Goodwillie calculus – approximation of homotopy theories by stable homotopy theories
For a locally presentable (∞,1)-category whose objects we think of as spaces of sorts, its tangent -category
is an (∞,1)-category over , whose objects may be thought of as spaces that are infinitesimal thickenings of those of .
More concretely, the tangent -category for is the fiberwise stabilization of the codomain fibration .
This generalizes – as discussed at deformation theory – the classical example of the bifibration Mod CRing of the category of all modules over the cateory CRing of all commutative rings:
the fiber of the tangent -category over an object may be thought of as the -category of square-0-extensions of , for a module over . Dually, in we may think of these as being infinitesimal neighbourhoods of 0-sections of vector bundles – or rather of quasicoherent sheaves – over whatever space is regarded to be the algebra of functions on.
A remarkable amount of information about the geometry of these spaces/objects in is encoded in the fiber of the tangent -category over them. Notably the left adjoint (∞,1)-functor
to the domain projection turns out to send each to its cotangent complex , to be thought of as the module of Kähler differentials on the space that is functions on.
A 1-categorical approximation to the notion of tangent -category is that of tangent category.
Let be a locally presentable (∞,1)-category.
(fiberwise stabilization)
For a categorical fibration, the fiberwise stabilization is – roughly – the fibration universal with the property that for each its fiber over is the stabilization of the fiber over .
This is (Lurie, section 1.1) formulated in view of (Lurie, remark 1.1.8). There is called the stable envelope .
(tangent -category)
Thetangent -category is the fiberwise stabilization of the codomain fibration :
This is DT, def 1.1.12.
For a maybe more explicit definition see below at Tangent ∞-topos – General.
Explicitly, the tangent -category is given as follows.
Given a presentable (∞,1)-category , the (∞,1)-functor
which classifies the codomain fibration under the (∞,1)-Grothendieck construction factors through the wide non-full inclusion
of (∞,1)-functors which are right adjoint (∞,1)-functors. For these the further (now full) inclusion
of the stable (∞,1)-categories has a right adjoint (∞,1)-functor
given by stabilization. (Note that this is not a functor on all of , where instead the obstructions to functoriality are given by Goodwillie calculus.)
So the classifying map of the codomain fibration factors through this and hence we can postcompose with the stabilization functor to obtain
This sends an object to the stabilization of the slice (∞,1)-category over :
Again by the (∞,1)-Grothendieck construction this classifies a Cartesian fibration over and this now is the tangent -category projection
This is the first part of the proof of DT. prop. 1.1.9.
The tangent -category of the locally presentable (∞,1)-category is itself a locally presentable -category.
In particular, it admits all (∞,1)-limits and (∞,1)-colimits.
This is (Lurie, prop. 1.1.13).
Moreover:
A diagram in the tangent -category is an (∞,1)-(co-)limit precisely if
it is a relative (∞,1)-(co-)limit with respect to the projection ;
its image under this projection is an (∞,1)-(co-)limit in .
(Lurie, HigherAlgebra, prop. 7.3.1.12)
We discuss how the tangent -category construction indeed generalizes the equivalence between the tangent category over CRing and the category Mod of all modules over commutative rings.
Let be a coherent (∞,1)-operad and let be a stable -monoidal (∞,1)-category.
Let
be an -algebra in . Then the stabilization of the over-(∞,1)-category over is canonically equivalent to
This is (Lurie, theorem 1.5.14).
Let be a coherent (∞,1)-operad and let be a presentable stable -monoidal (∞,1)-category. Then there is a canonical equivalence
of presentble fibrations over .
This is (Lurie, theorem, 1.5.19).
In words this says that under the given assumptions, objects of may be identified with pairs
where
From its definition as the fiberwise stabilization of the codomain fibration the tangent -category inherits a second -functor to , coming from the domain evaluation
(cotangent complex)
The domain evaluation admits a left adjoint (∞,1)-functor
that is also a section of in that
and which hence exhibits as a retract of .
This is the cotangent complex -functor : for the object is the cotangent complex of .
This is (Lurie, def. 1.2.2, remark 1.2.3).
In more detail this adjunction is the composite
where is the fiberwise stabilization relative adjunction, def. .
We discuss how the tangent -category of an (∞,1)-topos is itself an (∞,1)-topos over the tangent -category of the original base (∞,1)-topos.
In terms of Omega-spectrum spectrum objects this is due to (Joyal 08) joint with Georg Biedermann. In terms of excisive functors this is due to observations by Georg Biedermann, Charles Rezk and Jacob Lurie, see at -Excisive functor – Properties – -Excisive reflection
Let be the diagram category as follows:
Given an (∞,1)-topos , an (∞,1)-functor
is equivalently
a choice of object (the image of ]);
a sequence of objects in the slice (∞,1)-topos over ;
a sequence of morphisms from into the loop space object of in the slice.
This is a prespectrum object in the slice (∞,1)-topos .
A natural transformation between two such functors with components
is equivalently a morphism of base objects in together with morphisms into the (∞,1)-pullback of the components of along .
Therefore the (∞,1)-presheaf (∞,1)-topos
is the codomain fibration of with “fiberwise pre-stabilization”.
A genuine spectrum object is a prespectrum object for which all the structure maps are equivalences. The full sub-(∞,1)-category
on the genuine spectrum objects is therefore the “fiberwise stabilization” of the self-indexing, hence the tangent -category.
(spectrification is left exact reflective)
The inclusion of spectrum objects into is left reflective, hence it has a left adjoint (∞,1)-functor – spectrification – which preserves finite (∞,1)-limits.
Forming degreewise loop space objects constitutes an (∞,1)-functor and by definition of this comes with a natural transformation out of the identity
This in turn is compatible with in that
Consider then a sufficiently deep transfinite composition . By the small object argument available in the presentable (∞,1)-category this stabilizes, and hence provides a reflection .
Since transfinite composition is a filtered (∞,1)-colimit and since in an (∞,1)-topos these commute with finite (∞,1)-limits, it follows that spectrum objects are an left exact reflective sub-(∞,1)-category.
For an (∞,1)-topos over the base (∞,1)-topos , its tangent (∞,1)-category is an (∞,1)-topos over the base (and hence in particular also over itself).
By the the spectrification lemma has a geometric embedding into the (∞,1)-presheaf (∞,1)-topos , and this implies that it is an (∞,1)-topos (by the discussion there).
Moreover, since both adjoint (∞,1)-functor in the global section geometric morphism preserve finite (∞,1)-limits they preserve spectrum objects and hence their immediate (∞,1)-presheaf prolongation immediately restricts to the inclusion of spectrum objects
This statement also follows from the general theory of excisive functors, and in this form it is due to Charles Rezk. See at n-Excisive functor – Properties – n-Excisive reflection for the above fact and its generalization to “Goodwillie jet bundles”.
We may think of the tangent -topos as being an extension of by its stabilization :
Crucial for the internal interpretation in homotopy type theory is that the homotopy types in are stable homotopy types.
One may understand as the result of adjoining to a universal “stable object”
For details see at excisive (∞,1)-functor the section Characterization via a generic stable object.
Assume that is a cohesive (∞,1)-topos over ∞Grpd, in that there is an adjoint quadruple
with being full and faithful (∞,1)-functors and preserving finite (∞,1)-products.
Since (∞,1)-limits and (∞,1)-colimits in an (∞,1)-presheaf (∞,1)-topos are computed objectwise, this adjoint quadruple immediately prolongs to
Moreover, all three right adjoints preserves the (∞,1)-pullbacks involved in the characterization of spectrum objects and hence restrict to
But then we have a further left adjoint given as the composite
Again since is a left exact (∞,1)-functor this composite preserves finite (∞,1)-products.
So it follows in conclusion that if is a cohesive (∞,1)-topos then its tangent -category is itself a cohesive (∞,1)-topos over the tangent -category of the base (∞,1)-topos, which is an extension of the cohesion of the -topos over by the cohesion of the stable -category over :
For more on this see at tangent cohesive (∞,1)-topos.
Let be the (∞,1)-category of E-∞ rings and let . Then the stabilization of the over-(∞,1)-category over
is equivalent to the category of -module spectra.
We discuss here aspects of the tangent -categories of (∞,1)-toposes.
First consider the base (∞,1)-topos ∞Grpd.
For each ∞-groupoid/homotopy type . there is a natural equivalence of (∞,1)-categories
between the slice (∞,1)-category of ∞Grpd over and the (∞,1)-functor (∞,1)-category of maps .
By the (∞,1)-Grothendieck construction.
For each ∞-groupoid/homotopy type . there is a natural equivalence of (∞,1)-categories
between the fiber of the tangent (∞,1)-category of ∞Grpd over , def. , and the (∞,1)-category of parameterized spectra over .
Applying remark in remark yields that
The statement then follows with the “stable Giraud theorem”.
This means that the tangent -category is equivalently what in (Joyal 08, section 30.34) is denoted in the case that is the (∞,1)-category of spectra.
The tangent -category is itself an (∞,1)-topos.
With the above equivalence this is (Joyal 08, section 35.5, 35.6 (with Georg Biedermann)).
The terminal object in should be the zero spectrum regarded as a parameterized spectrum over the point
From this it follows that
The global elements/global sections functor (which forms the (∞,1)-categorical mapping space out of the terminal object)
sends an -parameterized spectrum to its base homotopy type .
This functor has a left and right adjoint (∞,1)-functor both given by sending to the zero spectrum bundle over .
So we have an infinite chain of adjoint (∞,1)-functors
The functor is a full and faithful (∞,1)-functor
and so the tangent -category is cohesive over ∞Grpd, hence by prop. is a cohesive (∞,1)-topos:
Recalling that here , we have one more adjunction, the cotangent complex adjunction due to prop.
For a general (∞,1)-topos the above discussion goes through essentially verbatim. If is itself cohesive, then we end up with
For a locally ∞-connected (∞,1)-topos (hence in particular for a cohesive (∞,1)-topos), there are canonical (∞,1)-functors
and such that covers the global section geometric morphism in that it fits into a square
By definition of stabilization, is the (∞,1)-Grothendieck construction of
Since the loop space object (∞,1)-functor is an (∞,1)-limit construction and since the right adjoint global section functor preserves all (∞,1)-limits, there is a homotopy-commuting diagram
in (∞,1)Cat. This induces a natural morphism
and hence a morphism
The morphism in question is the postcomposition of this with pullback/restriction of the (∞,1)-Grothendieck construction along the reflective inclusion (by assumption on )
where we used that by reflectivity .
When is an -topos it should carry another structure of a symmetric monoidal (∞,1)-category, induced by fiberwise smash product of spectrum objects….
Discussion of model category models is in
The -category theoretic definition and study of the notion of tangent -categories is from
and
The (infinity,1)-topos structure on tangent -categories (cf. Joyal loci):
with expository emphasis in:
Presentation by model categories is discussed in
Yonatan Harpaz, Joost Nuiten, Matan Prasma, Tangent categories of algebras over operads (arXiv:1612.02607)
Yonatan Harpaz, Joost Nuiten, Matan Prasma, The abstract cotangent complex and Quillen cohomology of enriched categories (arXiv:1612.02608)
Yonatan Harpaz, Joost Nuiten, Matan Prasma, The tangent bundle of a model category, Theory and Applications of Categories Vol. 34, 2019, No. 33, pp 1039-1072, journal web site
Vincent Braunack-Mayer, Combinatorial parametrised spectra (based on the PhD thesis),
Algebr. Geom. Topol. 21 (2021) 801-891 [arXiv:1907.08496, doi:10.2140/agt.2021.21.801]
Generalization to parameterized objects in any stable (∞,1)-category is discussed in:
A more general notion of a tangent structure on an -category was attempted to reformulate the Goodwillie calculus in:
Note that this is an extension to -categories of the concept of a tangent bundle category. In the following slides it is claimed that the tangent -category construction of this page is a particular case of their more general construction (see slide 3):
Two further examples of tangent structures on an -category, on (∞,1)-Topos and its opposite, are given in:
Exposition:
Last revised on July 13, 2024 at 07:56:13. See the history of this page for a list of all contributions to it.