The notions in these notes (for example accessible and presentable $\infty$-categories) will be the one relative to the size parameter $\varkappa$.

Definition 0.2. Let $\kappa$ be an infinite cardinal. We say that a simplicial set $X$ is $\kappa$-small if the cardinality of $X$’s nondegenerate simplexes is smaller than $\kappa$.

Caution 0.3. According to our convention, a $\varkappa$-simplicial set is a simplicial lying in the universe $V_{\varkappa }$, while a $\varkappa$-small simplicial set is a simplicial set that is isomorphic to a $\varkappa$-simplicial set.

Let $\mathcal{S}={\mathcal{S}}^{\varkappa }$ be the $\infty$-category of $\varkappa$-Kan complexes. (Depending on the context, sometimes $\mathcal{S}$ might denote the $\infty$-category of spaces having size parameter other than $\varkappa$.)

1Right Exact Functors

Definition 1.1. Let $\kappa$ be a regular cardinal. Let $\mathcal{C}$ be an $\infty$-category which admits $\kappa$-small colimits. Let $F:\mathcal{C}\to \mathcal{D}$ be a functor between $\infty$-categories, then $F$ is called $\kappa$-right exact if it preserves $\kappa$-small colimits.

This definition coincides with [1] Definition 5.3.2.1 by virtue of [1] Proposition 5.3.2.9.

2Filtered $\infty$-Categories

Definition 2.1. Let $\kappa$ be a regular cardinal and let $\mathcal{C}$ be an $\infty$-category. We say that $\mathcal{C}$ is $\kappa$-filtered if, for every $\kappa$-small simplicial set $K$ and every map $f:K\to \mathcal{C}$, there exists a map $\overline{f}:K^{▹}\to \mathcal{C}$ extending $f$.

We say that $\mathcal{C}$ is filtered if it is $\omega$-filtered.

Proposition 2.2 ([1] Proposition 5.3.3.3). Let $\kappa <\varkappa$ be a regular cardinal, and let $\mathcal{J}$ be a $\varkappa$-small $\infty$-category. Then $\mathcal{J}$ is $\kappa$-filtered if and only if the colimit functor $\mathrm{Fun}(\mathcal{J},\mathcal{S})\to \mathcal{S}$ preserves $\kappa$-small limits.

Corollary 2.3. Let $\kappa <\varkappa$ be a regular cardinal, $\mathcal{J}$ a $\varkappa$-small $\kappa$-filtered $\infty$-category, $K$ a $\kappa$-small simplicial set, $F:{\mathcal{J}}^{▹}\times K^{◃}\to \mathcal{S}$ a diagram such that for every $x\in \mathcal{J}$, $F{\mid }_{\{x\}\times K^{◃}}$ is a limit diagram, and for every $y\in K$, $F{\mid }_{{\mathcal{J}}^{▹}\times \{y\}}$ is a colimit diagram. Let $c_0\in {\mathcal{J}}^{▹}$, $c_1\in K^{◃}$ be the cone points, then $F{\mid }_{\{c_0\}\times K^{◃}}$ is a limit diagram if and only if $F{\mid }_{{\mathcal{J}}^{▹}\times \{c_1\}}$ is a colimit diagram.

3Compact Objects

Definition 3.1 ([1] Definition 5.3.4.5). Let $\kappa <\varkappa$ be a regular cardinal. Let $\mathcal{C}$ be an $\infty$-category which admits $\varkappa$-small $\kappa$-filtered colimits. Let $\mathcal{D}$ be an $\infty$-category, we say that a functor $f:\mathcal{C}\to \mathcal{D}$ is $\kappa$-continuous if it preserves $\varkappa$-small $\kappa$-filtered colimits.

Let $\mathcal{C}$ be as above, an object $C\in \mathcal{C}$ is said to be $\kappa$-compact if the functor $h^C:\mathcal{C}\to \mathcal{S}$ corepresented by $C$ is $\kappa$-continuous. We will use ${\mathcal{C}}^{\kappa }$ to denote the full subcategory of $\mathcal{C}$ spanned by the $\kappa$-compact objects.

Let $\mathcal{C}$ be an $\infty$-category which admits $\varkappa$-small filtered colimits. An object $C\in \mathcal{C}$ is said to be compact if it is $\omega$-compact.

Example 3.2. Let $\kappa <\varkappa$ be a regular cardinal. Let $\mathcal{C}$ be an $\infty$-category which admits $\varkappa$-small $\kappa$-filtered colimits. Let ${\mathcal{C}}_0\subseteq \mathcal{C}$ be a full subcategory which is stable under $\varkappa$-small $\kappa$-filtered colimits, and $C\in {\mathcal{C}}_0$ an object. Then $C$ is $\kappa$-compact in ${\mathcal{C}}_0$ if it is $\kappa$-compact in $\mathcal{C}$.

Proposition 3.3. Let $\kappa <\varkappa$ be a regular cardinal. Let $\mathcal{C}$ be an $\infty$-category which admits $\varkappa$-small colimits. Then the full subcategory ${\mathcal{C}}^{\kappa }\subseteq \mathcal{C}$ is stable under $\kappa$-small colimits.

Proof. Use Corollary 2.3.

$\square$

4$\infty$-Categories of Ind-Objects

Definition 4.1. Let $\kappa <\varkappa$ be a regular cardinal and let $\mathcal{C}$ be a $\varkappa$-small $\infty$-category. We let ${\mathrm{Ind}}_{\kappa }(\mathcal{C})$ denote the full subcategory of $P(\mathcal{C})=\mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathcal{S})$ spanned by those presheaves whose corresponding right fibration $\tilde{\mathcal{C}}\to \mathcal{C}$ satisfies that $\tilde{\mathcal{C}}$ is $\kappa$-filtered.

In the case $\kappa =\omega$, we write $\mathrm{Ind}(\mathcal{C})$ for ${\mathrm{Ind}}_{\kappa }(\mathcal{C})$.

Proposition 4.2 ([1] Proposition 5.3.5.3). Let $\kappa <\varkappa$ be a regular cardinal and let $\mathcal{C}$ be a $\varkappa$-small $\infty$-category. The full subcategory ${\mathrm{Ind}}_{\kappa }(\mathcal{C})\subseteq P(\mathcal{C})$ is stable under $\varkappa$-small $\kappa$-filtered colimits (and in paricular it admits $\varkappa$-small $\kappa$-filtered colimit).

Proposition 4.3 ([1] Corollary 5.3.5.4). Let $\mathcal{C}$ be a $\varkappa$-small $\infty$-category, let $\kappa <\varkappa$ be a regular cardinal, and let $F:{\mathcal{C}}^{\mathrm{op}}\to \mathcal{S}$ be an object of $\mathcal{P}(\mathcal{C})$. Then the following conditions are equivalent:

(i) There exists a $\varkappa$-small $\kappa$-filtered $\infty$-category $\mathcal{J}$ and a diagram $\mathcal{J}\to \mathcal{C}$ such that $F$ is a colimit of the composition $\mathcal{J}\to \mathcal{C}\to \mathcal{P}(\mathcal{C})$.

(ii) $F$ belongs to ${\mathrm{Ind}}_{\kappa }(\mathcal{C})$.

If $\mathcal{C}$ admits $\kappa$-small colimits, then (i) and (ii) are equivalent to

(iii) $F$ preserves $\kappa$-small limits.

Proposition 4.4 ([1] Proposition 5.3.5.10). Let $\kappa <\varkappa$ be a regular cardinal, let $\mathcal{C}$ be a $\varkappa$-small $\infty$-category, and let $\mathcal{D}$ be an $\infty$-category which admits $\varkappa$-small $\kappa$-filtered colimits. Then the Yoneda embedding $\mathcal{C}\to {\mathrm{Ind}}_{\kappa }(\mathcal{C})$ induces an equivalence ${\mathrm{Fun}}^{\prime }({\mathrm{Ind}}_{\kappa }(\mathcal{C}),\mathcal{D})\to \mathrm{Fun}(\mathcal{C},\mathcal{D})$, where ${\mathrm{Fun}}^{\prime }({\mathrm{Ind}}_{\kappa }(\mathcal{C}),\mathcal{D})$ denote the full subcategory of $\mathrm{Fun}({\mathrm{Ind}}_{\kappa }(\mathcal{C}),\mathcal{D})$ spanned by those $\kappa$-continuous functors.

Proposition 4.5 ([1] Proposition 5.3.5.11). Let $\kappa <\varkappa$ be a regular cardinal, let $\mathcal{C}$ be a $\varkappa$-small $\infty$-category, and let $\mathcal{D}$ be an $\infty$-category which admits $\varkappa$-small $\kappa$-filtered colimits. Let $F:{\mathrm{Ind}}_{\kappa }(\mathcal{C})\to \mathcal{D}$ be a $\kappa$-cocontinuous functor and let $f=F{\mid }_{\mathcal{C}}$ be the restriction of $F$ to $\mathcal{C}$ through the Yoneda embedding. Then

(i) If $f$ is fully faithful and its essential image consists of $\kappa$-compact objects of $\mathcal{D}$, then $F$ is fully faithful.

(ii) $F$ is an equivalence if and only if $f$ is fully faithful, the essential image of $f$ consists of $\kappa$-compact objects of $\mathcal{D}$, and the objects $\{f(C){\}}_{C\in \mathcal{C}}$ generate $\mathcal{D}$ under $\varkappa$-small $\kappa$-filtered colimits.

In particular, taking $F={\mathrm{id}}_{{\mathrm{Ind}}_{\kappa }(\mathcal{C})}$, we see that a representable presheaf is a $\kappa$-compact object in ${\mathrm{Ind}}_{\kappa }(\mathcal{C})$.

Proposition 4.6 ([1] Proposition 5.3.5.13). Let $\kappa <\varkappa$ be a regular cardinal, let $f:\mathcal{C}\to {\mathcal{C}}^{\prime }$ be a functor between $\varkappa$-small $\infty$-categories, such that $\mathcal{C}$ admits $\kappa$-small colimits. Then $f$ is $\kappa$-right exact if and only if ${\mathrm{Ind}}_{\kappa }(f)$ is a left adjoint, where ${\mathrm{Ind}}_{\kappa }(f):{\mathrm{Ind}}_{\kappa }(\mathcal{C})\to {\mathrm{Ind}}_{\kappa }({\mathcal{C}}^{\prime })$ is the functor induced by the universal property of ${\mathrm{Ind}}_{\kappa }$ (see Proposition 4.4). Moreover, in this situation a right adjoint of ${\mathrm{Ind}}_{\kappa }(f)$ is given by restriction along $f$ (and in particular the restriction functor $\mathcal{P}({\mathcal{C}}^{\prime })\to \mathcal{P}(\mathcal{C})$ brings ${\mathrm{Ind}}_{\kappa }({\mathcal{C}}^{\prime })$ into ${\mathrm{Ind}}_{\kappa }(\mathcal{C})$).

5Accessible $\infty$-Categories

Definition 5.1 ([1] Definition 5.4.2.1). Let $\mathcal{C}$ be an $\infty$-category and let $\kappa <\varkappa$ be a regular cardinal. $\mathcal{C}$ is called $\kappa$-accessible if there exists a $\varkappa$-small $\infty$-category ${\mathcal{C}}^0$ such that $\mathcal{C}$ is equivalent to ${\mathrm{Ind}}_{\kappa }({\mathcal{C}}^0)$. We say that $\mathcal{C}$ is accessible if it is $\kappa$-accessible for some regular cardinal smaller than $\varkappa$.

Example 5.2. Let $\kappa <\varkappa$ be a regular cardinal, let $\mathcal{C}$ be a $\varkappa$-small $\infty$-category. Then $\mathcal{P}(\mathcal{C})=\mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathcal{S})$ is $\kappa$-accessible by virtue of [1] Proposition 5.3.5.12.

Proposition 5.3 ([1] Proposition 5.4.2.2). Let $\mathcal{C}$ be an $\infty$-category and $\kappa <\varkappa$ be a regular cardinal. Then the following conditions are equivalent:

(i) $\mathcal{C}$ is $\kappa$-accessible.

(ii) $\mathcal{C}$ is locally small and admits $\varkappa$-small $\kappa$-filtered colimits, the full subcategory ${\mathcal{C}}^{\kappa }\subseteq \mathcal{C}$ is essentially small generates $\mathcal{C}$ under $\varkappa$-small $\kappa$-filtered colimits.

(iii) $\mathcal{C}$ admits $\varkappa$-small $\kappa$-filtered colimits and contains an essentially small full subcategory ${\mathcal{C}}^{\prime }\subseteq \mathcal{C}$ which consists of $\kappa$-compact objects and generates $\mathcal{C}$ under $\varkappa$-small $\kappa$-filtered colimits.

In this situation we have a natural equivalence ${\mathrm{Ind}}_{\kappa }({\mathcal{C}}^{\kappa })\to \mathcal{C}$, by virtue of Proposition 4.5.

Proposition 5.4 ([1] Proposition 5.4.4.3). Let $\mathcal{C}$ be an accessible $\infty$-category and let $K$ be a $\varkappa$-small simplicial set. Then $\mathrm{Fun}(K,\mathcal{C})$ is accessible.

Definition 5.5 ([1] Definition 5.4.2.5). Let $\mathcal{C}$ be an accessible $\infty$-category, and let ${\mathcal{C}}^{\prime }$ be an $\infty$-category. A functor $f:\mathcal{C}\to {\mathcal{C}}^{\prime }$ is called accessible if it is $\kappa$-continuous for some regular cardinal $\kappa <\varkappa$.

Proposition 5.6 ([1] Proposition 5.4.6.6). Let

be a categorical pullback diagram of infinity categories such that ${\mathcal{D}}^{\prime },\mathcal{D}$ and $\mathcal{C}$ are accessible, and $f$ and $g$ are accessible functors. Then ${\mathcal{C}}^{\prime }$ is accessible.

6Presentable $\infty$-Categories

Definition 6.1 ([1] Definition 5.5.0.1). An $\infty$-category $\mathcal{C}$ is presentable if it is accessible and admits $\varkappa$-small colimits.

Definition 6.2 ([1] Definition 5.5.7.1). Let $\kappa <\varkappa$ be a regular cardinal. We say that an $\infty$-category $\mathcal{C}$ is $\kappa$-compactly generated if it is presentable and $\kappa$-accessible. We say that it is compactly generated if it is $\omega$-compactly generated.

Example 6.3. $\mathcal{S}$ is presentable.

${\mathcal{S}}_{\ast }$ is presentable by virtue of [1] Proposition 5.5.3.11.

Proposition 6.4. Let $\kappa <\varkappa$ be a regular cardinal. Let $\mathcal{C}$ be a $\kappa$-compactly generated $\infty$-category. Let ${\mathcal{D}}_0\subseteq {\mathcal{C}}^{\kappa }$ be a full subcategory, let $\mathcal{D}\subseteq \mathcal{C}$ be the full subcategory generated by ${\mathcal{D}}_0$ under $\varkappa$-small colimits. Then $\mathcal{D}$ is $\kappa$-compactly generated, and the inclusion $\mathcal{D}\to \mathcal{C}$ preserves $\kappa$-compact objects.

Proof. We first prove that $\mathcal{D}$ is $\kappa$-compactly generated. We only need to prove $\mathcal{D}$ is $\kappa$-accessible since it admits $\varkappa$-small colimits. Let ${\mathcal{D}}_1=D\cap {\mathcal{C}}^{\kappa }$, then we have a natural functor $F:{\mathrm{Ind}}_{\kappa }({\mathcal{D}}_1)\to \mathcal{D}$ induced by the universal property of ${\mathrm{Ind}}_{\kappa }$. This functor is fully faithful by Proposition 4.5, and we only need to prove that it is essentially surjective. ${\mathcal{D}}_1$, when being regarded as a full subcategory of ${\mathcal{C}}^{\kappa }$, is stable under $\kappa$-small colimits, thus the composite functor ${\mathrm{Ind}}_{\kappa }({\mathcal{D}}_1)\to \mathcal{D}\to \mathcal{C}$ preserves $\varkappa$-small colimits by virtue of Proposition 4.6. But ${\mathrm{Ind}}_{\kappa }({\mathcal{D}}_1)$ is presentable by virtue of [1] Proposition 5.5.1.1, so the essential image of $F$, when being regarded as a full subcategory of $\mathcal{C}$, is stable under $\varkappa$-small colimits, so it is equal to $\mathcal{D}$ since it is contained in $\mathcal{D}$ and contains ${\mathcal{D}}_0$.

Now we prove that the inclusion $\mathcal{D}\to \mathcal{C}$ preserves $\kappa$-compact objects. In view of [1] Proposition 5.5.7.2, it will suffices to prove that the right adjoint to the inclusion $\mathcal{D}\to \mathcal{C}$ is $\kappa$-continuous. But since $\mathcal{D}\to \mathcal{C}$ is equivalent to the functor ${\mathrm{Ind}}_{\kappa }({\mathcal{D}}_1\to {\mathcal{C}}^{\kappa })$, so its right adjoint is given by restriction along ${\mathcal{D}}_1\to {\mathcal{C}}^{\kappa }$, by Proposition 4.6, so it is $\kappa$-continuous.

$\square$

Proposition 6.5. Let $\mathcal{C}$ be a presentable $\infty$-category and let $K$ be a $\varkappa$-small simplicial set. Then $\mathrm{Fun}(K,\mathcal{C})$ is presentable.

Proof. Immediate from Proposition 5.4 and [2] tag 02XA.

$\square$

Proposition 6.6 ([1] Corollary 5.5.2.4). A presentable $\infty$-category admits $\varkappa$-small limits.

References

[1]	Jacob Lurie, “Higher Topos Theory”, Available for download at https://www.math.ias.edu/ lurie/papers/HTT.pdf
[2]	Jacob Lurie, “Kerodon”, https://kerodon.net