Monoidal categories as coCartesian fibrations

Monoidal categories as coCartesian fibrations#

Last time we made the following definition.

Definition 1. A monoidal category is a coCartesian fibration $\mathcal{C} \to \Delta^\mathrm{op}$ such that $$ \mathcal{C}_ n \to \mathcal{C}_ 1 \times \dotsb \times \mathcal{C}_ 1$$ induced by $\rho_i \colon [1] \cong \lbrace i, i+1 \rbrace \subseteq [n]$ is an equivalence, for every $n \ge 0$.

In fact, we can define the category $\mathsf{Cat}^\mathrm{mon}$ of monoidal categories (with monoidal functors) as the full subcategory of $(\mathsf{coCart}_ {/\Delta^\mathrm{op}})_ \mathrm{str}$ consisting of the ones satisfying the above.

Definition 2. The category $\mathsf{Cat}^\mathrm{laxmon}$ of monoidal categories with lax monoidal functors is defined as the category where
objects are monoidal categories $\mathsf{C} \to \Delta^\mathrm{op}$,
morphisms are commutative diagrams $$ \begin{CD} \mathcal{C} @>{F}>> \mathcal{D} \br @VVV @VVV \br \Delta^\mathrm{op} @= \Delta^\mathrm{op} \end{CD} $$ where $F$ sends coCartesian lifts of $\rho_i$ to coCartesian arrows.

Now we can define algebra objects as just sections, because it was defined as lax monoidal functors $\ast \to \mathcal{C}$.

Definition 3. The category of algebra objects in a monoidal category $\mathcal{C}$ as the full subcategory of sections $\Delta^\mathrm{op} \to \mathcal{C}$ sending $\rho_i$ to Cartesian arrows.

For symmetric monoidal categories, we need to use the category $\mathsf{Fin}_ \ast$ of finite pointed sets.

Definition 4. A symmetric monoidal category is a coCartesian fibration $C^\otimes to \mathsf{Fin}_ \ast$ such that $$ \mathcal{C}_ I \cong \prod_{i \in I \setminus \lbrace \ast \rbrace}^{} C_i $$ where $C_I \to C_i$ are induced by $I \to \lbrace i, \ast \rbrace$ sending everything other than $i$ to $\ast$.

Then we can define $\mathsf{Cat}^\mathrm{sym}$ and $\mathsf{Cat}^\mathrm{laxsym}$ and $\mathsf{CAlg}(\mathcal{C})$ similarly.

Remark 5. There is a natural functor $\Delta^\mathrm{op} \to \mathsf{Fin}_ \ast$ sending each $[n]$ to $\lbrace 1/2, \dotsc, n-1/2, ast \rbrace$, where the fractions are to be interpreted as as the set of decompositions $[n] = S \amalg T$ such that $S$ and $T$ are nonempty convex subsets.

Enriched categories#

Let $\mathcal{A}$ be a monoidal category.

Definition 6. A category $\mathcal{C}$ enriched over $\mathcal{A}$ consists of
a collection of objects,
for every $X, Y \in \operatorname{Obj}(\mathcal{C})$, an object $\underline{\Hom}(X, Y) \in \mathcal{A}$,
for $X, Y, Z \in \operatorname{Obj}(\mathcal{C})$, a composition $$ \underline{\Hom}(X, Y) \otimes \underline{\Hom}(Y, Z) \to \underline{\Hom}(X, Z) $$ satisfying the pentagon axiom.

Example 7. Every category is enriched over $\mathsf{Set}^\times$. A closed symmetric monoidal category is canonically enriched over itself.

Given an category $\mathcal{C}$ enriched over $\mathcal{A}$, we can recover an ordinary category $\underline{\mathcal{C}}$ that has the same objects and having the morphisms $$ \Hom(X, Y) = \Hom_\mathcal{A}(1, \underline{\Hom}(X, Y)). $$

Example 8. In the above example, the closed symmetric monoidal category $\mathcal{C}$ enriched over itself has $\underline{\mathcal{C}}$ just $\mathcal{C}$.

Simplicial sets as ∞-groupoids#

Now we do a “higher categories for engineering students” so we know how to manipulate some symbols. The basic principle is that $=$ in the classical sense not really allowed and we should always interpret it as choosing an isomorphism. We will have $$ \mathsf{Sets} \subset \mathsf{Groupoids} \subset \dotsb \subset \infty\mathsf{-Groupoids} $$ and we want to work with $\infty$-groupoids instead of sets. These $\infty$-groupoids can also be understood as a space up to homotopy. To model these spaces, we use simplicial sets, functors $\Delta^\mathrm{op} \to \mathsf{Set}$.

Definition 9. The Yoneda embedding defines a simplicial set $$ \Delta^n = \Hom(-, [n]), $$ and then for each $0 \le i \le n$ we define the horn $\Lambda_i^n$ as a sub-simplicial set of $\Delta^n$ given by $$ \Lambda_i^n([m]) = \lbrace \alpha \colon [m] \to [n] : \alpha([m]) \cup \lbrace i \rbrace \neq [n] \rbrace. $$

Definition 10. A Kan complex is a simplicial set $S$ such that every diagram as below has an extension $\Delta^n \to S$ for all $0 \le i \le n$. $$ \begin{CD} \Lambda_i^n @>>> S \br @VVV \br \Delta^n \end{CD} $$

This is what models topological spaces. There is a geometric realization functor $$ \lvert - \rvert \colon \mathsf{sSet} \to \mathsf{CGTop} $$ sending $\Delta^n$ to the space $$ \lvert \Delta^n \rvert = \lbrace (t_0, \dotsc, t_n) \in [0,1]^{n+1} : t_0 + \dotsb + t_n = 1 \rbrace $$ and preserving colimits. The singular simplicial set construction is going to be a right adjoint to it. Both constructions preserve finite products.

The category $\mathsf{sSet}$ of simplicial sets has all small limits and colimits. Moreover, $\mathsf{sSet}^\times$ is a closed symmetric monoidal category, with internal Hom just given by $$ \underline{\Hom}(S,T)([n]) = \Hom(\Delta^n, \underline{\Hom}(S, T)) = \Hom(\Delta^n \times S, T). $$

Propostiion 11. The full subcategory $\mathsf{Kan}$ of Kan complexes inherits a closed symmetric monoidal structure, i.e., the internal Hom of two Kan complexes is still a Kan complex.

It’s also not difficult to see that the singular simplicial set of a (compactly generated) topological space is a Kan complex. These two functors are not inverses to each other, but they can be thought of as “equivalences” after applying the machinery of model categories.

Homotopy pullbacks and pushouts#

Let’s say we have a diagram $$ R \to S \leftarrow T $$ of Kan complexes. We define the simplicial set $$ R \times_S^h T = R \times_{S^{\lbrace 0 \rbrace}} \underline{\Hom}(\Delta^1, S) \times_{S^{\lbrace 1 \rbrace}} T. $$ Its geometric realization is homotopy equivalent to the homotopy pullback $$ \lvert R \times_S^h T \rvert \sim \lvert R \rvert \times_{\lvert S \rvert}^h \lvert T \rvert = \lbrace (r, t, \gamma) : r \in \lvert R \rvert, t \in \lvert T \rvert, \gamma \colon [0,1] \to \lvert S \rvert, r \mapsto \gamma(0), t \mapsto \gamma(1) \rbrace. $$

There is a similar thing we can do for the pushout. We have, for $$ R \leftarrow S \to T, $$ the homotopy pushout $$ R \amalg_{S,h} T = R \amalg_{\lbrace 0 \rbrace \times S} \Delta^1 \times S \amalg_{\lbrace 1 \rbrace \times S} T. $$ This will have $$ \lvert R \amalg_S^h T \rvert \cong \lvert R \rvert \amalg_{\lvert S \rvert}^h \lvert T \rvert $$ the homotopy pushout, which is just $R$ and $T$ glued by a cylinder.