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Simon Henry
Simon Henry
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This is probably not a full answer to your question, but I think it is a remark worth to make:

It is actually a couple of remarks:

1)If you have a weak factorization system where either the left class is made of epis or the right class is made of monos then you immediately get a unique (orthogonal) factorization system: any pair of diagonal filler in a lifting square will be either equalized by the mono on the right or co-equalized by the epi on the left.

  1. model structure where one of the weak factorizations are unique/orthogonal factorization system are not the interesting ones. They corresponds to trivial model structures on reflective/co-reflective subcategories.

So It makes sense that the model structure we study in practice are "as far as possible" of satisfying the first condition. and this tends to means that cofibration will be monos and fibration will be epis (but it is not always the case).

Also notes that it is always the case that it follows from the axiom of model categories that trivial fibrations with cofibrant target are split epis and trivial cofibrations with fibrant domain are split monos, so unless the only weak equivalences are the isos you always gets that some cofibrations are non trivial monos and some trivial fibrations are non trivial epis.

Edit: so to sum up, this is closely related with Chris comment: it boils down to the difference between unique factorization system and weak factorization systems.

This is probably not a full answer to your question, but I think it is a remark worth to make:

It is actually a couple of remarks:

1)If you have a weak factorization system where either the left class is made of epis or the right class is made of monos then you immediately get a unique (orthogonal) factorization system: any pair of diagonal filler in a lifting square will be either equalized by the mono on the right or co-equalized by the epi on the left.

  1. model structure where one of the weak factorizations are unique/orthogonal factorization system are not the interesting ones. They corresponds to trivial model structures on reflective/co-reflective subcategories.

So It makes sense that the model structure we study in practice are "as far as possible" of satisfying the first condition. and this tends to means that cofibration will be monos and fibration will be epis (but it is not always the case).

Also notes that it is always the case that it follows from the axiom of model categories that trivial fibrations with cofibrant target are split epis and trivial cofibrations with fibrant domain are split monos, so unless the only weak equivalences are the isos you always gets that some cofibrations are non trivial monos and some trivial fibrations are non trivial epis.

Edit: so to sum up, this is closely related with Chris comment: it boils down to the difference between unique factorization system and weak factorization systems.

This is probably not a full answer to your question, but I think it is a remark worth to make:

It is actually a couple of remarks:

1)If you have a weak factorization system where either the left class is made of epis or the right class is made of monos then you immediately get a unique (orthogonal) factorization system: any pair of diagonal filler in a lifting square will be either equalized by the mono on the right or co-equalized by the epi on the left.

  1. model structure where one of the weak factorizations are unique/orthogonal factorization system are not the interesting ones. They corresponds to trivial model structures on reflective/co-reflective subcategories.

So It makes sense that the model structure we study in practice are "as far as possible" of satisfying the first condition. and this tends to means that cofibration will be monos and fibration will be epis (but it is not always the case).

Also notes that it follows from the axiom of model categories that trivial fibrations with cofibrant target are split epis and trivial cofibrations with fibrant domain are split monos, so unless the only weak equivalences are the isos you always gets that some cofibrations are non trivial monos and some trivial fibrations are non trivial epis.

Edit: so to sum up, this is closely related with Chris comment: it boils down to the difference between unique factorization system and weak factorization systems.

Source Link
Simon Henry
Simon Henry
  • 42.4k
  • 5
  • 107
  • 205

This is probably not a full answer to your question, but I think it is a remark worth to make:

It is actually a couple of remarks:

1)If you have a weak factorization system where either the left class is made of epis or the right class is made of monos then you immediately get a unique (orthogonal) factorization system: any pair of diagonal filler in a lifting square will be either equalized by the mono on the right or co-equalized by the epi on the left.

  1. model structure where one of the weak factorizations are unique/orthogonal factorization system are not the interesting ones. They corresponds to trivial model structures on reflective/co-reflective subcategories.

So It makes sense that the model structure we study in practice are "as far as possible" of satisfying the first condition. and this tends to means that cofibration will be monos and fibration will be epis (but it is not always the case).

Also notes that it is always the case that it follows from the axiom of model categories that trivial fibrations with cofibrant target are split epis and trivial cofibrations with fibrant domain are split monos, so unless the only weak equivalences are the isos you always gets that some cofibrations are non trivial monos and some trivial fibrations are non trivial epis.

Edit: so to sum up, this is closely related with Chris comment: it boils down to the difference between unique factorization system and weak factorization systems.