# In what sense is the 'Yoneda embedding' of an $A_\infty$-category an embedding?

$$\def\A{\mathcal{A}} \def\J{\mathcal{J}}$$I am reading P. Seidel, Fukaya Categories and Picard-Lefschetz Theory, and in (1l) he defines the Yoneda embedding of a (non-unital) A$$_\infty$$-category $$\A$$ to be a certain (non-unital) A$$_\infty$$-functor $$\mathcal{J}:\A\to\operatorname{nu-mod}(\A)$$ (where $$\operatorname{nu-mod}(\A)$$ is the A$$_\infty$$-category of non-unital $$\A$$-modules).

I will copy here the definition of $$\mathcal{J}$$ as explained in Dirichlet Branes and Mirror Symmetry, 8.3.4, p. 612 (the notation has been adapted to Seidel's one):

We can now define something known as the Yoneda embedding, an A$$_{\infty}$$-functor $$\J:\mathcal{A} \rightarrow \operatorname{nu-mod}(\mathcal{A})$$. This has a simple definition, namely for $$Y \in \operatorname{Ob}(\mathcal{A})$$, we set $$\J(Y)$$ to be the module $$X \mapsto \operatorname{Hom}_{\mathcal{A}}(X, Y)$$. Of course, the latter is already a chain complex, coming from $$\mu^1_{\mathcal{A}}$$, and to define the module structure, we need to provide maps as in (8.9). However, as in this case $$\mathcal{M}(X_{d-1})=\operatorname{Hom}_{\mathcal{A}}(X_{d-1}, Y)$$ and $$\mathcal{M}\left(X_0\right)=\operatorname{Hom}_{\mathcal{A}}\left(X_0, Y\right)$$, these maps are just given by $$\mu^d_{\mathcal{A}}$$. So this gives the functor Yon as a map from objects of $$\mathcal{A}$$ to objects of $$\operatorname{nu-mod}(\mathcal{A})$$; however, we have to define the higher maps, $$\J^d: \operatorname{Hom}_{\mathcal{A}}(X_{d-1}, X_d) \otimes \cdots \otimes \operatorname{Hom}_{\mathcal{A}}(X_0, X_1) \rightarrow\\ \operatorname{Hom}_{\operatorname{nu-mod}(\mathcal{A})}\left(\J\left(X_0\right), \J\left(X_d\right)\right)[1-d] \text {. }$$

So given $$a_1, \ldots, a_d$$, we need to give a pre-homomorphism, i.e., a collection of maps $$\begin{gathered} t^n: \J(X_0)(Y_{n-1}) \otimes \operatorname{Hom}_{\mathcal{A}}(Y_{n-2}, Y_{n-1}) \otimes \cdots \otimes \operatorname{Hom}_{\mathcal{A}}(Y_0, Y_1) \rightarrow \\ \J\left(X_d\right)\left(Y_0\right)\left[2-d-n+\sum |a_i|\right], \end{gathered}$$ i.e., \begin{aligned} (\J^d)^n: \operatorname{Hom}_{\mathcal{A}}\left(Y_{n-1}, X_0\right) & \otimes \operatorname{Hom}_{\mathcal{A}}\left(Y_{n-2}, Y_{n-1}\right) \otimes \cdots \otimes \operatorname{Hom}_{\mathcal{A}}\left(Y_0, Y_1\right) \rightarrow \\ & \operatorname{Hom}_{\mathcal{A}}(Y_0, X_d)\left[2-d-n+\sum |a_i|\right]. \end{aligned}

There is an obvious choice, i.e., $$\left(\J^d\left(a_d, \ldots, a_1\right)\right)^n(b, b_{n-1}, \ldots, b_1)=\mu^{d+n}_{\mathcal{A}}(a_d, \ldots, a_1, b, b_{n-1}, \ldots, b_1).$$

My questions are:

1. Why is the Yoneda embedding an 'embedding'?

2. Is there a definition of an 'embedding of A$$_\infty$$-categories'? If so, does the Yoneda embedding satisfy this definition?

I guess an answer for 2 counts as an answer for 1.

An A$$_\infty$$-category $$\A$$ is said to be c-unital if $$H(\A)$$ is a unital category. A non-unital A$$_\infty$$-functor between non-unital A$$_\infty$$-categories $$\mathcal{F}:\mathcal{A}\to\mathcal{B}$$ is said to be cohomologically full and faithful if $$H(\mathcal{F}):H(\mathcal{A})\to H(\mathcal{B})$$ is full and faithful (Seidel, (1b)). It turns out that $$\J$$ is cohomologically full and faithful whenever $$\A$$ is c-unital (Seidel, Corollary 2.13). Is this the sense in which $$\J$$ is an embedding? (So maybe a more appropriate name for $$\J$$ would be the “Yoneda cohomological embedding.”)

Since you already understand the (c-)unital case, any failure of $$\mathcal J$$ to be an embedding is not going to come from the use of derived or $$A_\infty$$ algebras, but rather from the nonunitality. So it is a good strategy to think about the nonunital but associative / underived algebras.

Suppose that $$A$$ is a nonunital algebra. You can think of it as a nonunital category with one object, if you like. The Yoneda embedding, in this case, is the rank-1 free module: $$A$$ acting on itself by (right, say) multiplication. I'll call it $$A_A$$. Associativity supplies a nonunital homomorphism $$\mathcal J : A \to \mathrm{End}_A (A_A)$$.

If $$A$$ were unital, then this $$\mathcal J$$ would be an isomorphism. That is the full-faithfulness of the Yoneda embedding.

But if $$A$$ is nonunital, then $$\mathcal{J}$$ is definitely not surjective: $$\mathrm{End}_A(A_A)$$ is a unital algebra! $$\mathcal{J}$$ can also fail to be injective. Consider, for example, the zero multiplication, in which you declare that $$ab = 0$$ for any $$a,b \in A$$. This is a perfectly good associative, but nonunital, multiplication. Then $$\mathrm{End}_A(A_A) = \mathrm{End}(A)$$ is all linear maps from $$A$$ to itself, and $$\mathcal{J} : A \to \mathrm{End}_A(A_A)$$ is the zero map.

Since $$\mathcal{J}$$ typically fails to be any sort of "embedding" in the underived case, it will also fail to be an "embedding" in the more general derived-algebra setting. (Except if you are the type of formal person who decides to resolve every morphism to an injection, I guess that's sort of an "embedding", but then you have removed all meaning from the word "embedding.")

In the nonunital world, the correct term is simply "Yoneda functor" and not "Yoneda embedding".

• Thank you so much! I guess then that the name "Yoneda embedding" might come from the formal similarity with the Yoneda (true) embedding of a unital 1-cat (or of a unital dg-cat). Commented May 10 at 6:08