In Coalgebras in symmetric monoidal categories of spectra, Péroux and Shipley prove the following (Lemma 2.4):

Let $\mathcal{C}=\mathsf{Sets},\mathsf{Top}$, or $\mathsf{sSets}$. The free basepoint functor $$(-)^+\colon(\mathcal{C},\times,\mathrm{pt})\to(\mathcal{C}_*,\wedge,S^0)$$ lifts to an equivalence of categories $$(-)^+\colon\mathsf{CoMon}(\mathcal{C})\dashrightarrow\mathsf{CoMon}(\mathcal{C}_*).$$ As such, any comonoid in $\mathcal{C}_*$ is isomorphic to one of the form $(C^+,\Delta^+_{C},\epsilon^+_C)$ with $(C,\Delta_{C},\epsilon_C)$ a comonoid in $\mathcal{C}$.

• Is this result valid in general for $\mathcal{C}$ a bicomplete Cartesian closed symmetric monoidal category, where $(\mathcal{C}_*,\wedge,S^0)$ is now the symmetric monoidal category in Riehl, Categorical Homotopy Theory, Construction 3.3.14?
• Let $\mathcal{C}$ be a bicomplete Cartesian closed symmetric monoidal $\infty$-category.
  1. Can we similarly build a symmetric monoidal $\infty$-category structure $(\wedge,S^0)$ on the category $\mathcal{C}_*$ of pointed objects of $\mathcal{C}$ (i.e. the coslice $\infty$-category $\mathcal{C}_{*/}$)?
  2. If this is the case, do we have an analogue of Péroux–Shipley's result for $\mathbb{E}_{k}$-comonoids in $(\mathcal{C}_*,\wedge,S^0)$, where $1\leq k\leq\infty$?
  3. (If Item 1 fails, is Item 2 nevertheless true for $\mathcal{C}=\mathcal{S}$, the $\infty$-category of spaces?)

In Coalgebras in symmetric monoidal categories of spectra, Péroux and Shipley prove the following (Lemma 2.4):

Let $\mathcal{C}=\mathsf{Sets},\mathsf{Top}$, or $\mathsf{sSets}$. The free basepoint functor $$(-)^+\colon(\mathcal{C},\times,\mathrm{pt})\to(\mathcal{C}_*,\wedge,S^0)$$ lifts to an equivalence of categories $$(-)^+\colon\mathsf{CoMon}(\mathcal{C})\dashrightarrow\mathsf{CoMon}(\mathcal{C}_*).$$ As such, any comonoid in $\mathcal{C}_*$ is isomorphic to one of the form $(C^+,\Delta^+_{C},\epsilon^+_C)$ with $(C,\Delta_{C},\epsilon_C)$ a comonoid in $\mathcal{C}$.

• Is this result valid in general for $\mathcal{C}$ a bicomplete Cartesian closed symmetric monoidal category, where $(\mathcal{C}_*,\wedge,S^0)$ is now the symmetric monoidal category in Riehl, Categorical Homotopy Theory, Construction 3.3.14?
• Let $\mathcal{C}$ be a bicomplete Cartesian closed symmetric monoidal $\infty$-category.
  1. Can we similarly build a symmetric monoidal $\infty$-category structure $(\wedge,S^0)$ on the category $\mathcal{C}_*$ of pointed objects of $\mathcal{C}$ (i.e. the coslice $\infty$-category $\mathcal{C}_{*/}$)?
  2. If this is the case, do we have an analogue of Péroux–Shipley's result for $\mathbb{E}_{k}$-comonoids in $(\mathcal{C}_*,\wedge,S^0)$, where $1\leq k\leq\infty$?
  3. (If Item 1 fails, is Item 2 nevertheless true for $\mathcal{C}=\mathcal{S}$, the $\infty$-category of spaces?)

In Coalgebras in symmetric monoidal categories of spectra, Péroux and Shipley prove the following (Lemma 2.4):

Let $\mathcal{C}=\mathsf{Sets},\mathsf{Top}$, or $\mathsf{sSets}$. The free basepoint functor $$(-)^+\colon(\mathcal{C},\times,\mathrm{pt})\to(\mathcal{C}_*,\wedge,S^0)$$ lifts to an equivalence of categories $$(-)^+\colon\mathsf{CoMon}(\mathcal{C})\dashrightarrow\mathsf{CoMon}(\mathcal{C}_*).$$ As such, any comonoid in $\mathcal{C}_*$ is isomorphic to one of the form $(C^+,\Delta^+_{C},\epsilon^+_C)$ with $(C,\Delta_{C},\epsilon_C)$ a comonoid in $\mathcal{C}$.

• Is this result valid in general for $\mathcal{C}$ a bicomplete Cartesian closed symmetric monoidal category, where $(\mathcal{C}_*,\wedge,S^0)$ is now the symmetric monoidal category in Riehl, Categorical Homotopy Theory, Construction 3.3.14?
• Let $\mathcal{C}$ be a bicomplete Cartesian closed symmetric monoidal $\infty$-category.
  1. Can we similarly build a symmetric monoidal $\infty$-category structure $(\wedge,S^0)$ on the category $\mathcal{C}_*$ of pointed objects of $\mathcal{C}$ (i.e. the coslice $\infty$-category $\mathcal{C}_{*/}$)?
  2. If this is the case, do we have an analogue of Péroux–Shipley's result for $\mathbb{E}_{k}$-comonoids in $(\mathcal{C}_*,\wedge,S^0)$, where $1\leq k\leq\infty$?
  3. (If Item 1 fails, is Item 2 nevertheless true for $\mathcal{C}=\mathcal{S}$, the $\infty$-category of spaces?)

In Coalgebras in symmetric monoidal categories of spectra, Péroux and Shipley prove the following (Lemma 2.4):

Let $\mathcal{C}=\mathsf{Sets},\mathsf{Top}$, or $\mathsf{sSets}$. The free basepoint functor $$(-)^+\colon(\mathcal{C},\times,\mathrm{pt})\to(\mathcal{C}_*,\wedge,S^0)$$ lifts to an equivalence of categories $$(-)^+\colon\mathsf{CoMon}(\mathcal{C})\dashrightarrow\mathsf{CoMon}(\mathcal{C}_*).$$ As such, any comonoid in $\mathcal{C}_*$ is isomorphic to one of the form $(C^+,\Delta^+_{C},\epsilon^+_C)$ with $(C,\Delta_{C},\epsilon_C)$ a comonoid in $\mathcal{C}$.

• Is this result valid in general for $\mathcal{C}$ a bicomplete Cartesian closed symmetric monoidal category, where $(\mathcal{C}_*,\wedge,S^0)$ is now the symmetric monoidal category in Riehl, Categorical Homotopy Theory, Construction 3.3.14?
• Let $\mathcal{C}$ be a bicomplete Cartesian closed symmetric monoidal $\infty$-category.
  1. Can we similarly build a symmetric monoidal $\infty$-category structure $(\wedge,S^0)$ on the category $\mathcal{C}_*$ of pointed objects of $\mathcal{C}$ (i.e. the coslice $\infty$-category $\mathcal{C}_{*/}$)?
  2. If this is the case, do we have an analogue of Péroux–Shipley's result for $\mathbb{E}_{k}$-comonoids in $(\mathcal{C}_*,\wedge,S^0)$, where $1\leq k\leq\infty$?
  3. (If Item 1 fails, is Item 2 nevertheless true for $\mathcal{C}=\mathcal{S}$, the $\infty$-category of spaces?)

In Coalgebras in symmetric monoidal categories of spectra, Péroux and Shipley prove the following (Lemma 2.4):

Let $\mathcal{C}=\mathsf{Sets},\mathsf{Top}$, or $\mathsf{sSets}$. The free basepoint functor $$(-)^+\colon(\mathcal{C},\times,\mathrm{pt})\to(\mathcal{C}_*,\wedge,S^0)$$ lifts to an equivalence of categories $$(-)^+\colon\mathsf{CoMon}(\mathcal{C})\dashrightarrow\mathsf{CoMon}(\mathcal{C}_*).$$ As such, any comonoid in $\mathcal{C}_*$ is isomorphic to one of the form $(C^+,\Delta^+_{C},\epsilon^+_C)$ with $(C,\Delta_{C},\epsilon_C)$ a comonoid in $\mathcal{C}$.

Originally, this question asked if there's an analogue of Péroux–Shipley's result for $\mathbb{E}_{k}$-comonoids in $(\mathcal{S}_*,\wedge,S^0)$, for $1\leq k\leq\infty$.

At least in its most basic sense (i.e. that $\mathbb{E}_{k}$-comonoids in $\mathcal{S}_*$ are just pointed spaces), there isn't, as pointed by Denis Nardin here.

• Is there some more refined analogue of Péroux–Shipley's result in this context? That is, is there some sense in which $\mathbb{E}_{k}$-comonoids in $\mathcal{S}_*$ are "not too far" from being just pointed spaces?
• What are some natural examples of such objects?

In Coalgebras in symmetric monoidal categories of spectra, Péroux and Shipley prove the following (Lemma 2.4):

Let $\mathcal{C}=\mathsf{Sets},\mathsf{Top}$, or $\mathsf{sSets}$. The free basepoint functor $$(-)^+\colon(\mathcal{C},\times,\mathrm{pt})\to(\mathcal{C}_*,\wedge,S^0)$$ lifts to an equivalence of categories $$(-)^+\colon\mathsf{CoMon}(\mathcal{C})\dashrightarrow\mathsf{CoMon}(\mathcal{C}_*).$$ As such, any comonoid in $\mathcal{C}_*$ is isomorphic to one of the form $(C^+,\Delta^+_{C},\epsilon^+_C)$ with $(C,\Delta_{C},\epsilon_C)$ a comonoid in $\mathcal{C}$.

• Is this result valid in general for $\mathcal{C}$ a bicomplete Cartesian closed symmetric monoidal category, where $(\mathcal{C}_*,\wedge,S^0)$ is now the symmetric monoidal category in Riehl, Categorical Homotopy Theory, Construction 3.3.14?
• Let $\mathcal{C}$ be a bicomplete Cartesian closed symmetric monoidal $\infty$-category.
  1. Can we similarly build a symmetric monoidal $\infty$-category structure $(\wedge,S^0)$ on the category $\mathcal{C}_*$ of pointed objects of $\mathcal{C}$ (i.e. the coslice $\infty$-category $\mathcal{C}_{*/}$)?
  2. If this is the case, do we have an analogue of Péroux–Shipley's result for $\mathbb{E}_{k}$-comonoids in $(\mathcal{C}_*,\wedge,S^0)$, where $1\leq k\leq\infty$?
  3. (If Item 1 fails, is Item 2 nevertheless true for $\mathcal{C}=\mathcal{S}$, the $\infty$-category of spaces?)