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You must allow the weak equivalences to be unpointed homotopy equivalences. These become honest pointed homotopy equivalences between fibrant-cofibrant objects by the generalized whitehead theorem. Strom's mistake, I think, was that he didn't realize that unbased $Top$ with his model structure has the very special property that all objects are fibrant-cofibrant. This is not the case for based spaces, however (all objects are fibrant but not cofibrant). The cofibrant objects are precisely the nondegenerately based spaces.

By restricting to nondegenerately based spaces, he restricted himself to working in the category of cofibrant objects of the actual model category $*\downarrow Top$ equipped with the relative Strom model structure. It shouldn't be too surprising that this subcategory does not admit a (compatible) model structure.

(Proof that all objects in Strom's model structure on $Top$ are fibrant-cofibrant):

Lemma: A Hurewicz fibration $A\to B$ has the RLP with respect to every inclusion of a space $X$ into its cylinder $X\times I$. When $B$ is the terminal object, let $X\times I\to X$ be the map killing the cylinder. Then we get a lift to $X\times I$ of any map $X\to A$ by composing $X\times I\to X\to A$.

Lemma: A closed Hurewicz cofibration is a closed inclusion $A\hookrightarrow B$ such that $A\times I \coprod_{A\times \{0\}} B\times \{0\}\hookrightarrow B\times I$ has the LLP with respect to any map $Y\to *$. When $A$ is empty, this reduces to finding an extension of a map $B\to Y$ to a homotopy extending this, but again, this is immediately possible by composition with the trivial homotopy $B\times I\to B$.

The theorem of Strom is that there exists a model structure on Top with

• $C$ = closed hurewicz cofibrations
• $W$ = homotopy equivalences
• $F$ = hurewicz fibrations

From which it follows that all objects are fibrant-cofibrant.

(Source of definitions: Dwyer-Spalinski example 3.6)

You must allow the weak equivalences to be unpointed homotopy equivalences. These become honest pointed homotopy equivalences between fibrant-cofibrant objects by the generalized whitehead theorem. Strøm's mistake, I think, was that he didn't realize that unbased $Top$ with his model structure has the very special property that all objects are fibrant-cofibrant. This is not the case for based spaces, however (all objects are fibrant but not cofibrant). The cofibrant objects are precisely the nondegenerately based spaces.

By restricting to nondegenerately based spaces, he restricted himself to working in the category of cofibrant objects of the actual model category $*\downarrow Top$ equipped with the relative Strøm model structure. It shouldn't be too surprising that this subcategory does not admit a (compatible) model structure.

(Proof that all objects in Strøm's model structure on $Top$ are fibrant-cofibrant):

Lemma: A Hurewicz fibration $A\to B$ has the RLP with respect to every inclusion of a space $X$ into its cylinder $X\times I$. When $B$ is the terminal object, let $X\times I\to X$ be the map killing the cylinder. Then we get a lift to $X\times I$ of any map $X\to A$ by composing $X\times I\to X\to A$.

Lemma: A closed Hurewicz cofibration is a closed inclusion $A\hookrightarrow B$ such that $A\times I \coprod_{A\times \{0\}} B\times \{0\}\hookrightarrow B\times I$ has the LLP with respect to any map $Y\to *$. When $A$ is empty, this reduces to finding an extension of a map $B\to Y$ to a homotopy extending this, but again, this is immediately possible by composition with the trivial homotopy $B\times I\to B$.

The theorem of Strøm is that there exists a model structure on Top with

• $C$ = closed hurewicz cofibrations
• $W$ = homotopy equivalences
• $F$ = hurewicz fibrations

From which it follows that all objects are fibrant-cofibrant.

(Source of definitions: Dwyer-Spalinski example 3.6)

You must allow the weak equivalences to be unpointed homotopy equivalences. These become honest pointed homotopy equivalences between fibrant-cofibrant objects by the generalized whitehead theorem. Strom's mistake, I think, was that he didn't realize that unbased $Top$ with his model structure has the very special property that all objects are fibrant-cofibrant. This is not the case for based spaces, however (all objects are fibrant but not cofibrant). The cofibrant objects are precisely the nondegenerately based spaces.

By restricting to nondegenerately based spaces, he restricted himself to working in the category of cofibrant objects of the actual model category $*\downarrow Top$ equipped with the relative Strom model structure. It shouldn't be too surprising that this subcategory does not admit a model structure.

(Proof that all objects in Strom's model structure on $Top$ are fibrant-cofibrant):

Lemma: A Hurewicz fibration $A\to B$ has the RLP with respect to every inclusion of a space $X$ into its cylinder $X\times I$. When $B$ is the terminal object, let $X\times I\to X$ be the map killing the cylinder. Then we get a lift to $X\times I$ of any map $X\to A$ by composing $X\times I\to X\to A$.

Lemma: A closed Hurewicz cofibration is a closed inclusion $A\hookrightarrow B$ such that $A\times I \coprod_{A\times \{0\}} B\times \{0\}\hookrightarrow B\times I$ has the LLP with respect to any map $Y\to *$. When $A$ is empty, this reduces to finding an extension of a map $B\to Y$ to a homotopy extending this, but again, this is immediately possible by composition with the trivial homotopy $B\times I\to B$.

The theorem of Strom is that there exists a model structure on Top with

• $C$ = closed hurewicz cofibrations
• $W$ = homotopy equivalences
• $F$ = hurewicz fibrations

From which it follows that all objects are fibrant-cofibrant.

(Source of definitions: Dwyer-Spalinski example 3.6)

You must allow the weak equivalences to be unpointed homotopy equivalences. These become honest pointed homotopy equivalences between fibrant-cofibrant objects by the generalized whitehead theorem. Strom's mistake, I think, was that he didn't realize that unbased $Top$ with his model structure has the very special property that all objects are fibrant-cofibrant. This is not the case for based spaces, however (all objects are fibrant but not cofibrant). The cofibrant objects are precisely the nondegenerately based spaces.

By restricting to nondegenerately based spaces, he restricted himself to working in the category of cofibrant objects of the actual model category $*\downarrow Top$ equipped with the relative Strom model structure. It shouldn't be too surprising that this subcategory does not admit a (compatible) model structure.

(Proof that all objects in Strom's model structure on $Top$ are fibrant-cofibrant):

Lemma: A Hurewicz fibration $A\to B$ has the RLP with respect to every inclusion of a space $X$ into its cylinder $X\times I$. When $B$ is the terminal object, let $X\times I\to X$ be the map killing the cylinder. Then we get a lift to $X\times I$ of any map $X\to A$ by composing $X\times I\to X\to A$.

Lemma: A closed Hurewicz cofibration is a closed inclusion $A\hookrightarrow B$ such that $A\times I \coprod_{A\times \{0\}} B\times \{0\}\hookrightarrow B\times I$ has the LLP with respect to any map $Y\to *$. When $A$ is empty, this reduces to finding an extension of a map $B\to Y$ to a homotopy extending this, but again, this is immediately possible by composition with the trivial homotopy $B\times I\to B$.

The theorem of Strom is that there exists a model structure on Top with

• $C$ = closed hurewicz cofibrations
• $W$ = homotopy equivalences
• $F$ = hurewicz fibrations

From which it follows that all objects are fibrant-cofibrant.

(Source of definitions: Dwyer-Spalinski example 3.6)

You must allow the weak equivalences to be unpointed homotopy equivalences. These become honest pointed homotopy equivalences between fibrant-cofibrant objects by the generalized whitehead theorem. Strom's mistake, I think, was that he didn't realize that unbased $Top$ with his model structure has the very special property that all objects are fibrant-cofibrant. This is not the case for based spaces, however (all objects are fibrant but not cofibrant). The cofibrant objects are precisely the nondegenerately based spaces.

By restricting to nondegenerately based spaces, he restricted himself to working in the category of cofibrant objects of the actual model category $*\downarrow Top$ equipped with the relative Strom model structure. It shouldn't be too surprising that this subcategory does not admit a model structure.

(Proof that all objects in Strom's model structure are fibrant-cofibrant):

A Hurewicz fibration $A\to B$ has the RLP with respect to every inclusion of a space $X$ into its cylinder $X\times I$. When $B$ is the terminal object, let $X\times I\to X$ be the map killing the cylinder. Then we get a lift to $X\times I$ of any map $X\to A$ by composing $X\times I\to X\to A$.

A closed Hurewicz cofibration is a closed inclusion $A\hookrightarrow B$ such that $A\times I \coprod_{A\times \{0\}} B\times \{0\}\hookrightarrow B\times I$ has the LLP with respect to any map $Y\to *$. When $A$ is empty, this reduces to finding an extension of a map $B\to Y$ to a homotopy extending this, but again, this is immediately possible.

You must allow the weak equivalences to be unpointed homotopy equivalences. These become honest pointed homotopy equivalences between fibrant-cofibrant objects by the generalized whitehead theorem. Strom's mistake, I think, was that he didn't realize that unbased $Top$ with his model structure has the very special property that all objects are fibrant-cofibrant. This is not the case for based spaces, however (all objects are fibrant but not cofibrant). The cofibrant objects are precisely the nondegenerately based spaces.

By restricting to nondegenerately based spaces, he restricted himself to working in the category of cofibrant objects of the actual model category $*\downarrow Top$ equipped with the relative Strom model structure. It shouldn't be too surprising that this subcategory does not admit a model structure.

(Proof that all objects in Strom's model structure on $Top$ are fibrant-cofibrant):

Lemma: A Hurewicz fibration $A\to B$ has the RLP with respect to every inclusion of a space $X$ into its cylinder $X\times I$. When $B$ is the terminal object, let $X\times I\to X$ be the map killing the cylinder. Then we get a lift to $X\times I$ of any map $X\to A$ by composing $X\times I\to X\to A$.

Lemma: A closed Hurewicz cofibration is a closed inclusion $A\hookrightarrow B$ such that $A\times I \coprod_{A\times \{0\}} B\times \{0\}\hookrightarrow B\times I$ has the LLP with respect to any map $Y\to *$. When $A$ is empty, this reduces to finding an extension of a map $B\to Y$ to a homotopy extending this, but again, this is immediately possible by composition with the trivial homotopy $B\times I\to B$.

The theorem of Strom is that there exists a model structure on Top with

• $C$ = closed hurewicz cofibrations
• $W$ = homotopy equivalences
• $F$ = hurewicz fibrations

From which it follows that all objects are fibrant-cofibrant.

(Source of definitions: Dwyer-Spalinski example 3.6)