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Let $\mathbf{Top}$ the category of (nice) topological spaces. For any space $Z$, define $\mathbf{End}_{operad}(Z)$ as the endomorphism operad.

Is there always a map of operad $$\mathbf{End}_{operad}(Z\times Z)\rightarrow \mathbf{End}_{operad}(Z) $$

Edit: Intuitively, I would say the answer should be NO i.e. there is NOT ALWAYS such a map of operad. But I don't have a concrete counterexample.

Let $\mathbf{Top}$ be the category of (nice) topological spaces. For any space $Z$, define $\mathbf{End}_{\text{operad}}(Z)$ as the endomorphism operad.

Is there always a map of operads $$\mathbf{End}_{\text{operad}}(Z\times Z)\rightarrow \mathbf{End}_{\text{operad}}(Z) $$

Edit: Intuitively, I would say the answer should be NO i.e. there is NOT ALWAYS such a map of operads. But I don't have a concrete counterexample.

Let $\mathbf{Top}$ the category of (nice) topological spaces. For any space $Z$, define $\mathbf{End}_{operad}(Z)$ as the endomorphism operad.

Given two spaces $X,Y$, is there always a map of operad $$\mathbf{End}_{operad}(X\times Y)\rightarrow \mathbf{End}_{operad}(X) $$

Edit: Intuitively, I would say the answer should be NO i.e. there is NOT ALWAYS such a map of operad. But I don't have a concrete counterexample.

Let $\mathbf{Top}$ the category of (nice) topological spaces. For any space $Z$, define $\mathbf{End}_{operad}(Z)$ as the endomorphism operad.

Is there always a map of operad $$\mathbf{End}_{operad}(Z\times Z)\rightarrow \mathbf{End}_{operad}(Z) $$

Edit: Intuitively, I would say the answer should be NO i.e. there is NOT ALWAYS such a map of operad. But I don't have a concrete counterexample.

Let $\mathbf{Top}$ the category of (nice) topological spaces. For any space $Z$, define $\mathbf{End}_{operad}(Z)$ as the endomorphism operad.

Given two spaces $X,Y$, is there always a map of operad $$\mathbf{End}_{operad}(X\times Y)\rightarrow \mathbf{End}_{operad}(X) $$

Let $\mathbf{Top}$ the category of (nice) topological spaces. For any space $Z$, define $\mathbf{End}_{operad}(Z)$ as the endomorphism operad.

Given two spaces $X,Y$, is there always a map of operad $$\mathbf{End}_{operad}(X\times Y)\rightarrow \mathbf{End}_{operad}(X) $$

Edit: Intuitively, I would say the answer should be NO i.e. there is NOT ALWAYS such a map of operad. But I don't have a concrete counterexample.