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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 24th 2010
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexBack [here](http://golem.ph.utexas.edu/category/2007/08/more_on_tangent_categories.html) Urs wrote on tangent spaces, differentials and curvature for $n$-categories. I can see the entry for tangent spaces, but did $n$-curvature find a place at nLab? I can't see it. More generally, should we expect much of differential geometry to translate to n-categories? I'd like to be able to understand this from [Goodwillie calculus](http://ncatlab.org/nlab/show/Goodwillie+calculus) >There are in some sense exactly two tangent connections on the category of spaces (or should we say on any model category?). Both are flat and torsion-free. There is a map between them, so it is meaningful to subtract them. As is well-known in differential geometry, the difference between two connections is a 1-form with values in endomorphisms (whereas the curvature is a 2-form with values in endomorphisms). Thus there is a way of discussing the discrepancy between pushouts and pullbacks in the language of differential geometry, but it is a tensor field of a different type from what I had guessed. I could see how the tangent space at an object could be pulled back along a morphism. Hmm, is there something a little like adjoints to the [base change](http://ncatlab.org/nlab/show/base+change) functor going on?

   Back here Urs wrote on tangent spaces, differentials and curvature for $n$-categories. I can see the entry for tangent spaces, but did $n$-curvature find a place at nLab? I can’t see it.

   More generally, should we expect much of differential geometry to translate to n-categories? I’d like to be able to understand this from Goodwillie calculus

   There are in some sense exactly two tangent connections on the category of spaces (or should we say on any model category?). Both are flat and torsion-free. There is a map between them, so it is meaningful to subtract them. As is well-known in differential geometry, the difference between two connections is a 1-form with values in endomorphisms (whereas the curvature is a 2-form with values in endomorphisms). Thus there is a way of discussing the discrepancy between pushouts and pullbacks in the language of differential geometry, but it is a tensor field of a different type from what I had guessed.

   I could see how the tangent space at an object could be pulled back along a morphism. Hmm, is there something a little like adjoints to the base change functor going on?

2. • CommentRowNumber2.
   • CommentAuthorEric
   • CommentTimeJul 24th 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexHere is a reference I'm currently looking at that is a few links deep following the links you posted, so I thought I would bump it up a little higher to make it more easily accessible: >[Arrow-theoretic differential theory](http://www.math.uni-hamburg.de/home/schreiber/tanyon.pdf) If there is a reason for me to learn category theory, understanding this paper is a big one. The opening paragraphs are inspiring.

   Here is a reference I’m currently looking at that is a few links deep following the links you posted, so I thought I would bump it up a little higher to make it more easily accessible:

   Arrow-theoretic differential theory

   If there is a reason for me to learn category theory, understanding this paper is a big one. The opening paragraphs are inspiring.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 26th 2010
   • (edited Jul 26th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexMaybe the notions that I was after were somewhat different than those needed in Goodwillie calculus . What i called "tangent category" was the codomain fibration _without_ fiberwise stabilization, which is however the crucial aspect in the Goodwillie-context. I'd like to ask anyone interested in these ideas of mine to not look at these old notes, but into what has become of these ideas meanwhile. An approximation to the state of my art is being typed up at <a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos">differential cohomology in an (oo,1)-topos</a>. The upshot of essentially all I tried to do back then has condensed into this abstract statement: for $A$ an object in a $\infty$-connected $(\infty,1)$-topos that is once deloopable, its _curvature characteristic class_ is a canonical morphism $$ A \to \mathbf{\flat}_{dR} \mathbf{B}A $$ where $\mathbf{\flat} := LConst \Gamma $ is the monad of the terminal global section geometric morphism and $\mathbf{\flat}_{dR} Q := \mathbf{\flat}Q \times_Q *$. It is then a theorem that this morphism may be _modeled_ in a model category presentation of the $(\infty,1)$-topos as the map that takes the top two-thirds of the following diagram to the bottom third $$ \array{ Y &\to& A &&& underlying\;cocycle \\ \downarrow && \downarrow \\ \mathbf{\Pi}_R(Y) &\to& \mathbf{E}A &&& connection\;and\;curvature \\ \downarrow && \downarrow \\ \mathbf{\Pi}_R(Y) &\to& \mathbf{B}A &&& curvature\;classes } \,. $$ (See the entry for the detailed statement). Here $\mathbf{E}A = A^I \times_A *$ is what I used to call "tangent category". It does indeed pick up the curvature of an $A$-cocycle in the sense of differential cohomology. I can't claim to have thought at all about how this might relate to topics in Goodwillie calculus. Maybe it's not inconceivable that there is some kind of relation. But I suppose that would require much more thinking.

   Maybe the notions that I was after were somewhat different than those needed in Goodwillie calculus . What i called “tangent category” was the codomain fibration without fiberwise stabilization, which is however the crucial aspect in the Goodwillie-context.

   I’d like to ask anyone interested in these ideas of mine to not look at these old notes, but into what has become of these ideas meanwhile. An approximation to the state of my art is being typed up at differential cohomology in an (oo,1)-topos.

   The upshot of essentially all I tried to do back then has condensed into this abstract statement:

   for $A$ an object in a $\infty$-connected $(\infty,1)$-topos that is once deloopable, its curvature characteristic class is a canonical morphism

   $$A \to \mathbf{\flat}_{dR} \mathbf{B}A$$

   where $\mathbf{\flat} := LConst \Gamma$ is the monad of the terminal global section geometric morphism and $\mathbf{\flat}_{dR} Q := \mathbf{\flat}Q \times_Q *$.

   It is then a theorem that this morphism may be modeled in a model category presentation of the $(\infty,1)$-topos as the map that takes the top two-thirds of the following diagram to the bottom third

   $$\array{ Y &\to& A &&& underlying\;cocycle \\ \downarrow && \downarrow \\ \mathbf{\Pi}_R(Y) &\to& \mathbf{E}A &&& connection\;and\;curvature \\ \downarrow && \downarrow \\ \mathbf{\Pi}_R(Y) &\to& \mathbf{B}A &&& curvature\;classes } \,.$$

   (See the entry for the detailed statement). Here $\mathbf{E}A = A^I \times_A *$ is what I used to call “tangent category”. It does indeed pick up the curvature of an $A$-cocycle in the sense of differential cohomology.

   I can’t claim to have thought at all about how this might relate to topics in Goodwillie calculus. Maybe it’s not inconceivable that there is some kind of relation. But I suppose that would require much more thinking.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 26th 2010
   • (edited Jul 26th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexDavid, you made me think. Recall that I keep claiming that * while _every_ $(\infty,1)$-topos comes with its intrinsic notion of cohomology; * if it is in addition an <a href="http://ncatlab.org/nlab/show/locally+n-connected+(n,1)-topos#InfCase">oo-connected</a> $(\infty,1)$-topos then it comes moreover with a notion of <a href="http://ncatlab.org/schreiber/show/Lie+theory+in+an+(∞%2C1)-topos">intrinsic Lie theory</a> and intrinsic _differential_ cohomology. That involves notions of _linearization_ , notably of Lie differentiation. I haven't thought about how that could relate to Goodwillie calculus, but now that you are pushing me, it begins to seem not unplausible that maybe it _should_ be related. But one needs to be careful. There are other oo-categorical notions of "linearizations", too. For instance there is the notion of [[rational homotopy theory in an (infinity,1)-topos]], which also is about linearization in an (oo,1)-topos in a sense. But I think quite differently from Goodwillie-linearization (or at least so it seems on first sight!). For a bit of time I thought that I should build all of intrinsic differential cohomology on that linearization notion of "rational homotopy theory on an (oo,1)-topos". But then I noticed that this is, while possible, not _necessary_ : the assumption of $\infty$-connectedness alone induces apparently all one needs for a notion of Lie theory, Lie differentiation, etc. So I am not sure how all these notions of linearization relate. But I am beginning to think that you are right and that this deserves more thinking.

   David,

   you made me think.

   Recall that I keep claiming that

   • while every $(\infty,1)$-topos comes with its intrinsic notion of cohomology;

   • if it is in addition an oo-connected $(\infty,1)$-topos then it comes moreover with a notion of intrinsic Lie theory and intrinsic differential cohomology.

   That involves notions of linearization , notably of Lie differentiation.

   I haven’t thought about how that could relate to Goodwillie calculus, but now that you are pushing me, it begins to seem not unplausible that maybe it should be related.

   But one needs to be careful. There are other oo-categorical notions of “linearizations”, too. For instance there is the notion of rational homotopy theory in an (infinity,1)-topos, which also is about linearization in an (oo,1)-topos in a sense. But I think quite differently from Goodwillie-linearization (or at least so it seems on first sight!).

   For a bit of time I thought that I should build all of intrinsic differential cohomology on that linearization notion of “rational homotopy theory on an (oo,1)-topos”. But then I noticed that this is, while possible, not necessary : the assumption of $\infty$-connectedness alone induces apparently all one needs for a notion of Lie theory, Lie differentiation, etc.

   So I am not sure how all these notions of linearization relate. But I am beginning to think that you are right and that this deserves more thinking.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 27th 2010
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexMore on calculus of functors at [MO](http://mathoverflow.net/questions/33455/seeing-stacks-in-the-calculus-of-functors). Mike Shulman says there: >The "homotopy calculus" of functors from Top to Top (or to Spectra) doesn't look a whole lot like stacks to me, but the "manifold calculus" of space-valued functors on some poset of subspaces of a manifold M does look very much like stacks to me. When I last thought about this (which was during Tom's talks at the Georgia topology conference), it looked kind of as though there was a hierarchy of different Grothendieck topologies on that poset of subspaces, and an nth degree polynomial functor was a stack relative to the nth topology. There seem to be three parts to the calculus (orthogonal, homotopy, manifold). I wonder how 'linear' plays itself out in each. In the third, immersions seem to form the derivative of embeddings. It would be good to understand this point from the [Goodwillie calculus](http://ncatlab.org/nlab/show/Goodwillie+calculus) page: >In these theories, covariant functors are analogous to presheaves and linear functors are analogous to sheaves (The definition of a linear functor is essentially a homotopy-invariant version of the definition of a sheaf). The process of approximating a general functor by a linear one is analogous to sheafification, and so forth.

   More on calculus of functors at MO. Mike Shulman says there:

   The “homotopy calculus” of functors from Top to Top (or to Spectra) doesn’t look a whole lot like stacks to me, but the “manifold calculus” of space-valued functors on some poset of subspaces of a manifold M does look very much like stacks to me. When I last thought about this (which was during Tom’s talks at the Georgia topology conference), it looked kind of as though there was a hierarchy of different Grothendieck topologies on that poset of subspaces, and an nth degree polynomial functor was a stack relative to the nth topology.

   There seem to be three parts to the calculus (orthogonal, homotopy, manifold). I wonder how ’linear’ plays itself out in each. In the third, immersions seem to form the derivative of embeddings.

   It would be good to understand this point from the Goodwillie calculus page:

   In these theories, covariant functors are analogous to presheaves and linear functors are analogous to sheaves (The definition of a linear functor is essentially a homotopy-invariant version of the definition of a sheaf). The process of approximating a general functor by a linear one is analogous to sheafification, and so forth.