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A 1991 paper of Lewis, title “Is there a convenient category of spectra?” proves that there is no category $\mathrm{Sp}$ satisfying the following desiderata$^1$:

1. There is a symmetric monoidal smash product $\wedge$;
2. There is an adjunction $\Sigma^\infty\colon\mathrm{Top}_*^\mathrm{CGWHaus}\rightleftarrows\mathrm{Sp}:\Omega^\infty$;
3. The sphere spectrum $\mathbb{S}$ is a unit for $\wedge$;
4. There is either a natural transformation $$(\Omega^\infty E)\wedge(\Omega^\infty F)\Rightarrow\Omega^\infty(E\wedge F)$$ or a natural transformation $$\Sigma^\infty(E\wedge F)\Rightarrow(\Sigma^\infty E)\wedge(\Sigma^\infty F).$$ Furthermore, these natural transformations are required to commute with the unity, commutativity, and associativity isomorphisms of $\mathrm{Top}_*^\mathrm{CGWHaus}$ and $\mathrm{Sp}$;
5. There is a natural weak equivalence $\Omega^\infty\Sigma^\infty X\xrightarrow{\cong}\varinjlim(\Omega^n\Sigma^nX)$.

Question: The theorem is proved for (ordinary) categories. But what happens for $\infty$-categories? Is the $\infty$-category of spectra defined in Higher Algebra “convenient”?

The $\infty$-category $\mathrm{Sp}$ satisfies (in an appropriate sense) [1] and [3] (HA 4.8.2.19), and [2] (HA 1.1.2.8). What about [4] and [5]?

$^1$Which I am paraphrasing partly from the 2017 Talbot notes and partly from Lewis.

A 1991 paper of Lewis, titled “Is there a convenient category of spectra?” proves that there is no category $\mathrm{Sp}$ satisfying the following desiderata$^1$:

1. There is a symmetric monoidal smash product $\wedge$;
2. There is an adjunction $\Sigma^\infty\colon\mathrm{Top}_*^\mathrm{CGWHaus}\rightleftarrows\mathrm{Sp}:\Omega^\infty$;
3. The sphere spectrum $\mathbb{S}$ is the monoidal unit for $\wedge$;
4. There is either a natural transformation $$(\Omega^\infty E)\wedge(\Omega^\infty F)\Rightarrow\Omega^\infty(E\wedge F)$$ or a natural transformation $$\Sigma^\infty(E\wedge F)\Rightarrow(\Sigma^\infty E)\wedge(\Sigma^\infty F).$$ Furthermore, these natural transformations are required to commute with the unity, commutativity, and associativity isomorphisms of $\mathrm{Top}_*^\mathrm{CGWHaus}$ and $\mathrm{Sp}$;
5. There is a natural weak equivalence $\Omega^\infty\Sigma^\infty X\xrightarrow{\cong}\varinjlim(\Omega^n\Sigma^nX)$.

Question: The theorem is proved for (ordinary) categories. But what happens for $\infty$-categories? Is the $\infty$-category of spectra defined in Higher Algebra “convenient”?

The $\infty$-category $\mathrm{Sp}$ satisfies (in an appropriate sense) [1] and [3] (HA 4.8.2.19), and [2] (HA 1.1.2.8). What about [4] and [5]?

$^1$Which I am paraphrasing partly from the 2017 Talbot notes and partly from Lewis.

A 1991 paper of Lewis, title “Is there a convenient category of spectra?” proves that there is no category $\mathrm{Sp}$ satisfying the following desiderata$^1$:

1. There is a symmetric monoidal smash product $\wedge$;
2. There is an adjunction $\Sigma^\infty\colon\mathrm{Top}_*^\mathrm{CGWHaus}\rightleftarrows\mathrm{Sp}:\Omega^\infty$;
3. The sphere spectrum $\mathbb{S}$ is a unit for $\wedge$;
4. There is either a natural transformation $$(\Omega^\infty E)\wedge(\Omega^\infty F)\Rightarrow\Omega^\infty(E\wedge F)$$ or a natural transformation $$\Sigma^\infty(E\wedge F)\Rightarrow(\Sigma^\infty E)\wedge(\Sigma^\infty F).$$ Furthermore, these natural transformations are required to commute with the unity, commutativity, and associativity isomorphisms of $\mathrm{Top}_*^\mathrm{CGWHaus}$ and $\mathrm{Sp}$;
5. There is a natural weak equivalence $\Omega^\infty\Sigma^\infty X\xrightarrow{\cong}\varinjlim(\Omega^n\Sigma^nX)$.

Question: The theorem is proved for (ordinary) categories. But what happens for $\infty$-categories? Is the $\infty$-category of spectra defined in Higher Algebra “convenient”?

The $\infty$-category $\mathrm{Sp}$ satisfies (in an appropriate sense) [1] and [3] (HA 4.8.2.19), and [2] (HA 1.1.2.8). What about [4] and [5]?

$^1$Which I am partly paraphrasing from the 2017 Talbot notes and partly paraphrasing from Lewis.

A 1991 paper of Lewis, title “Is there a convenient category of spectra?” proves that there is no category $\mathrm{Sp}$ satisfying the following desiderata$^1$:

1. There is a symmetric monoidal smash product $\wedge$;
2. There is an adjunction $\Sigma^\infty\colon\mathrm{Top}_*^\mathrm{CGWHaus}\rightleftarrows\mathrm{Sp}:\Omega^\infty$;
3. The sphere spectrum $\mathbb{S}$ is a unit for $\wedge$;
4. There is either a natural transformation $$(\Omega^\infty E)\wedge(\Omega^\infty F)\Rightarrow\Omega^\infty(E\wedge F)$$ or a natural transformation $$\Sigma^\infty(E\wedge F)\Rightarrow(\Sigma^\infty E)\wedge(\Sigma^\infty F).$$ Furthermore, these natural transformations are required to commute with the unity, commutativity, and associativity isomorphisms of $\mathrm{Top}_*^\mathrm{CGWHaus}$ and $\mathrm{Sp}$;
5. There is a natural weak equivalence $\Omega^\infty\Sigma^\infty X\xrightarrow{\cong}\varinjlim(\Omega^n\Sigma^nX)$.

Question: The theorem is proved for (ordinary) categories. But what happens for $\infty$-categories? Is the $\infty$-category of spectra defined in Higher Algebra “convenient”?

The $\infty$-category $\mathrm{Sp}$ satisfies (in an appropriate sense) [1] and [3] (HA 4.8.2.19), and [2] (HA 1.1.2.8). What about [4] and [5]?

$^1$Which I am paraphrasing partly from the 2017 Talbot notes and partly from Lewis.

A 1991 paper of Lewis, title “Is there a convenient category of spectra?” proves that there is no category $\mathrm{Sp}$ satisfying the following desiderata$^1$:

1. There is a symmetric monoidal smash product $\wedge$;
2. There is an adjunction $\Sigma^\infty\colon\mathrm{Top}_*^\mathrm{CGWHaus}\rightleftarrows\mathrm{Sp}:\Omega^\infty$;
3. The sphere spectrum $\mathbb{S}$ is a unit for $\wedge$;
4. There is either a natural transformation $$(\Omega^\infty E)\wedge(\Omega^\infty F)\Rightarrow\Omega^\infty(E\wedge F)$$ or a natural transformation $$\Sigma^\infty(E\wedge F)\Rightarrow(\Sigma^\infty E)\wedge(\Sigma^\infty F).$$ Furthermore, these natural transformations are required to commute with the unity, commutativity, and associativity isomorphisms of $\mathrm{Top}_*^\mathrm{CGWHaus}$ and $\mathrm{Sp}$;
5. There is a natural weak equivalence $\Omega^\infty\Sigma^\infty X\xrightarrow{\cong}\varinjlim(\Omega^n\Sigma^nX)$.

Question: The theorem is proved for (ordinary) categories. But what happens for $\infty$-categories? Is the $\infty$-category of spectra defined in Higher Algebra “convenient”?

The $\infty$-category $\mathrm{Sp}$ satisfies (in an appropriate sense) [1] and [3] (see HA 4.8.2.19), and [2](see HA 1.1.2.8). What about [4] and [5]?

$^1$Which I am partly paraphrasing from the 2017 Talbot notes and partly paraphrasing from Lewis.

A 1991 paper of Lewis, title “Is there a convenient category of spectra?” proves that there is no category $\mathrm{Sp}$ satisfying the following desiderata$^1$:

1. There is a symmetric monoidal smash product $\wedge$;
2. There is an adjunction $\Sigma^\infty\colon\mathrm{Top}_*^\mathrm{CGWHaus}\rightleftarrows\mathrm{Sp}:\Omega^\infty$;
3. The sphere spectrum $\mathbb{S}$ is a unit for $\wedge$;
4. There is either a natural transformation $$(\Omega^\infty E)\wedge(\Omega^\infty F)\Rightarrow\Omega^\infty(E\wedge F)$$ or a natural transformation $$\Sigma^\infty(E\wedge F)\Rightarrow(\Sigma^\infty E)\wedge(\Sigma^\infty F).$$ Furthermore, these natural transformations are required to commute with the unity, commutativity, and associativity isomorphisms of $\mathrm{Top}_*^\mathrm{CGWHaus}$ and $\mathrm{Sp}$;
5. There is a natural weak equivalence $\Omega^\infty\Sigma^\infty X\xrightarrow{\cong}\varinjlim(\Omega^n\Sigma^nX)$.

Question: The theorem is proved for (ordinary) categories. But what happens for $\infty$-categories? Is the $\infty$-category of spectra defined in Higher Algebra “convenient”?

The $\infty$-category $\mathrm{Sp}$ satisfies (in an appropriate sense) [1] and [3] (HA 4.8.2.19), and [2]  (HA 1.1.2.8). What about [4] and [5]?

$^1$Which I am partly paraphrasing from the 2017 Talbot notes and partly paraphrasing from Lewis.