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Suppose $\mathcal{T}$ is a triangulated category. What are the conditions $\mathcal{T}$ must satisfy in order to have a t-structure? If a t-structure exists, which further conditions would ensure that $\mathcal{T}$ is the derived category of its heart?

My question is motivated by the ongoing search for an abelian category of mixed motives for which several constructions of triangulated categories exist. In this context, is it the case that

(1) the aforementioned conditions are met by one or all of the existing triangulated categories so the existence of the abelian category is assured and the remaining issue is one of construction of a t-structure, or

(2) the conditions are not known to be satisfied by any of the existing triangulated categories so even the existence of a t-structure is unknown, or

(3) no such conditions are known, i.e., the answer to my questions in the first paragraph is "don't know!", at least in thisthat generality.

I believe from my reading that option (1) is not true, but I've included it just to make sure. Thanks!

Suppose $\mathcal{T}$ is a triangulated category. What are the conditions $\mathcal{T}$ must satisfy in order to have a t-structure? If a t-structure exists, which further conditions would ensure that $\mathcal{T}$ is the derived category of its heart?

My question is motivated by the ongoing search for an abelian category of mixed motives for which several constructions of triangulated categories exist. In this context, is it the case that

(1) the aforementioned conditions are met by one or all of the existing triangulated categories so the existence of the abelian category is assured and the remaining issue is one of construction of a t-structure, or

(2) the conditions are not known to be satisfied by any of the existing triangulated categories so even the existence of a t-structure is unknown, or

(3) no such conditions are known, i.e., the answer to my questions in the first paragraph is "don't know!", at least in this generality.

I believe from my reading that option (1) is not true, but I've included it just to make sure. Thanks!

Suppose $\mathcal{T}$ is a triangulated category. What are the conditions $\mathcal{T}$ must satisfy in order to have a t-structure? If a t-structure exists, which further conditions would ensure that $\mathcal{T}$ is the derived category of its heart?

My question is motivated by the ongoing search for an abelian category of mixed motives for which several constructions of triangulated categories exist. In this context, is it the case that

(1) the aforementioned conditions are met by one or all of the existing triangulated categories so the existence of the abelian category is assured and the remaining issue is one of construction of a t-structure, or

(2) the conditions are not known to be satisfied by any of the existing triangulated categories so even the existence of a t-structure is unknown, or

(3) no such conditions are known, i.e., the answer to my questions in the first paragraph is "don't know!", at least in that generality.

I believe from my reading that option (1) is not true, but I've included it just to make sure. Thanks!

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user163840
user163840

Suppose $\mathcal{T}$ is a triangulated category. What are the conditions $\mathcal{T}$ must satisfy in order to have a t-structure? If a t-structure exists, which further conditions would ensure that $\mathcal{T}$ is the derived category of its heart?

My question is motivated by the ongoing search for an abelian category of mixed motives for which several constructions of triangulated categories exist. In this context, is it the case that

(1) the aforementioned conditions are met by one or all of the existing triangulated categories so the existence of the abelian category is assured and the remaining issue is one of construction of a t-structure, or

(2) the conditions are not known to be satisfied by any of the existing triangulated categories so even the existence of a t-structure is unknown, or

(3) theno such conditions are not known, i.e., the answer to my questions in the first paragraph is "don't know!", at least in this generality.

I believe from my reading that option (1) is not true, but I've included it just to make sure. Thanks!

Suppose $\mathcal{T}$ is a triangulated category. What are the conditions $\mathcal{T}$ must satisfy in order to have a t-structure? If a t-structure exists, which further conditions would ensure that $\mathcal{T}$ is the derived category of its heart?

My question is motivated by the ongoing search for an abelian category of mixed motives for which several constructions of triangulated categories exist. In this context, is it the case that

(1) the aforementioned conditions are met by one or all of the existing triangulated categories so the existence of the abelian category is assured and the remaining issue is one of construction of a t-structure, or

(2) the conditions are not known to be satisfied by any of the existing triangulated categories so even the existence of a t-structure is unknown, or

(3) the conditions are not known, i.e., the answer to my questions in the first paragraph is "don't know!", at least in this generality.

I believe from my reading that (1) is not true, but I've included it just to make sure. Thanks!

Suppose $\mathcal{T}$ is a triangulated category. What are the conditions $\mathcal{T}$ must satisfy in order to have a t-structure? If a t-structure exists, which further conditions would ensure that $\mathcal{T}$ is the derived category of its heart?

My question is motivated by the ongoing search for an abelian category of mixed motives for which several constructions of triangulated categories exist. In this context, is it the case that

(1) the aforementioned conditions are met by one or all of the existing triangulated categories so the existence of the abelian category is assured and the remaining issue is one of construction of a t-structure, or

(2) the conditions are not known to be satisfied by any of the existing triangulated categories so even the existence of a t-structure is unknown, or

(3) no such conditions are known, i.e., the answer to my questions in the first paragraph is "don't know!", at least in this generality.

I believe from my reading that option (1) is not true, but I've included it just to make sure. Thanks!

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user163840
user163840

When does a triangulated category have a heart?

Suppose $\mathcal{T}$ is a triangulated category. What are the conditions $\mathcal{T}$ must satisfy in order to have a t-structure? If a t-structure exists, which further conditions would ensure that $\mathcal{T}$ is the derived category of its heart?

My question is motivated by the ongoing search for an abelian category of mixed motives for which several constructions of triangulated categories exist. In this context, is it the case that

(1) the aforementioned conditions are met by one or all of the existing triangulated categories so the existence of the abelian category is assured and the remaining issue is one of construction of a t-structure, or

(2) the conditions are not known to be satisfied by any of the existing triangulated categories so even the existence of a t-structure is unknown, or

(3) the conditions are not known, i.e., the answer to my questions in the first paragraph is "don't know!", at least in this generality.

I believe from my reading that (1) is not true, but I've included it just to make sure. Thanks!