Colored operads as finitely essentially algebraic theory.

Question

I call a planar operad what is also called planar (multi-)coloured operad or multicategory and symmetric operad a symmetric multicategory or symmetric (multi-)colored operad.

I have two questions regarding them.

1. I read that a theory of planar operad can be (finitely) essentially algebraic. This would imply that the category of small planar operads is finitely presentable and, in particular, it is small (co)complete.

However, I cannot figure out how to describe such a theory, in order for it to be essentially algebraic. In particular, I don't understand how to cope with the domain symbol (if it has to be present). Is there any reference or can anybody suggest me a way?

(Notice that the description presented here is not even a theory, formally).

2. What about the theory of symmetric operads?

Answer 1

I will use the language of logic rather than sketches.

• There is a sort $O$ of objects/colours.
• For each natural number $n$, there is a sort $A_n$ of $n$-ary operations.
• We have function symbols $\operatorname{dom}^n_k : A_n \to O$ for each $n$ and $1 \le k \le n$.
• We have a function symbol $\operatorname{codom}^n : A_n \to O$ for each $n$.
• We have a function symbol $u : O \to A_1$.
• We have a relation symbol $c_{m, n_1, \ldots, n_m} : A_m \times A_{n_1} \times \cdots \times A_{n_m} \times A_{n_1 + \cdots + n_m}$ for each $m$ and each $n_1, \ldots, n_m$.
• (For symmetric operads only.) We have a function symbol $\sigma : A_n \to A_n$ for each $\sigma \in \mathrm{Sym}(n)$.

There are lots of axioms (or more precisely, axiom schemes):

• $\operatorname{dom}^1_1 (u (t)) = t$ and $\operatorname{codom}^1 (u (t)) = t$.
• For any $(f, g_1, \ldots, g_m)$ (of sort $A_m \times A_{n_1} \times \cdots \times A_{n_m}$), if $$\operatorname{dom}^m_k (f) = \operatorname{codom}^{n_k} (g_k)$$ then there is a unique $h$ (of sort $A_{n_1 + \cdots + n_m}$) such that $c_{m, n_1, \ldots, n_m} (f, g_1, \ldots, g_m, h)$.
• $c_{m, n_1, \ldots, n_m} (f, g_1, \ldots, g_m, h)$ only if the following conditions hold: $$\operatorname{dom}^m_k (f) = \operatorname{codom}^{n_k} (g_k)$$ $$\operatorname{codom}^{n_1 + \cdots + n_m} (h) = \operatorname{codom}^m (f)$$ $$\operatorname{dom}^{n_1 + \cdots + n_m}_{n_1 + \cdots + n_k + j} = \operatorname{dom}^{n_{k+1}}_j (g_{k+1})$$
• Unit and associative laws for ($u$ and) $c$.
• (For symmetric operads only.) For all $\sigma \in \mathrm{Sym}(n)$, $$\operatorname{dom}^n_{\sigma (k)} (\sigma (f)) = \operatorname{dom}^n_{k} (f)$$
• (For symmetric operads only.) For the identity $e_n \in \mathrm{Sym}_n$, $$e_n (f) = f$$ and for all $\sigma \in \mathrm{Sym}(n)$ and $\tau \in \mathrm{Sym}(n)$, $$\sigma (\tau (f)) = (\sigma \tau) (f)$$
• (For symmetric operads only.) A compatibility condition between $\sigma$ and $c$.

It is all straightforward, honest!

Answer 2

This is a note on Zhen Lin's answer, too long to be a comment.

According to J. Adámek, J. Rosicky, "Locally_Presentable and Accessible Categories"

Definition 3.34. An essentially algebraic theory is a quadruple $\Gamma = (\Sigma, E, \Sigma_t, \text{Def})$ consisting on a many-sorted signature $\Sigma$ of algebras (so, no relation symbols), a set $E$ of $\Sigma$-equations, a set $\Sigma_t\subseteq \Sigma$ of "total" operation symbols, and a function $\text{Def}$ assigning to each "partial" operation symbol $\sigma\colon \prod_{i \in I} s_i \to s$ in $\Sigma\setminus \Sigma_t$ a set $\text{Def}(\sigma)$ of $\Sigma_t$-equations in the standard variables $x_i \in V_i$ ($i \in I$).

A essentially algebraic theory $\Gamma$ is $\lambda$-ary, for a regular cardinal $\lambda$, provided that $\Sigma$ is $\lambda$-ary (i.e., each operation symbol has arity less than $\lambda$), each of the equation of $E$ and $\text{Def}(\sigma)$ uses less than $\lambda$ standard variables, and each $\text{Def}(\sigma)$ contains less than $\lambda$ equations.

Theorem 3.36. A category is locally $\lambda$-presentable iff it is equivalent to the category of models of a $\lambda$-ary essentially algebraic theory.

In the case of operads, $\Sigma$ is the signature described with care in Zhen Lin'answer but for the relation symbols $c_{m, n_1, \dots, n_m}$, which become "partial" operation symbols $c_{m, n_1, \dots, n_m}\colon A_m\times A_{n_1}\times\dots \times A_{n_M}\to A_{n_1+\dots, n_m}$. The second axiom scheme (in Zhen Lin's post) corresponds to the set $\text{Def}(c_{m, n_1, \dots, n_m})$, while the other axioms correspond to the elements (equations) of $E$.

In particular, the category of (planar or symmetric) operads is locally finitely presentable and thus bicomplete.