Monoidal ∞-categories

• The Yoneda embedding
• Adjoint functors
• Monoidal ∞-categories
• Presentable ∞-categories

We say that $\mathcal{C} \hookrightarrow \mathcal{D}$ is a full subcategory if it is fully faithful, meaning all maps between Hom spaces is an equivalence. This would imply that $\operatorname{h}\mathcal{C} \to \operatorname{h}\mathcal{D}$ is fully faithful.

Definition 1. We say that a functor $\mathcal{C} \to \mathcal{D}$ is $1$-full if it fits into a Cartesian square $$ \begin{CD} \mathcal{C} @>>> \mathcal{D} \br @VVV @VVV \br \mathcal{C}^0 @>>> \operatorname{h}\mathcal{D} \end{CD} $$ where $\mathcal{C}^0 \hookrightarrow \operatorname{h}\mathcal{D}$ is a subcategory (of ordinary categories).

Then we can define the $1$-full subcategory $$ (\mathsf{Cart}_ {/D})_ \mathrm{str} \hookrightarrow \widehat{\mathsf{Cat}}_ {/\mathcal{D}}$$ where objects are Cartesian fibrations $\mathcal{C} \to \mathcal{D}$ and morphisms are precisely the $1$-full subspaces of functors sending Cartesian arrows to Cartesian arrows.

Theorem 2. There is an equivalence $$ (\mathsf{Cart}_ {/\mathcal{D}})_ \mathrm{str} \simeq \mathsf{Fun}(\mathcal{D}^\mathrm{op}, \widehat{\mathsf{Cat}}), $$ and similarly $$ (\mathsf{CoCart}_ {/\mathcal{D}})_ \mathrm{str} \simeq \mathsf{Fun}(\mathcal{D}, \widehat{\mathsf{Cat}}). $$

The Yoneda embedding#

Theorem 3. There exists a fully faithful embedding $$ \mathcal{C} \to \mathsf{Fun}(\mathcal{C}^\mathrm{op}, \mathsf{Spc}). $$

This isn’t that easy to construct. Morally we want to write down $$ C \mapsto \operatorname{Maps}_ \mathcal{C}(-, C), $$ but since composition is not strict, it takes some care to make sense of $\operatorname{Maps}(-, C)$ as a functor, and also talk about functoriality in $C$.

Instead, we note that this functor is the same as $$ \mathcal{C}^\mathrm{op} \times \mathcal{C} \to \mathsf{Spc}, $$ and so this will correspond to some Grothendieck fibration $$ \mathcal{D} \to \mathcal{C} \times \mathcal{C}^\mathrm{op}. $$ One way is to write down the twisted arrow category, using the quasi-category language. But this really uses the model of quasi-categories, so we will take a different approach.

There is a diagram $$ \begin{CD} \mathsf{Fun}([1], \mathcal{C}) @>>> \mathcal{C} \times \mathcal{C} \br @VVV @VVV \br \mathcal{C} @= \mathcal{C} \end{CD} $$ where both vertical arrows are Cartesian fibrations. This corresponds to a diagram $$ \begin{CD} \widehat{\mathsf{Cat}} @>>> \widehat{\mathsf{Cat}} \br @AAA @AAA \br \mathcal{C}^\mathrm{op} @= \mathcal{C}^\mathrm{op} \end{CD} $$ where the left map is $C \mapsto \mathcal{C}_ {C/}$ and the right map is $C \mapsto \mathcal{C}$. This gives an object of $$ \mathsf{Fun}(\mathcal{C}^\mathrm{op}, \widehat{\mathsf{Cat}}_ {/\mathcal{C}}). $$ But we note that this actually lands in $\mathsf{CoCart}_ {/\mathcal{C}}$. So we get a functor $$ \mathcal{C}^\mathrm{op} \to \mathsf{CoCart}_ {/\mathcal{C}} \simeq \mathsf{Fun}(\mathcal{C}, \widehat{\mathsf{Cat}}) $$ corresponding to $$ \mathcal{C}^\mathrm{op} \times \mathcal{C} \to \widehat{\mathsf{Cat}}. $$ Then we can check that this actually lands in ∞-groupoids.

Similarly, we can do the following.

Theorem 4. There exists a functor $$ \widehat{\mathsf{Cat}}^\mathrm{op} \times \widehat{\mathsf{Cat}} \to \widehat{\mathsf{Cat}} $$ given by $(C, D) \mapsto \mathsf{Fun}(C, D)$ at the level of objects.

Once we have this Yoneda embedding, we can now say the following. Let $F \colon I \to \mathcal{C}$ be a diagram. If $C = \varinjlim_I F$, then for all $D \in \mathcal{C}$ we have $$ \operatorname{Maps}(C, D) \simeq \varprojlim_I \operatorname{Maps}(F(C_i), D), $$ and if $C = \varprojlim_I F$, then for all $D \in \mathcal{C}$ we have $$ \operatorname{Maps}(D, C) \simeq \varprojlim_I \operatorname{Maps}(D, F(C_i)). $$

Adjoint functors#

Theorem 5. Let $\mathcal{C}$ and $\mathcal{D}$ be two ∞-categories, let $F \colon \mathcal{C} \to \mathcal{D}$ and $G \colon \mathcal{D} \to \mathcal{C}$ be functors. The following are equivalent:

1. there exists a functor $\mathcal{M} \to [1]$ that is both a Cartesian fibration and a coCartesian fibration (also called a bifibration), such that $\mathcal{M}_ {\lbrace 0 \rbrace} \cong \mathcal{C}$, $\mathcal{M}_ {\lbrace 1 \rbrace} \cong \mathcal{D}$, and the corresponding $[1] \to \widehat{\mathsf{Cat}}$ is homotopic to $F$ and the corresponding to $[1]^\mathrm{op} \to \widehat{\mathsf{Cat}}$ is homotopic to $G$,
2. there exist natural transformations $e \colon F \circ G \to \id_\mathcal{D}$ and $u \colon \id_\mathcal{C} \to G \circ F$ such that $$ F \to F \circ (G \circ F) \cong (F \circ G) \circ F \to F $$ and $$ G \to (G \circ F) \circ G \cong G \circ (F \circ G) \to G $$ are homotopic to the identity,
3. there exist a natural transformation $u \colon \id_\mathcal{C} \to G \circ F$ such that for all $C \in \mathcal{C}$ and $D \in \mathcal{D}$, the map $$ \operatorname{Maps}_ \mathcal{D}(F(C), D) \to \operatorname{Maps}_ \mathcal{C}(G(F(C)), G(D)) \to \operatorname{Maps}_ \mathcal{C}(C, G(D)) $$ is a homotopy equivalence,
4. $(F, G)$ form an adjoint pair between $\mathcal{H}$-enriched categories $\operatorname{h}\mathcal{C}$ and $\operatorname{h}\mathcal{D}$.

Somehow you don’t need to specify all these higher data to define an adjunction.

Monoidal ∞-categories#

Definition 6. A symmetric monoidal category (or monoidal category) is a coCartesian fibration $\mathcal{C}^\otimes \to \mathsf{Fin}_ \ast$ (or over $\Delta^\mathrm{op}$) such that $$ \mathcal{C}_ I \to \prod_{i \in I}^{} \mathcal{C}_ i $$ is an equivalence.

We can also define the category $\widehat{\mathsf{Cat}}^\mathrm{sym}$ of symmetric monoidal categories with symmetric monoidal functors as a full subcategory of $(\mathsf{coCart}_ {/\mathsf{Fin}_ \ast})_ \mathrm{str}$. We can also define $\widehat{\mathsf{Cat}}^\mathrm{laxsym}$ in a similar way.

Proposition 7. Let $\mathcal{C}$ be an ∞-category with finite products. Then it admits a natural Cartesian symmetric monoidal structure $\mathcal{C}^\times$.

Proposition 8. We have $$ \mathsf{Alg}(\widehat{\mathsf{Cat}}^\times) \simeq \widehat{\mathsf{Cat}}^\mathrm{mon} $$ and similarly $$ \mathsf{CAlg}(\widehat{\mathsf{Cat}}^\times) \simeq \widehat{\mathsf{Cat}}^\mathrm{sym}. $$

The category $\widehat{\mathsf{Cat}}$ has all limits and colimits. The categories $\widehat{\mathsf{Cat}}^\mathrm{mon}$ and $\widehat{\mathsf{Cat}}^\mathrm{sym}$ will also have all limits and colimits. The forgetful functors $$ \widehat{\mathsf{Cat}}^\mathrm{mon}, \widehat{\mathsf{Cat}}^\mathrm{sym} \to \widehat{\mathsf{Cat}} $$ will preserve all limits but not colimits.

Presentable ∞-categories#

Definition 9. Suppose $\mathcal{C}$ admits all filtered colimits.

1. We say that an object $C \in \mathcal{C}$ is compact when $$ \operatorname{Maps}(C, \varinjlim_I D_i) \simeq \varinjlim_{i \in I} \operatorname{Maps}(C, D_i). $$
2. We say that $\mathcal{C}$ is compactly generated if there exists a set of compact objects $\lbrace C_i \rbrace$ such that every object $C \in \mathcal{C}$ can be written as a filtered colimit of $C_i$.

In general, assume $\mathcal{C}$ has all filtered colimits, and let $\lbrace C_i \rbrace$ be a set of compact objects. Let $\mathcal{C}^0 \subseteq \mathcal{C}$ be the full subcategory spanned by $\lbrace C_i \rbrace$. There is the Yoneda embedding $$ \iota \colon \mathcal{C} \to \mathcal{P}(\mathcal{C}^0) = \mathsf{Fun}((\mathcal{C}^0)^\mathrm{op}, \mathsf{Spc}). $$ Then $\mathcal{C}$ is compactly generated by $\lbrace C_i \rbrace$ if and only if

• $\iota$ is fully faithful, and
• the essential image of $\iota$ consists of those functors $(\mathcal{C}^0)^\mathrm{op} \to \mathsf{Spc}$ that preserve finite limits.

Example 10. The category $\mathsf{Spc}$ is compactly generated; the finite CW-complexes are compact, and they generate everything. For $A$ a commutative ring, $\mathsf{Mod}_ A^\heartsuit$ is compactly generated, with finite modules being compact.

Let $\kappa$ be a regular cardinal. We can define the notion of an object being $\kappa$-compact, using $\kappa$-filtered diagrams.

Definition 11. A category $\mathcal{C}$ is presentable when

• it has all small colimits, and
• for some $\kappa$, there exists a set of $\kappa$-compact objects that generate $\mathcal{C}$ under $\kappa$-filtered colimits.