# Parsing the definition of center of an algebra in a higher-categorical setting

I'm having trouble parsing a definition in Lurie's "Rotation Invariance in Algebraic $K$-Theory". The definition os for the notion of center of an associative algebra object, and occurs in Remark 2.1.3.

The setting is as follows. We have a symmetric monoidal $\infty$-category $\mathcal{C}$. We write $\mathrm{Alg}(\mathcal{C})$ for the $\infty$-category of associative algebra objects in $\mathcal{C}$ and $\mathrm{LMod}(\mathcal{C})$ for the $\infty$-category of left modules in $\mathcal{C}$. Informally, we think of objects in $\mathrm{LMod}(\mathcal{C})$ as pairs $(A, M)$ where $A$ is an associative algebra object and $M$ is a left $A$-module.

Noting that $\mathrm{Alg}(\mathcal{C})$ and $\mathrm{LMod}(\mathcal{C})$ inherit symmetric monoidal structures, we make the following definitions. We define $\mathrm{Alg}^{(2)}(\mathcal{C}) =\mathrm{Alg}(\mathrm{Alg}(\mathcal{C}))$ and $\mathrm{LMod}^{(2)}(\mathcal{C}) = \mathrm{Alg}(\mathrm{LMod}(\mathcal{C}))$. It is known that $\mathrm{Alg}^{(2)}(\mathcal{C})$ is equivalent to the category of $\mathbb{E}_2$-algebra objects. Informally, we think of objects in $\mathrm{LMod}^{(2)}(\mathcal{C})$ as pairs $(A,M)$ where $A$ is an $\mathbb{E}_2$-algebra object and $M$ is an $A$-algebra. We call $\mathrm{LMod}^{(2)}(\mathcal{C})$ the category of central actions in $\mathcal{C}$.

Finally, we arrive at the definition I am stuck on. Fix an associative algebra object $M \in \mathrm{Alg}(\mathcal{C})$. We say that a central action $(A,M) \in \mathrm{LMod}^{(2)}(\mathcal{C})$ exhibits $A$ as a center of $M$ if, for every $\mathbb{E}_2$-algebra $B \in \mathrm{Alg}^{(2)}(\mathcal{C})$, the canonical map $$\mathrm{Map}_{\mathrm{Alg}^{(2)}(\mathcal{C})}(B, A) \to \mathrm{LMod}^{(2)}(\mathcal{C}) \times_{\mathrm{Alg}(\mathcal{C})} \{M\}$$ is a homotopy equivalence.

It is not stated what exactly the canonical map is, but it is probably thought of as follows. The factor $\mathrm{Map}_{\mathrm{Alg}^{(2)}(\mathcal{C})}(B, A) \to \mathrm{LMod}^{(2)}(\mathcal{C})$ sends a map $\phi : B \to A$ of $\mathbb{E}_2$-algebras to the central action $(B,M)$ where $B$ acts via $\phi$. The other factor is crystal clear.

Perhaps I'm not seeing clearly, but I'm not sure how to parse this definition. It seems to be saying the following. Fix an $\mathbb{E}_2$-algebra $B$. The the data of a central action $(R,M)$ of any $\mathbb{E}_2$-algebra $R$ on $M$ is the same as the data of a map $B \to A$ of $\mathbb{E}_2$-algebras.

This doesn't seem quite right to me. Naively translating this into ordinary algebra, it feels quite bizarre. Moreover, Lurie gives another definition for center in Higher Algebra, Definition 5.3.1.6. It is not clear to me that these two definitions are equivalent.

Am I just confused? Does this indeed produce a sensible notion of center of an associative algebra? A very vague informal argument will suffice.

Let us try to figure out what's happening on discrete rings, where $E_2=E_\infty$. The category $\mathrm{LMod}^{(2)}$ is, as you surmised, the category of pairs $(A,B)$ where $A$ is a commutative algebra and $B$ is an associative $A$-algebra (i.e. an algebra object in the monoidal category of $A$-modules). That is we require the multiplication in $B$ to be $A$-bilinear, hence for every elements $b_1,b_2\in B$ and $a_1,a_2\in A$ we have $$(a_1b_1)(a_2b_2)=(a_1a_2)(b_1b_2)$$ You can rewrite this data as a triple $(A,B,\phi)$ where $A$ is a commutative ring, $B$ is an associative ring and $\phi:A\to B$ is a central ring homomorphism, that is a ring homomorphism whose image is contained in the center.

Then for every associative ring $B$ the category $$\mathrm{LMod}^{(2)}\times_{\mathrm{Alg}}\{B\}$$ is simply the category of commutative rings $A$ with a central ring homomorphism to $B$. Lurie is then defining the center of $B$ as the terminal object of this category. It is natural now that this corresponds to the classical notion of the center.

Of course, as soon as you pass to $\infty$-categories more complications arise, in particular the center is not going to be a commutative ring anymore but simply an $E_2$-ring. You can however iterate this procedure and obtain the $E_{n+1}$-center of an $E_n$-ring and so obtain as much commutativity as you want.

EDIT Actually, I believe that the rotation invariance paper has a typo (as you can see the right hand side of the so-called equivalence is independent of $B$ and in fact it's not even an ∞-groupoid). I think that the correct definition is that the canonical map

$$\mathrm{Alg}^{(2)}(\mathcal{C})_{/A}\to \mathrm{LMod}^{(2)}(\mathcal{C})\times_{\mathrm{Alg}(\mathcal{C})}\{B\}$$

sending $[A'\to A]$ to $(A',B)$ where $A'$ acts centrally on $B$ via $A'\to A$ is an equivalence. This is equivalent to the description I gave above.

• thank you for the response! This seems to be closer to the definition given at HA 5.3.1.6, which I agree is a very sensible notion for center. I suppose my confusion is about the connection between the definition given at HA 5.3.1.6 and the definition given in "Rotation Invariance" 2.1.3. One thing I find particulary peculiar is the fact that the definition in "Rotation Invariance" makes reference to a canonical map, where the LHS makes reference to an arbitrary E_2 ring B, but the RHS is independent of B. – Brian Shin Aug 19 '18 at 18:09
• @BrianShin Actually I think that the definition in question in "Rotation invariance" is a typo. I suspect Lurie wanted to give a "fiberwise" definition (comparing $\mathrm{Map}(A',A)$ with the space of central actions of $A'$ on $B$ given by $\{A'\}\times_{\mathrm{Alg}^{(2)}}\mathrm{LMod}^{(2)}\times_{\mathrm{Alg}}\{B\}$) and he changed mind halfway through. – Denis Nardin Aug 19 '18 at 18:38
• this seems probable, and now the equivalence of definitions is apprent. Thank you! – Brian Shin Aug 19 '18 at 18:58