Algebra objects | Dongryul Kim

We want to define the trace formalism to higher categories. In the next three lectures, starting next time, we will discuss the foundations of higher categories. Today, we will discuss an alternative definition of a symmetric monoidal category.

Lax monoidal functors#

Definition 1. A lax (symmetric) monoidal functor between (symmetric) monoidal categories $\mathcal{C}$ and $\mathcal{D}$ is a pair $(F, \lambda)$ where
$F \colon \mathcal{C} \to \mathcal{D}$ is a functor,
$\lambda$ is a natural transformation from the lower left to the upper right path in $$ \begin{CD} \mathcal{C} \times \mathcal{C} @>>> \mathcal{C} \br @V{F \times F}VV @V{F}VV \br \mathcal{D} \times \mathcal{D} @>>> \mathcal{D}, \end{CD} $$ i.e., a morphism $$ \lambda \colon F(X) \otimes F(Y) \to F(X \otimes Y) $$ functorial in $X$ and $Y$,
such that
$\lambda$ is compatible with $a$ (and $c$), and
there exists a morphism $\epsilon \colon 1_\mathcal{D} \to F(1_\mathcal{C})$ such that the composition $$ 1_\mathcal{D} \otimes F(X) \to F(1_\mathcal{C}) \otimes F(X) \to F(1_\mathcal{C} \otimes X) \cong F(X) $$ is $l_{F(X)}$,
and similarly for $F(X) \otimes 1_\mathcal{D}$.

Such an $\epsilon$, if exists, is going to be unique. We won’t prove it though.

Example 2. The forgetful functor $$ (\mathsf{Vect}_ k^\heartsuit)^\otimes \to \mathsf{Set}^\times $$ is lax symmetric monoidal; there is a natural map $V \times W \to V \otimes W$ of sets for vector spaces $V$ and $W$. More generally, given any (symmetric) monoidal category $\mathcal{C}$, the functor $$ \mathcal{C} \to \mathsf{Set}^\times; \quad X \mapsto \Hom(1, X) $$ is a lax (symmetric) monoidal functor.

Definition 3. Let $(F, \lambda)$ and $(G, \mu)$ be two (symmetric) monoidal functors. A natural monoidal transformation is a natural transformation $\varphi \colon F \Rightarrow G$ such that $$ \begin{CD} F(X) \otimes F(Y) @>{\lambda}>> F(X \otimes Y) \br @V{\varphi \otimes \varphi}VV @V{\varphi}VV \br G(X) \otimes G(Y) @>{\mu}>> G(X \otimes Y) \end{CD} $$ commutes.

Using this, we can now form functor categories $$ \mathsf{Fun}^\mathrm{lax mon}(\mathcal{C},\mathcal{D}) \supseteq \mathsf{Fun}^\mathrm{mon}(\mathcal{C}, \mathcal{D}), $$ where the right side is a full subcategory of the left, and similarly a version for symmetric monoidal categories $$ \mathsf{Fun}^\mathrm{lax symm mon}(\mathcal{C}, \mathcal{D}) \supseteq \mathsf{Fun}^\mathrm{lax symm}(\mathcal{C}, \mathcal{D}). $$ We can organize all monoidal categories into a $2$-category $$ \mathsf{Cat}^\mathrm{lax mon} \supset \mathsf{Cat}^\mathrm{mon}, $$ where the right hand side is not a full subcategory but just a $1$-full subcategory (meaning fullness only at $2$-morphisms).

Algebra objects#

Definition 4. Let $\mathcal{C}$ be a monoidal category. The category of algebra objects is defined to be $\mathsf{Alg}(\mathcal{C}) = \mathsf{Fun}^\mathrm{lax mon}(\ast, \mathcal{C})$, where $\ast$ is the category with one object and one morphism, with the trivial monoidal structure.
Similarly, if $\mathcal{C}$ is a symmetric monoidal category, we define a category of commutative algebra object as $\mathsf{CAlg}(\mathcal{C}) = \mathsf{Fun}^\mathrm{lax symm}(\ast, \mathcal{C})$.

Concretely, an algebra object is an object $A \in \mathcal{C}$ with a morphism $m \colon A \otimes A \to A$ satisfying associativity $$ \begin{CD} A \otimes A \otimes A @>{m \otimes \id}>> A \otimes A \br @V{\id \otimes m}VV @V{m}VV \br A \otimes A @>{m}>> A \end{CD} $$ such that there exists a unit $\epsilon \colon 1 \to A$ satisfying certain properties.

Remark 5. We have symmetric monoidal functors $$ \mathsf{CAlg}(\mathcal{C}) \to \mathsf{Alg}(\mathcal{C}) \to \mathcal{C} $$ for $\mathcal{C}$ a symmetric monoidal category. For ordinary categories, the left functor $\mathsf{CAlg}(\mathcal{C}) \to \mathsf{Alg}(\mathcal{C})$ is fully faithful, but this will no longer be true in the higher categorical setting.

Example 6. Here are some examples of algebra objects.
When $\mathcal{C} = \mathsf{Vect}_ k^\heartsuit$, algebra objects are just $k$-algebras and commutative algebras are commutative $k$-algebras.
When $\mathcal{C} = \mathsf{Ch}_ k$, the category of chain complexes of $k$-vector spaces, algebra objects are dg-algebras.
When $\mathcal{C} = \mathsf{Cat}^\times$, algebra objects are strict monoidal categories, with morphisms being strict monoidal functors.

Monoid objects#

We are now going to talk about an alternative definition of a monoidal category.

Definition 7. We write $\Delta$ for the category of finite ordered sets with morphisms being non-decreasing maps. This is equivalent to $$ \lbrace [0] = \lbrace 0 \rbrace, [1] = \lbrace 0, 1 \rbrace, \dotsc \rbrace $$ where morphisms are again non-decreasing maps.

Definition 8. For $\mathcal{C}$ a category, we define the category $$ \mathcal{C}^{\Delta^\mathrm{op}} = \mathsf{Fun}(\Delta^\mathrm{op}, \mathcal{C}) $$ of simplicial objects in $\mathcal{C}$.

Definition 9. Let $\mathcal{C}$ be a category with finite products. We define the category of monoid objects of $\mathcal{C}$ to be the full subcategory of $\mathsf{Fun}(\Delta^\mathrm{op}, \mathcal{C})$ consisting of objects $X_\bullet$ satisfying
$X_0$ is the final object in $\mathcal{C}$,
the map $X_n \to X_1 \times \dotsb \times X_1$ induced by maps $[1] \cong \lbrace i, i+1 \rbrace \subseteq [n]$ is an isomorphism.

Remark 10. Since $\mathcal{C}$ is an ordinary category, a monoid object is uniquely determined by $X_0, X_1, X_2$, and maps between them. The multiplication map is just given by $$ X_1 \times X_1 \cong X_2 \to X_1. $$

Example 11. A monoid object in $\mathsf{Set}$ is going to be a usual monoid.

Remark 12. Assume $\mathcal{C}$ has finite products. Then there is a natural Cartesian monoidal structure $\mathcal{C}^\times$ with products as tensor products. Then $\mathsf{Alg}(\mathcal{C}^\times)$ is the same as $\mathsf{Mon}(\mathcal{C})$.

Cartesian fibrations#

Now the idea is to define $\mathsf{Cat}^\mathrm{mon}$ as a monoid object in $\mathsf{Cat}^\times$. But what is a monoid object in $\mathsf{Cat}^\times$? We need to take into account the $2$-categorical structure and ask that $X_n \to X_1 \times \dotsb \times X_1$ is just an equivalence rather than an isomorphism. But it still should be a full subcategory of $\mathsf{Delta}^\mathrm{op} \to \mathsf{Cat}$.

To deal with these $2$-categorical issues, we use Grothendieck’s constructions. This says that a $2$-categorical functor $F \colon \mathcal{C}^\mathrm{op} \to \mathsf{Cat}$ somehow can be encoded by a category $\tilde{C}$ living over $\mathcal{C}$.

The objects of $\tilde{C}$ are pairs $(C, \eta)$ where $C \in \mathcal{C}$ and $\eta \in F(C)$.
A morphism $(C, \eta) \to (D, \lambda)$ is a map $C \to D$ together with a map $\eta \to F(\varphi)(\lambda)$ in $F(C)$.

Proposition 13. The map $\tilde{F} \colon \tilde{C} \to \mathcal{C}$ is a Cartesian fibration.

Definition 14. Let $F \colon \mathcal{D} \to \mathcal{C}$ be a functor. A map $D_1 \to D_2$ in $\mathcal{D}$ is said to be Cartesian if for every $D \in \mathcal{D}$, the diagram $$ \begin{CD} \Hom_\mathcal{D}(D, D_1) @>>> \Hom_\mathcal{D}(D, D_2) \br @VVV @VVV \br \Hom_\mathcal{C}(F(D), F(D_1)) @>>> \Hom_\mathcal{C}(F(D), F(D_2)) \end{CD} $$ is a fiber product.

Definition 15. We say that $F \colon \mathcal{D} \to \mathcal{C}$ is a Cartesian fibration when for every $C_1 \to C_2$ in $\mathcal{C}$ and $D_2$ with $F(D_2) = C_2$, there exists a Cartesian arrow $D_1 \to D_2$ over $C_1 \to C_2$.

Remark 16. This lift $D_1 \to D_2$ will be unique up to unique isomorphism, given that it exists.

It’s quite easy to verify that the Grothendieck construction $\tilde{C} \to C$ is a Cartesian fibration; for any $\varphi \colon C_1 \to C_2$ and $\eta_2 \in F(C_2)$, we see that $(C_1, F(\varphi)(\eta_2)) \to (C_2, \eta_2)$ is a Cartesian lift.

Definition 17. Given a category $\mathcal{C}$, we can define the category $\mathsf{Cart}_ {/\mathcal{C}}^\mathrm{str}$ of Cartesian fibrations over $\mathcal{C}$, where
objects are Cartesian fibrations $\mathcal{D} \to \mathcal{C}$,
morphisms are functors $\mathcal{D}_ 1 \to \mathcal{D}_ 2$ such that $$ \begin{CD} \mathcal{D}_ 1 @>>> \mathcal{D}_ 2 \br @VVV @VVV \br \mathcal{C} @= \mathcal{C} \end{CD} $$ (strictly) commutes, and sends Cartesian arrows to Cartesian arrows.

Theroem 18. The Grothendieck construction $$ \mathsf{Fun}(\mathcal{C}^\mathrm{op}, \mathsf{Cat}) \to \mathsf{Cart}_ {/\mathcal{C}}^\mathrm{str} $$ is an equivalence.

Remark 19. We can also reverse the arrows and get a notion of coCartesian fibrations.

Monoidal categories as Cartesian fibrations#

For a functor $\mathcal{C} \to \Delta$, we denote $$ \mathcal{C}_ n = \mathcal{C} \times_{\Delta} \lbrace [n] \rbrace, $$ meaning the objects are objects mapping to $[n]$ and morphisms are those that map to the identity map.

Definition 20. A monoidal category is a Cartesian fibration $\mathcal{C} \to \Delta$, such that for every $n \ge 0$, the map $$ \mathcal{C}_ n \to \mathcal{C}_ 1 \times \dotsb \times \mathcal{C}_ 1 $$ induced by $[1] \cong \lbrace i, i+1 \rbrace \subseteq [n]$ is an equivalence of categories.