Author: Dmitri Pavlov Format: TextSuppose C is a [[combinatorial]] [[stable model category]] equipped with an accessible [[t-structure]].
It seems to me that in this context an accessible t-structure
can be defined simply as a pair of accessible full subcategories (C_{≥0}, C_{<0})
such that X∈C_{≥0} if and only if RHom(X,Y) is contractible for any Y∈C_{<0}
and vice versa, Y∈C_{<0} if and only if RHom(X,Y) is contractible for any X∈C_{≥0}.
This implies that C_{≥0} is closed under homotopy colimits,
in particular, under suspensions, so this renders unnecessary one of the conditions in the definition
of a t-structure.
The other condition concerns the existence of a homotopy exact sequence X_{≥0}→X→X_{<0}.
By passing to the underlying quasicategory one can invoke Lurie's results in Higher Algebra, 1.4.4, to construct
the corresponding exact sequence in a quasicategory,
and then lift it to the level of model categories, possibly after (co)fibrantly replacing X.
On the level of quasicategories these factorizations are functorial, but what about model categories?
It seems to me that one could try to construct a truncation functor τ_{≥0}: C→C_{≥0}
by taking all cofibrations S between cofibrant λ-small objects with cofiber in C_{≥0}
and apply the small object argument to the resulting set (which is a set by combinatoriality and accessibility).
(Here λ is such that the model category is λ-combinatorial.)
The small object argument will produce a functorial factorization such that
when we apply it to ∅→X we get ∅→A→X,
where ∅→A is a transfinite composition of cobase changes of cofibrations between cofibrant objects,
and because such cobase changes and transfinite compositions are also homotopy cobase changes
and homotopy transfinite compositions, the morphism ∅→A has a connective homotopy cofiber,
so A∈C_{≥0}.
The small object argument guarantees that the other morphism, A→X, has a right lifting property
with respect to all morphisms in the generating set S used in the small object argument.
It seems to me that this should imply that A→X has a homotopy cofiber in C_{<0}:
indeed, from the quasicategorical version we know that it suffices to have the homotopy right
lifting property with respect to the homotopy generators of C_{≥0}, for which we can take the (co)domains of S.
Assuming that X is fibrant, any morphism in the underlying quasicategory
from a cofibrant object to X can be presented by a morphism in the model category,
so the homotopy right lifting property follows from the ordinary (strict) right lifting property.
If we perform a _bifibrant_ replacement of X, then it seems to me like
the other truncation functor, τ_{<0} can then be constructed by taking
the “cone” of A→X (recall that A and X are cofibrant, even though A→X need not be a cofibration),
using a functorial simplicial framing, e.g., a simplicial enrichment.
All of this feels very much like reinventing the wheel, so I'm wondering whether
anything like this has appeared in the literature before.
It seems to me that in this context an accessible t-structure can be defined simply as a pair of accessible full subcategories (C_{≥0}, C_{<0}) such that X∈C_{≥0} if and only if RHom(X,Y) is contractible for any Y∈C_{<0} and vice versa, Y∈C_{<0} if and only if RHom(X,Y) is contractible for any X∈C_{≥0}.
This implies that C_{≥0} is closed under homotopy colimits, in particular, under suspensions, so this renders unnecessary one of the conditions in the definition of a t-structure.
The other condition concerns the existence of a homotopy exact sequence X_{≥0}→X→X_{<0}. By passing to the underlying quasicategory one can invoke Lurie's results in Higher Algebra, 1.4.4, to construct the corresponding exact sequence in a quasicategory, and then lift it to the level of model categories, possibly after (co)fibrantly replacing X.
On the level of quasicategories these factorizations are functorial, but what about model categories?
It seems to me that one could try to construct a truncation functor τ_{≥0}: C→C_{≥0} by taking all cofibrations S between cofibrant λ-small objects with cofiber in C_{≥0} and apply the small object argument to the resulting set (which is a set by combinatoriality and accessibility). (Here λ is such that the model category is λ-combinatorial.)
The small object argument will produce a functorial factorization such that when we apply it to ∅→X we get ∅→A→X, where ∅→A is a transfinite composition of cobase changes of cofibrations between cofibrant objects, and because such cobase changes and transfinite compositions are also homotopy cobase changes and homotopy transfinite compositions, the morphism ∅→A has a connective homotopy cofiber, so A∈C_{≥0}.
The small object argument guarantees that the other morphism, A→X, has a right lifting property with respect to all morphisms in the generating set S used in the small object argument. It seems to me that this should imply that A→X has a homotopy cofiber in C_{<0}: indeed, from the quasicategorical version we know that it suffices to have the homotopy right lifting property with respect to the homotopy generators of C_{≥0}, for which we can take the (co)domains of S.
Assuming that X is fibrant, any morphism in the underlying quasicategory from a cofibrant object to X can be presented by a morphism in the model category, so the homotopy right lifting property follows from the ordinary (strict) right lifting property.
If we perform a _bifibrant_ replacement of X, then it seems to me like the other truncation functor, τ_{<0} can then be constructed by taking the “cone” of A→X (recall that A and X are cofibrant, even though A→X need not be a cofibration), using a functorial simplicial framing, e.g., a simplicial enrichment.
All of this feels very much like reinventing the wheel, so I'm wondering whether anything like this has appeared in the literature before.