Recollections and loose ends

• The nPOV
• Two homotopy theories of simplicial sets
• Fibrations of simplicial sets
• Left fibrations
• Inner fibrations
• Cocartesian fibrations
• Accessibility and presentability
• Filtered categories and compactness
• Essential and local smallness
• Accessibility
• Presentability

We review some of the concepts introduced last quarter, tie up some loose ends, and discuss where we are headed this quarter.

The nPOV#

The central philosophy of this entire ∞-category business is that all instances of “sets” should be replaced by “spaces.” Instead of categories, where mapping sets are sets, we have ∞-categories, where mapping spaces are spaces. Instead of sheaves, which are assignments of a set to each open set, we look at ∞-sheaves, which are assignments of a space to each open.

Here is a table of analogies.

Two homotopy theories of simplicial sets#

The way we modeled ∞-categories was through simplicial sets. There are mainly two different ways we should think about simplicial sets.

• A simplicial set as a space: in this point of view, a simplicial set $S$ is identified with its geometric realization $\lvert S \rvert$. For instance, the simplicial set $\Delta^1 \coprod_{\partial \Delta^1} \Delta^1$ is thought of as $S^1$. When we restrict to Kan complexes, we have a good homotopy theory.
• A simplicial set as a ∞-category: in this point of view, each $0$-simplex in a simplicial set $S$ is an object, each $1$-simplex (or a chain of $1$-simplices) in $S$ is a morphism, each $2$-simplex in $S$ defines a path in the morphism space, and so on. The simplicial set $\Delta^1 \coprod_{\partial \Delta^1} \Delta^1$ can be thought of as a category with two objects, and two non-invertible morphisms between them. To get a good homotopy theory, we restrict to simplicial sets that are ∞-categories.

If you know about model categories, the first corresponds to the Quillen model structure, and the second corresponds to the Joyal model structure.

Kan complexes are special kinds of ∞-categories. Here is a proposition we’ve seen last quarter.

Proposition 1 (HTT 1.2.5.1). If $\mathcal{C}$ is an $\infty$-category with all morphisms invertible up to homotopy (i.e., its homotopy category $\mathrm{h}\mathcal{C}$ is a groupoid) then $\mathcal{C}$ is a Kan complex.

This is the reason another name for a Kan complex is an ∞-groupoid. Given a simplicial set $S$, we can view it as a ∞-category and also as a ∞-groupoid. The two are related through the procedure of “formally inverting all morphisms.”

Fibrations of simplicial sets#

We’ve seen many different definitions of “fibrations” between simplicial sets.

• Trivial Kan fibrations: those that have the right lifting property with respect to $\partial \Delta^n \hookrightarrow \Delta^n$. The fibers are contractible Kan complexes, and it is an “equivalence” in the strongest possible sense.
• Kan fibrations: those that have the right lifting property with respect to all horn inclusions $\Lambda_k^n \hookrightarrow \Delta^n$. The fibers are all Kan complexes, and moreover for every edge $s \to t$ in the base, there is a homotopy equivalence between the fibers over $s$ and $t$.
• Left/right fibrations: those that have the right lifting property with respect to left/right horn inclusions $\Lambda_k^n \hookrightarrow \Delta^n$ for $0 \le k \lt n$ or $0 \lt k \le n$. The fibers are still all Kan complexes, due to a theorem of Joyal.
• Inner fibrations: those that have the right lifting property with respect to all inner horn inclusions $\Lambda_k^n \hookrightarrow \Delta^n$ for $0 \lt k \lt n$. The fibers are ∞-categories.

I want to argue that these different notions of fibrations, including others that we will see soon, are not just formalities and have actual meaning.

Left fibrations#

Let’s start with left fibrations (right fibrations are just dual to that). If you’re familiar with the theory of categories (co)fibered in groupoids, this is exactly the same. Here is a technical lemma.

Proposition 2 (HTT 2.1.2.7). If $A \to A^\prime$ is left anodyne and $B \subseteq B^\prime$ is an inclusion, then $$ (A \times B^\prime) \coprod_{A \times B} (A^\prime \times B) \to A^\prime \times B^\prime $$ is left anodyne.

Let $X \to S$ be a left fibration, and let $s \to t$ be an edge in $S$. We know that the fibers $X_s$ and $X_t$ are Kan complexes. Now the point is that the inclusion $$ X_s \times \{0\} \hookrightarrow X_s \times \Delta^1 $$ is left anodyne by the above lemma. This implies that this has the left lifting property with respect to $X \to S$, and so we can solve the following lifting problem. $$ \begin{CD} X_s \times \{0\} @>>> X \br @VVV @VVV \br X_s \times \Delta^1 @>>> S. \end{CD} $$ At the end, we obtain a map $X_s \to X_t$ of Kan complexes.

But this is a choice. Can we show that this is well-defined up to homotopy? Suppose we have two different lifts. Then we can glue them to form the diagram $$ \begin{CD} X_s \times \Lambda_0^2 @>>> X \br @VVV @VVV \br X_s \times \Delta^2 @>>> S, \end{CD} $$ where $\Delta^2 \to S$ is given by $s \to t \to t$. The left vertical map is again left anodyne, so we can lift it, and restrict to $\Delta^1 \subset \Delta^2$ to get a homotopy between the two maps $X_s \to X_t$.

This suggests that a left fibration $X \to S$ is basically the same thing as a functor $S \to \mathcal{S}$, where $\mathcal{S}$ is the ∞-category of spaces.

Theorem 3. Let $\mathcal{C}$ be an $\infty$-category. Then there is an equivalence between $\mathsf{Fun}(\mathcal{C}, \mathcal{S})$ and the homotopy coherent nerve of the simplicial category $\mathsf{LFib}(\mathcal{C})$, the full subcategory of $(\mathsf{Set} _ \Delta) _ {/\mathcal{C}}$ consisting of left fibrations.

Inner fibrations#

Inner fibrations don’t really have a straightening/unstraightening interpretation as we’ve seen with left and right fibrations. But we can still think about the functoriality on fibers.

Let’s say we have an inner fibration $X \to \Delta^1$. If you think about it, this map being an inner fibration is equivalent to $X$ being an ∞-category. Now the fibers $X_0$ and $X_1$ are also ∞-categories. Moreover, there are mapping spaces from objects of $X_0$ to objects of $X_1$. These can be arranged into a functor $$ M \colon X_0^\mathrm{op} \times X_1 \to \mathcal{S}. $$ This is called a correspondence of categories.

What happens if we have a inner fibration $X \to \Delta^2$? Again, $X$ will be an ∞-category. Let $F, G, H$ be the correspondences for $\Delta_{0,1}^1$, $\Delta_{1,2}^1$, and $\Delta_{0,2}^1$. For objects $C_i \in X_i$, we get a composition map $$ F(C_0, C_1) \times G(C_1, C_2) \to H(C_0, C_2). $$ Another way we can interpret this is that there a natural “composition” of the correspondences $F$ and $G$, given by $$ (F \ast G)(C_0, C_2) = \int_{C_1 \in X_1} F(C_0, C_1) \times G(C_1, C_2), $$ and then we have a map $F \ast G \to H$.

So roughly speaking, an inner fibration over $S$ is a lax functor from $S$ to the $(\infty,2)$-category of $(\infty, 1)$-categories with correspondences. Making this precise is definitely outside the scope of this seminar.

Cocartesian fibrations#

Note that a functor is a type of a correspondence, because given a functor $f \colon \mathcal{C} \to \mathcal{D}$, we can define $$ M(C, D) = \Hom_\mathcal{D}(f(C), D). $$ We even see that $f$ can be recovered from $M$, by moving around $D$ and applying Yoneda. Between inner fibrations and left fibrations, there should be a functors $S \to \mathsf{QCat}$. Cocartesian fibrations do precisely that. Whatever these are, they will at least be inner fibrations.

Definition 4 (HTT 2.4.1.1). Let $p \colon X \to S$ be an inner fibration of simplicial sets. We say that an edge $f \colon x \to y$ in $X$ is cocartesian with respect to $p$ if the induced map $$ X_{f/} \to X_{x/} \times_{S_{p(x)/}} S_{p(f)/}, $$ which is always a left fibration (HTT 2.1.2.1), is moreover a trivial Kan fibration.

What does this mean? Fix a $2$-simplex $p(x) \to p(y) \to p(z)$ in $S$ and a lift $z$. Since we have an inner fibration, there are correspondences $F \colon X_{p(x)} \to X_{p(z)}$ and $G \colon X_{p(y)} \to X_{p(z)}$. If you think about the cocartesian condition and look at the fiber over $z \in X$, we see that $$ G(y, z) \to F(x, z) $$ is a homotopy equivalence. In particular, if we apply to $p(y) = p(z)$, we see that $y$ must be the corepresenting object of $F(x, -)$.

Definition 5 (HTT 2.4.2.1). A map $p \colon X \to S$ is a cocartesian fibration if it is an inner fibration, and for every edge $f \colon x \to y$ of $S$ and a lift $\tilde{x} \in X$, there exists a $p$-cocartesian edge $\tilde{f} \colon \tilde{x} \to \tilde{y}$ in $X$ lifting $f$.

At this point, all correspondences must actually be functors, and the cocartesian lifts are telling us where each object should map to!

Theorem 6. Let $\mathcal{C}$ be an $\infty$-category. Then there is an equivalence between $\mathsf{Fun}(\mathcal{C}, \mathsf{QCat})$ and the $\infty$-category of cocartesian fibrations over $\mathcal{C}$ with cocartesian functors. (The latter is constructed as the homotopy coherent nerve of the simplicial category of cocartesian fibrations over $\mathcal{C}$ with cocartesian morphisms.)

Accessibility and presentability#

So far we’ve been ignoring a lot of set-theoretic issues. We now address some of it.

First, we assume that there is a strongly inaccessible cardinal $\kappa_G$, and fix one. Then the collection of sets having rank less than $\kappa_G$ satisfies ZFC, and the set of such sets $V_{\kappa_G}$ is called a Grothendieck universe. We say that a set is small when it is in $V_{\kappa_G}$. Using this, we can define what it means for any mathematical object to be small (since every mathematical object is a set). When we were talking about $\mathcal{S}$ the category of spaces or Kan complexes, we really mean the category of small Kan complexes. Similarly, $\mathsf{QCat}$ the ∞-category of quasicategories is really the category of small quasicategories.

Even if we have a notion of “small,” it is useful to have a finer control on sizes.

Definition 7. A cardinal $\kappa$ is called regular if $\lvert I \rvert \lt \kappa$ and $\lvert S_i \rvert \lt \kappa$ for all $i \in I$ then $$ \biggl\lvert \bigcup_{i \in I} S_i \biggr\rvert \lt \kappa. $$ A set is said to be $\kappa$-small when its cardinality is strictly smaller than $\kappa$.

Filtered categories and compactness#

Here is the definition.

Definition 8 (HTT 5.3.1.7). Let $\kappa$ be a regular cardinal. An ∞-category $\mathcal{C}$ is said to be $\kappa$-filtered if for every $\kappa$-small simplicial set $K$, every map $K \to \mathcal{C}$ has an extension to $K^\vartriangleright \to \mathcal{C}$. We say that $\mathcal{C}$ is filtered when it is $\omega$-filtered.

This notion of filtered really agrees with what we know from $1$-category theory.

Definition 9. Let $\mathcal{C}$ be a $\infty$-category which admits small $\kappa$-filtered colimits. We say that a functor $\mathcal{C} \to \mathcal{D}$ is $\kappa$-continuous if it preserves small $\kappa$-filtered colimits. We say that an object $C \in \mathcal{C}$ is $\kappa$-compact if $\Hom(-, C)$ is $\kappa$-continuous.

Recall the category $\mathcal{P}_\Sigma(\mathcal{C})$, which had two interpretations:

• the category obtained by freely adjoining sifted colimits to $\mathcal{C}$,
• the category of presheaves that preserves finite products.

Proposition 10 (HTT 5.3.5.4). Let $\kappa$ be a regular cardinal and $\mathcal{C}$ be a small $\infty$-category. For a functor $F \colon \mathcal{C}^\mathrm{op} \to \mathcal{S}$, the following are equivalent:

• there exists a small $\kappa$-filtered $\infty$-category $\mathcal{J}$ and a diagram $p \colon \mathcal{J} \to \mathcal{C}$ such that $F$ is a colimit of $j \circ p$,
• it corresponds to a right fibration $\tilde{\mathcal{C}} \to \mathcal{C}$ with $\tilde{\mathcal{C}}$ a $\kappa$-filtered category.

If $\mathcal{C}$ has all $\kappa$-small colimits, then the above conditions are equivalent to the following as well:

• $F$ preserves $\kappa$-small limits.

Definition 11 (HTT 5.3.5.1). For $\mathcal{C}$ a small $\infty$-category with $\kappa$-small colimits, we define the full subcategory $$ \operatorname{Ind}_\kappa(\mathcal{C}) \subseteq \mathcal{P}(\mathcal{C}) $$ consisting of those functors that preserves $\kappa$-small limits.

Essential and local smallness#

Proposition 12 (HTT 5.4.1.2). Let $\mathcal{C}$ be an $\infty$-category and $\kappa$ be an uncountable regular cardinal. The following are equivalent:

• the set of equivalence classes of $\mathcal{C}$ is $\kappa$-small, and for every morphism $f \colon C \to D$, the homotopy set $\pi_n(\Hom(C, D), f)$ is $\kappa$-small,
• the category $\mathcal{C}$ is $\kappa$-compact as an object of $\mathsf{QCat}$,
• there exists a $\kappa$-small simplicial set $\mathcal{C}^\prime$ that is an $\infty$-category, and a map $\mathcal{C}^\prime \to \mathcal{C}$ that is an equivalence.

In such a case, we say that $\mathcal{C}$ is essentially $\kappa$-small. We say that $\mathcal{C}$ is essentially small if it is essentially $\kappa$-small for some $\kappa \lt \kappa_G$.

We can then restrict this notion to ∞-groupoids.

Proposition 13 (HTT 5.4.1.7). Let $\mathcal{C}$ be an $\infty$-category. The following conditions are equivalent:

• for every pair of objects $X, Y \in \mathcal{C}$, the space $\Hom_\mathcal{C}(X, Y)$ is essentially small,
• for every small collection $S$ of objects of $\mathcal{C}$, the full subcategory of $\mathcal{C}$ spanned by $S$ is essentially small.

In this case, we say that $\mathcal{C}$ is locally small.

Accessibility#

We can’t always talk about small categories, because the category of presheaves, for instance, is not small. But we still want to have the benefits of being controlled by a small amount of information.

Definition 14 (HTT 5.4.2.1). For $\kappa$ a regular cardinal, we say that an $\infty$-category $\mathcal{C}$ is $\kappa$-accessible if it is equivalent to $\operatorname{Ind}_\kappa(\mathcal{C}_0)$ for a small $\infty$-category $\mathcal{C}_0$.

Proposition 15 (HTT 5.4.2.2). Let $\mathcal{C}$ be an $\infty$-category and $\kappa$ a regular cardinal. Then $\mathcal{C}$ if $\kappa$-accessible if and only if the following conditions hold:

• $\mathcal{C}$ is locally small,
• $\mathcal{C}$ admits $\kappa$-filtered colimits,
• the full subcategory $\mathcal{C}^\kappa$ of $\kappa$-compact objects is essentially small,
• $\mathcal{C}^\kappa$ generates $\mathcal{C}$ under small $\kappa$-filtered colimits.

Definition 16 (HTT 5.4.2.5). Let $\mathcal{C}$ be an accessible $\infty$-category. We say that a functor $F \colon \mathcal{C} \to \mathcal{D}$ is accessible if it is $\kappa$-continuous for some regular cardinal $\kappa$.

Presentability#

Definition 17 (HTT 5.5.0.1). An $\infty$-category is called presentable when it is accessible and admits small colimits.

Theoerem 18 (Simpson, HTT 5.5.1.1). Let $\mathcal{C}$ be an $\infty$-category. The following are equivalent:

• $\mathcal{C}$ is presentable,
• $\mathcal{C}$ is accessible, and for every regular cardinal $\kappa$ the full subcategory $\mathcal{C}^\kappa$ admits $\kappa$-small colimits,
• there exists a regular cardinal $\kappa$, a small $\infty$-category $\mathcal{D}$ which admits $\kappa$-small colimits, and an equivalence $\operatorname{Ind}_\kappa(\mathcal{D}) \to \mathcal{C}$.

There are nice things that happen when a category is presentable. The following is basically the adjoint functor theorem.

Proposition 19 (HTT 5.5.2.2). Let $\mathcal{C}$ be a presentable $\infty$-category. Then a functor $\mathcal{C}^\mathrm{op} \to \mathcal{S}$ is representable if and only if it preserves small limits. Similarly, a functor $\mathcal{C} \to \mathcal{S}$ is corepresentable if and only if it is accessible and preserves small limits.

Presentable categories also have a nice theory of localization.