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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.
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 .
The tangent -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.
This is the first part of the proof of DT. prop. 1.1.9.
Presentability and limits
This is (Lurie, prop. 1.1.13).
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. 220.127.116.11)
Relation to modules
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.
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
From its definition as the fiberwise stabilization of the codomain fibration the tangent -category inherits a second -functor to , coming from the domain evaluation
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. 1.
Tangent -topos of an -topos
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 n-Excisive functor – Properties – n-Excisive reflection.
Let be the diagram category as follows:
(Joyal 08, section 35.5)
(Joyal 08, section 35.1)
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.
(Joyal 08, section 35.5)
By the the spectrification lemma 1 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
Cohesive tangent -topos of a cohesive -topos
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.
(Lurie, cor. 1.5.15).
Of an -topos
We discuss here aspects of the tangent -categories of (∞,1)-toposes.
First consider the base (∞,1)-topos ∞Grpd.
Applying remark 5 in remark 1 yields that
The statement then follows with the ”stable Giraud theorem”.
The tangent -category is itself an (∞,1)-topos.
From this it follows that
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 .
The definition and study of the notion of tangent -categories is from
and section 7.3 of
The (infinity,1)-topos structure on tangent -categories is discussed in 35.5 of