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Daniel Bruegmann
Daniel Bruegmann
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I believe that Definition 2 and Definition 3 are equivalent. This involves that Definition 2 implies that F is multiplicative ("for each pairwise disjoint open sets $U_{1},\dots,U_{k}\subset M$, the map $m_{U_{1},\dots,U_{k};\bigcup_{i}U_{i}}$ is a quasi-isomorphism"), that every Weiss cover can be refined by a cover consisting of disjoint unions of small balls, and that (as you say) that there is a relationship between homotopy colimits and simplicial objects.

(In a cocomplete category every colimit can be written as a specific coequalizer of two coproducts. In an \infty-category like the derived \infty-category of a field, this generalizes to a colimit over a simplicial object. There is an appendix in [CG] which explains parts of this story and contains some references - not sure if you already saw this and wanted more or not.)

Definition 1 + multiplicative is equivalent to Definition 3.

The examples in [CG] satisfy Definition 1 but most of them are not multiplicative for $A$ the category of vector spaces with the ordinary tensor product of vector spaces. Therefore Definition 1 is not equivalent to Definition 3. These examples are multiplicative if one uses a completed versions of the tensor product, but then one would still need to do homological algebra with some flavor of complete vector spaces which usually do not form an abelian category. Maybe what one should do is use the symmetric monoidal category of liquid vector spaces, because they have a good tensor product and form an abelian category at the same time.

I believe that Definition 2 and Definition 3 are equivalent. This involves that Definition 2 implies that F is multiplicative ("for each pairwise disjoint open sets $U_{1},\dots,U_{k}\subset M$, the map $m_{U_{1},\dots,U_{k};\bigcup_{i}U_{i}}$ is a quasi-isomorphism"), that every Weiss cover can be refined by a cover consisting of disjoint unions of small balls, and that (as you say) that there is a relationship between homotopy colimits and simplicial objects.

(In a cocomplete category every colimit can be written as a specific coequalizer of two coproducts. In an \infty-category like the derived \infty-category of a field, this generalizes to a colimit over a simplicial object. There is an appendix in [CG] which explains parts of this story and contains some references - not sure if you already saw this and wanted more or not.)

Definition 1 + multiplicative is equivalent to Definition 3.

The examples in [CG] satisfy Definition 1 but most of them are not multiplicative for $A$ the category of vector spaces with the ordinary tensor product of vector spaces. Therefore Definition 1 is not equivalent to Definition 3. These examples are multiplicative if one uses a completed versions of the tensor product, but would still need to do homological algebra with some flavor of complete vector spaces which usually do not form an abelian category. Maybe what one should do is use the symmetric monoidal category of liquid vector spaces, because they have a good tensor product and form an abelian category at the same time.

I believe that Definition 2 and Definition 3 are equivalent. This involves that Definition 2 implies that F is multiplicative ("for each pairwise disjoint open sets $U_{1},\dots,U_{k}\subset M$, the map $m_{U_{1},\dots,U_{k};\bigcup_{i}U_{i}}$ is a quasi-isomorphism"), that every Weiss cover can be refined by a cover consisting of disjoint unions of small balls, and that (as you say) that there is a relationship between homotopy colimits and simplicial objects.

(In a cocomplete category every colimit can be written as a specific coequalizer of two coproducts. In an \infty-category like the derived \infty-category of a field, this generalizes to a colimit over a simplicial object. There is an appendix in [CG] which explains parts of this story and contains some references - not sure if you already saw this and wanted more or not.)

Definition 1 + multiplicative is equivalent to Definition 3.

The examples in [CG] satisfy Definition 1 but most of them are not multiplicative for $A$ the category of vector spaces with the ordinary tensor product of vector spaces. Therefore Definition 1 is not equivalent to Definition 3. These examples are multiplicative if one uses a completed versions of the tensor product, but then one would need to do homological algebra with some flavor of complete vector spaces which usually do not form an abelian category. Maybe what one should do is use the symmetric monoidal category of liquid vector spaces, because they have a good tensor product and form an abelian category at the same time.

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Daniel Bruegmann
Daniel Bruegmann
  • 890
  • 5
  • 14

I believe that Definition 2 and Definition 3 are equivalent. This involves that Definition 2 implies that F is multiplicative ("for each pairwise disjoint open sets $U_{1},\dots,U_{k}\subset M$, the map $m_{U_{1},\dots,U_{k};\bigcup_{i}U_{i}}$ is a quasi-isomorphism"), that every Weiss cover can be refined by a cover consisting of disjoint unions of small balls, and that (as you say) that there is a relationship between homotopy colimits and simplicial objects.

(In a cocomplete category every colimit can be written as a specific coequalizer of two coproducts. In an \infty-category like the derived \infty-category of a field, this generalizes to a colimit over a simplicial object. There is an appendix in [CG] which explains parts of this story and contains some references - not sure if you already saw this and wanted more or not.)

Definition 1 + multiplicative is equivalent to Definition 3.

The examples in [CG] satisfy Definition 1 but most of them are not multiplicative for $A$ the category of vector spaces with the ordinary tensor product of vector spaces. Therefore Definition 1 is not equivalent to Definition 3. These examples are multiplicative if one uses a completed versions of the tensor product, but would still need to do homological algebra with some flavor of complete vector spaces which usually do not form an abelian category. Maybe what one should do is use the symmetric monoidal category of liquid vector spaces, because they have a good tensor product and form an abelian category at the same time.