# 6.5 Operads and their algebras#

## 6.5.2 Operads from symmetric monoidal categories#

### Exercise 6.96#

For part 1.:

For part 2.:

For part 3. see part (iii) of Rough Definition 6.91, which is more detailed than part (iii) of Example 6.94. In this case we have $$i=1$$. The variable $$i$$ can be interpreted as โthe index where we want to substitute the first tupleโ (assuming the variable $$s$$ is used to mean โsubstitutedโ) so that:

$โ_1: ๐(s_1,...,s_m;t_1)ร๐(t_1,...,t_n;t)โ๐(s_1,...,s_m,t_2,...,t_n;t)$

Weโll have to assume:

\begin{split} \begin{align} f & โ ๐(s_1,...,s_m;t_1) = \textbf{Cospan}(2,2;2) \\ g & โ ๐(t_1,...,t_n;t) = \textbf{Cospan}(2,2,2;0) \end{align} \end{split}

So that:

\begin{align} g โ_1 f โ ๐(s_1,...,s_m,t_2,...,t_n;t) = \textbf{Cospan}(2,2,2,2;2) \end{align}

We must make this assumption because the definition in part (iii) of Definition 6.91 is unclear about order, and because the opposite substitution just doesnโt work. This assumption also fits Example 6.93, where the author is also using the variable names $$f$$ and $$g$$ (suggesting that is also what he is looking at). In Example 6.93, notice the author flips the order of $$f$$ and $$g$$ between the equation that defines $$g โ_i f$$ and the sentence below it.

Turning the cospans on their sides to compute the composite in the connected-components style of (6.47) (and adding colors only to help visually distinguish connected components):

For part 4.: