Algebras and modules | Dongryul Kim

Operads are confusing because they embody different notions.

Operads are categories with “many-to-one morphisms” structure. In particular, every operad $\mathcal{O}^\otimes$ has an underlying category $\mathcal{O}$.
Operads can be used for prescribing a type of monoidal structure. For every operad $\mathcal{O}^\otimes$, there is a notion of an $\mathcal{O}^\otimes$-monoidal category.
Operads define a notion of an “algebra” inside a symmetric monoidal category. For a symmetric monoidal category $\mathcal{C}$ and an operad, there is a category $\operatorname{Alg}_ \mathcal{O}(\mathcal{C})$ of $\mathcal{O}$-algebra objects in $\mathcal{C}$.

In fact, one can combine (2) and (3). If $\mathcal{O}^{\prime\otimes} \to \mathcal{O}^\otimes$ is morphism of operads, Lurie defines a notion of $\mathcal{O}^\prime$-algebra objects in an $\mathcal{O}$-monoidal category. But let’s not do that for the sake of our sanity.

Algebra objects from operads#

As motivation, let’s contemplate on what unital commutative rings (those rings we use in algebraic geometry) are. We start with the symmetric monoidal category $(\mathsf{Ab}, \otimes)$ of abelian groups with tensor product over $\mathbb{Z}$. A unital commutative ring is an abelian group $A$ together with morphisms $$ u \colon \mathbb{Z} \to A, \quad m \colon A \otimes A \to A $$ satisfying a bunch of axioms. The idea is that the axioms can be encoded neatly by specifying a “space of maps” $A^{\otimes n} \to A$ for each $n \ge 0$.

Definition 1. A unital commutative ring is the data of
an abelian group $A$,
for each finite set $I$, a map $m_I \colon A^{\otimes I} \to A$,
satisfying the conditions that
if $\lvert I \rvert = 1$ then $m_I = \id_A$,
if $f \colon I \to J$ is a map between finite sets then $m_I$ is equal to the composition $$ A^{\otimes I} = \bigotimes_ {j \in J} A^{\otimes f^{-1}(j)} \xrightarrow{\bigotimes m_ {f^{-1}(j)}} \bigotimes_{j \in J} A = A^{\otimes J} \xrightarrow{m_J} A. $$

The point is that $m_ \emptyset$ gives the unit map, $m_{\lbrace 1, 2 \rbrace}$ gives the multiplication map. Then the coherence condition turn out to give the unital axiom, the associativity axiom, and also the commutativity axiom.

This is flexible enough to accommodate other notions of algebras. Let’s try unital associative algebras. For a finite set $I$, let $\operatorname{ord}(I)$ denote the ways of linearly ordering $I$. Note that for $f \colon I \to J$, an ordering of each $J$ plus an ordering of all fibers of $f$ will give an ordering of $I$. So there is a natural map $$ \mathrm{comb} \colon \operatorname{ord}(J) \times \prod_{j \in J}^{} \operatorname{ord}(f^{-1}(j)) \to \operatorname{ord}(I). $$

Definition 2. A unital associative ring is the data of
an abelian group $A$,
for each finite set $I$ and ordering $o \in \operatorname{ord}(I)$, a map $m_{I,o} \colon A^{\otimes I} \to A$,
satisfying the conditions that
if $\lvert I \rvert = 1$ then $m_{I,o} = \id_A$,
if $f \colon I \to J$ is a map between finite sets, $o \in \operatorname{ord}(J)$, and $o_j \in \operatorname{ord}(f^{-1}(j))$, then then $m_ {I, \operatorname{comb}(o, (o_j))}$ is equal to the composition $$ A^{\otimes I} = \bigotimes_ {j \in J} A^{\otimes f^{-1}(j)} \xrightarrow{\bigotimes m_ {f^{-1}(j), o_j}} \bigotimes_{j \in J} A = A^{\otimes J} \xrightarrow{m_ {J,o}} A. $$

If you want to drop the assumption that there is a unit, we got you covered. Let $\operatorname{ord}^\prime(I)$ be the same as $\operatorname{ord}(I)$ when $I$ is nonempty, but $\operatorname{ord}^\prime(\emptyset) = \emptyset$. There is still $$ \mathrm{comb} \colon \operatorname{ord}^\prime(J) \times \prod_{j \in J}^{} \operatorname{ord}^\prime(f^{-1}(j)) \to \operatorname{ord}^\prime(I), $$ which is meaningful only when $f$ is surjective. Now we can literally copy-paste the above definition and replace all instances of $\mathrm{ord}$ with $\mathrm{ord}^\prime$.

Definition 3. A non-unital associative ring is the data of
an abelian group $A$,
for each finite set $I$ and ordering $o \in \operatorname{ord}^\prime(I)$, a map $m_{I,o} \colon A^{\otimes I} \to A$,
satisfying the conditions that
if $\lvert I \rvert = 1$ then $m_{I,o} = \id_A$,
if $f \colon I \to J$ is a map between finite sets, $o \in \operatorname{ord}^\prime(J)$, and $o_j \in \operatorname{ord}^\prime(f^{-1}(j))$, then then $m_ {I, \operatorname{comb}(o, (o_j))}$ is equal to the composition $$ A^{\otimes I} = \bigotimes_ {j \in J} A^{\otimes f^{-1}(j)} \xrightarrow{\bigotimes m_ {f^{-1}(j), o_j}} \bigotimes_{j \in J} A = A^{\otimes J} \xrightarrow{m_ {J,o}} A. $$

Now let’s look at a pair of a unital associative algebra $A$ and a left module $M$ over $A$. Again, we can think about what kind of maps $A^{\otimes n} \otimes M^{\otimes m}$ to $A$ or $M$ there are. Let $C = \lbrace a, m \rbrace$ be the set of “colors.” For a finite set $I$, a map $\varphi \colon I \to C$, and an element $c \in C$, we define $$ \operatorname{Mul}(\varphi, c) = \begin{cases} \operatorname{ord}(I) & c = a, \lvert \varphi^{-1}(m) \rvert = 0, \br \operatorname{ord}(\varphi^{-1}(a)) & c = m, \lvert \varphi^{-1}(m) \rvert = 1, \br \emptyset & \text{otherwise.} \end{cases} $$ You can probably guess how composition works. The point is that given a colored operad as such, we can define an algebra object as follows.

Definition 4. Let $\mathcal{O}$ be a colored operad and $\mathcal{C}$ be a symmetric monoidal category. An $\mathcal{O}$-algebra object in $\mathcal{C}$ is the data of
an object $A_X \in \mathcal{C}$ for each color $X \in \mathcal{O}$,
for each $I$ a finite set, $(X_i)_ {i in I} \in \mathcal{O}^I$, $Y \in \mathcal{O}$, and $m \in \operatorname{Mul}_ \mathcal{O}((X_i)_ {i \in I}, Y)$, a map $$ \bigotimes_{i \in I} X_i \to Y, $$
satisfying
for $\lvert I \rvert = 1$, $X = Y$, and $m = \id_X$, the corresponding map is the identity map $X \to X$,
composition behave as we expect.

In the ∞-version of the story,

there is a space of colors rather than a set of colors,
the set $\mathrm{Mul}$ of “many-to-one” maps is replaced with a space of such maps,
the composition laws hold up to coherent homotopy.

But the essence is the same.

CoCartesian formalism#

Since this is an ∞-category seminar, I feel obligated to at least state the actual definition of the category of algebra objects. Recall how an operad was defined last time. The idea was to package all information about $\mathsf{Mul}_ \mathcal{O}$ into a single category lying over $N(\mathsf{Fin}_ \ast)$.

Definition 5 (HA 2.1.1.10, paraphrased). An ∞-operad is a functor $\mathcal{O}^\otimes \to N(\mathsf{Fin}_\ast)$ such that
inert maps in $N(\mathsf{Fin}_\ast)$ have all coCartesian lifts,
the coCartesian lifts over $\langle n \rangle \to \langle 1 \rangle$ induce an equivalence $\mathcal{O}_ {\langle n \rangle}^\otimes \simeq \mathcal{O}^n$,
for $(C_i)_ {i \in I^\circ} \in \mathcal{O}_ I^{\otimes}$, $(C_j^\prime)_ {j \in J^\circ} \in \mathcal{O}_ J^\otimes$, and $f \colon I \to J$, a map $(C_i) \to (C_j^\prime)$ lying over $f$ is the same as a collection of maps $$ (C_i)_ {i \in f^{-1}(j)} \to C_j $$ for $j \in J^\circ$.

Of course, the mapping spaces $(C_i)_ {i \in I} \to C_j$ should be thought of as the sets $\mathsf{Mul}_ \mathcal{O}$ in the colored operad setting. Once we have this definition, it was easy to define a symmetric monoidal category. Here, the rough idea is that in a colored operad, if all $$ \operatorname{Mul}_ \mathcal{O}((X_i)_ {i \in I}, -) $$ are corepresentable, we may define $\bigotimes_{i \in I} X_i$ to be this corepresenting object.

Definition 6 (HA 2.0.0.7). A symmetric monoidal category is an operad $\mathcal{C}^\otimes \to N(\mathsf{Fin}_ \ast)$ that is a coCartesian fibration.

Now, given an operad $\mathcal{O}^\otimes$ and a symmetric monoidal category $\mathcal{C}^\otimes$, how do we define an $\mathcal{O}$-algebra object in $\mathcal{C}$? For every color $X \in \mathcal{O}$, we need an object of $\mathcal{C}$, so we need at least a functor $F \colon \mathcal{O} \to \mathcal{C}$. But then, if we decide to send the object $(X_i)_ {i \in I} \in \mathcal{O}^\otimes$ to $(F(X_i))_ {i \in I} \in \mathcal{C}^\otimes$, the algebra structure maps $$ \operatorname{Mul}_ \mathcal{O}((X_i), Y) \to \Hom_\mathcal{C}({\textstyle \bigotimes_{i \in I}} X_i, Y) $$ define a functor $\mathcal{O}^\otimes \to \mathcal{C}^\otimes$ lying over $N(\mathsf{Fin}_\ast)$.

Definition 7. Let $\mathcal{O}^\otimes$ be an operad and $\mathcal{C}^\otimes$ be a symmetric monoidal category. An $\mathcal{O}$-algebra object in $\mathcal{C}$ is a functor $\mathcal{O}^\otimes \to \mathcal{C}^\otimes$ such that $$ \begin{CD} \mathcal{O}^\otimes @>>> \mathcal{C}^\otimes \br @VVV @VVV \br N(\mathsf{Fin}_ \ast) @= N(\mathsf{Fin}_ \ast) \end{CD} $$ commutes, and
coCartesian lifts of inert maps in $\mathsf{Fin}_\ast$ are sent to coCartesian morphisms.
The category of $\mathcal{O}$-algebra objects in $\mathcal{C}$ is just the full subcategory $\mathsf{Alg}_ \mathcal{O}(\mathcal{C}) \subseteq \mathsf{Fun}_ {N(\mathsf{Fin}_ \ast)}(\mathcal{O}^\otimes, \mathcal{C}^\otimes)$ of those functors satisfying the above condition.

Little cubes operads#

We have already seen some examples of operads. Here is one important class. Consider the closed $n$-disk $D^n \subset \mathbb{R}^n$. Denote by $$ \operatorname{Mul}_ {\mathbb{E}_ n}(I, \ast) $$ the space of embeddings $D^n \times I \to D^n$ where each $D^n \to D^n$ is scaling composed with translation. This space will be a nice locally closed subspace of $\mathbb{R}^{(n+1) \lvert I \rvert}$. Given a map $f \colon I \to J$ of finite sets, there is a natural continuous composition map $$ \operatorname{Mul}_ {\mathbb{E}_ n}(J, \ast) \times \prod_{j \in J}^{} \operatorname{Mul}_ {\mathbb{E}_ n}(f^{-1}(j), \ast) \to \operatorname{Mul}_ {\mathbb{E}_ n}(I, \ast) $$ given by composing the embeddings. So this defines a “topological” colored operad with a single color.

Definition 8. For $0 \le n \lt \infty$, the operad $\mathbb{E}_ n^\otimes$ is defined as the homotopy coherent nerve of the topological operad above. For $n = \infty$, the operad $\mathbb{E}_ \infty^\otimes$ is defined as the identity functor $N(\mathsf{Fin}_ \ast) \to N(\mathsf{Fin}_ \ast)$.

This is actually something not so crazy.

Proposition 9. Let $\mathcal{C}$ be a symmetric monoidal $1$-category.
An $\mathbb{E}_0$-algebra is the same thing as an object $A \in \mathcal{C}$ together with a unit map $1 \to A$.
An $\mathbb{E}_1$-algebra is the same thing as a unital associative algebra.
For $2 \le n \le \infty$, an $\mathbb{E}_n$-algebra is the same thing as a unital commutative algebra.

Here is a more interesting fact. The category $\mathcal{S}$ of spaces has a Cartesian symmetric monoidal structure $(\mathcal{S}, \times)$, where tensor product is the Cartesian product. Given an $\mathbb{E}_ n$-algebra $X$, with $n \ge 1$, there is a natural monoid structure on $\pi_0(X)$.

Theorem 10 (May recognition theorem). The construction $X \mapsto \Omega^n X$ defines an equivalence $$ \mathcal{S}_ {\ast, \ge n} \simeq \mathsf{Alg}_ {\mathbb{E}_ n}^\mathrm{gp}(\mathcal{S}), $$ where the left hand side denotes the full subcategory of pointed space with $\pi_{\le n-1} = 0$, and the right hand side denotes the full subcategory of $\mathbb{E}_ n$-algebras for which $\pi_0$ is a group. When $n = \infty$, we get an equivalence $$ \mathsf{Sp}_ {\ge 0} \simeq \varprojlim (\mathcal{S}_ {\ast, \ge 0} \xleftarrow{\Omega} \mathcal{S}_ {\ast, \ge 1} \xleftarrow{\Omega} \dotsb) \simeq \mathsf{Alg}_ {\mathbb{E}_ \infty}^\mathrm{gp}(\mathcal{S}) $$ where $\mathsf{Sp}_ {\ge 0}$ is the full subcategory of spectra $X$ with $\pi_n X = 0$ when $n \lt 0$.

There is a sequence $$ \mathbb{E}_ 1 \to \mathbb{E}_ 2 \to \dotsb \to \mathbb{E}_ \infty $$ of operads. A consequence is that if $n \le m$ then an $\mathbb{E}_ n$-algebra naturally has a structure of an $\mathbb{E}_ m$-algebra.

Also, when $\mathcal{C}$ is a symmetric monoidal category, we will write $$ \mathsf{Alg}_ {\mathbb{E}_ \infty}(\mathcal{C}) = \mathsf{CAlg}(\mathcal{C}). $$

Limits and colimits of algebras#

Let $\mathcal{O}$ be an operad and $\mathcal{C}$ be a symmetric monoidal category. Let’s first talk about limits and colimits in the category $\mathsf{Alg}_ \mathcal{O}(\mathcal{C})$.

Proposition 11 (HA 3.2.2.1). Let $f \colon K \to \mathsf{Alg}_ \mathcal{O}(\mathcal{C})$ be a diagram, and assume that for every $X \in \mathcal{O}$, the induced functor $f_X \colon K \to \mathcal{C}$ has a limit. Then
$f$ has a limit, and
an extension $$ \bar{f} \colon K^\vartriangleleft \to \mathsf{Alg}_ \mathcal{O}(\mathcal{C}) $$ of $f$ is a limit if and only if $\bar{f}_ X \colon K^\vartriangleleft \to \mathcal{C}$ is a limit for every $X \in \mathcal{O}$.

Colimits are much harder to compute. The idea is that we can treat sifted colimits and finite coproducts, and then combine the two together in the general case. Surprisingly, Lurie gives up some degree of generality and starts talking about only $\mathcal{O}$-algebra objects in an $\mathcal{O}$-monoidal category. But because we don’t want to talk about $\mathcal{O}$-monoidal categories, let’s assume $\mathcal{O} = \mathbb{E}_ \infty$.

Proposition 12 (HA 3.2.3.3, 3.2.4.7). Let $\mathcal{C}$ be a symmetric monoidal category, and let $\kappa$ be an uncountable regular cardinal. Assume that $\mathcal{C}$ admits $\kappa$-small colimits and $X \otimes -$ preserves $\kappa$-small colimits for all $X \in \mathcal{C}$. Then the following are true.
The category $\mathsf{CAlg}(\mathcal{C})$ has $\kappa$-small colimits.
When $K$ is sifted and $\kappa$-small, a diagram $K^\vartriangleright \to \mathsf{CAlg}(\mathcal{C})$ is a colimit diagram if and only if the composition $K^\vartriangleright \to \mathsf{CAlg}(\mathcal{C}) \to \mathcal{C}$ is.
Finite coproducts in $\mathsf{CAlg}(\mathcal{C})$ are given by $\coprod_i A_i = \bigotimes_i A_i$.

Module objects#

We now define a notion of a module over an $\mathcal{O}$-algebra in an $\mathcal{O}$-monoidal category. When $\mathcal{O}$ is something called a coherent operad, the category of modules is supposed to have an extra structure of an operad. I really couldn’t parse the definition of this category for the life of me, even though Lurie calls it “fairly straightforward.” Let me only state the result of the construction.

Definition 13 (HA 3.3.3.8). Let $\mathcal{C}$ be a symmetric monoidal ∞-category. There is an ∞-category $\mathsf{Mod}(\mathcal{C})^\otimes$ with a categorical fibration $$ \mathsf{Mod}(\mathcal{C})^\otimes \to N(\mathsf{Fin}_ \ast) \times \mathsf{CAlg}(\mathcal{C}) $$ such that for each $A \in \mathsf{CAlg}(\mathcal{C})$ the fiber product $$ \mathsf{Mod}_ A(\mathcal{C})^\otimes = \mathsf{Mod}(\mathcal{C})^\otimes \times_{\mathsf{CAlg}(\mathcal{C})} \lbrace A \rbrace \to N(\mathsf{Fin}_ \ast) $$ is an ∞-operad.

Morally, this should have objects $$ (A, (M_i)_ {i \in I}) $$ where $I$ is a finite set, $A$ is a commutative algebra, and $M_i$ are $A$-modules. A morphism $$ (A, (M_i)_ {i \in I}) \to (B, (N_j)_ {j \in J}) $$ should be given by

an algebra homomorphism $f \colon A \to B$,
a partially defined map $\varphi \colon I \to J$,
for each $j \in J$ a map $g_j \colon \bigotimes_{i \in \varphi^{-1}(j)} M_i \to N_j$,
satisfying the condition that the diagrams $$ \begin{CD} A \otimes \bigotimes_{i \in \varphi^{-1}(j)} M_i @>{f \otimes g_j}>> B \otimes N_j \br @VVV @VVV \br \bigotimes_{i \in \varphi^{-1}(j)} M_i @>{g_j}>> N_j \end{CD} $$ commute, where the left vertical map is any one of the $\lvert f^{-1}(j) \rvert$ possible multiplication maps.

Unfortunately, it is not true in general that $\mathsf{Mod}_ A(\mathcal{C})$ is a symmetric monoidal category, because there is no way of building $M \otimes_A N$ from the absolute tensor product. But we can still talk about $A$-multilinear maps, so that we at least have an operad.

Generalities on modules#

We state some facts about modules and commutative algebras in a symmetric monoidal category.

Proposition 14 (HA 3.4.1.7, 3.4.1.9). Let $\mathcal{C}$ be a symmetric monoidal category, and let $A \in \mathsf{CAlg}(\mathcal{C})$. Then there is a canonical equivalence $$ \mathsf{CAlg}(\mathsf{Mod}_ A(\mathcal{C})) \simeq \mathsf{CAlg}_ {A/}. $$ Moreover, if $\bar{B}$ on the left hand side corresponds to $B \in \mathsf{CAlg}(\mathcal{C})$ on the right hand side, then there is an equivalence $$ \mathsf{Mod}_ {\bar{B}}(\mathsf{Mod}_ A(\mathcal{C})) \simeq \mathsf{Mod}_ B(\mathcal{C}) $$ of operads.

This is supposed to be a bit confusing because $\mathsf{Mod}_ A(\mathcal{C})$ is only an operad. But we can still define algebra objects in an operad, so this makes sense.

Proposition 15 (HA 3.4.3.6). Let $\mathcal{C}$ be a symmetric monoidal category that admits $K$-indexed limits.
For every $A \in \mathsf{CAlg}(\mathcal{C})$ the category $\mathsf{Mod}_ A(\mathcal{C})$ admits $K$-indexed limits.
A diagram $K^\vartriangleleft \to \mathsf{Mod}_ A(\mathcal{C})$ is a limit if and only if the induced $K^\vartriangleleft \to \mathcal{C}$ is a limit.

Definition 16. A presentable symmetric monoidal category is a symmetric monoidal category $\mathcal{C}$ such that
$\mathcal{C}$ is presentable,
the tensor product preserves all small colimits.

Theorem 17 (HA 3.4.4.2). If $\mathcal{C}$ is a presentable symmetric monoidal category and $A \in \mathsf{CAlg}(\mathcal{C})$, then the operad $\mathsf{Mod}_ A(\mathcal{C})$ is a presentable symmetric monoidal category.