nForum

to my surprise I notice that with current nominal count of 3930 nLab entries we are quite close to number 4001. So maybe, to establish a tradition we might want to post a followup to 3000 and One Things to Think About but in the style of David Corfield’s nLab Digest last time (see there to get an impression for what I mean).

So, I invite everyone to post here within the next weeks a little blurb in this style, if desired, which we can then all collect together into a blog post.

Apart from the seminar notes on (oo,1)-topos theory that I keep writing (but I think (oo,1)-category theory - contents and (oo,1)-topos theory - contents are beginning to look good) I am mainly trying to bring on my personal web the entry differential cohomology in an (oo,1)-topos, and the entries relating to it, into shape.

Progress is unfortunately much slower than I would hope. But some things improve, slowly but surely. For instance I am glad that I finally cleaned up and started to expand Lie infinity-algebroid with some serious indications of and examples for the incarnation as infinitesimal oo-Lie groupoids. In that context I also started telling more details of the story at Lie infinity-groupoid, but that is still pretty stubby.

In the course of this I keep adding bits and pieces here and there. For instance this morning I posted some proofs at model structure on simplicial presheaves on the Localization of (oo,1)-presheaves at a coverage (instead of at a full Grothendieck topology). This I am using in the discussion at locally n-connected (n,1)-topos as a tool for constructing examples of those. The central motivating example is the category of (oo,1)-sheaves on the site CartSp equipped with its good open cover coverage – and finally I had recently found the time to leave some comments at CartSp as to the relevance of this innocent-looking little category.

All this, of course, is still an outgrowth of the development we had a while ago at homotopy groups in an (infinity,1)-topos. After some back and forth it looks like the following generalization of the picture there is beginning to stabilize finally: oo-Lie theory is about geometric homotopy theory in a locally oo-connected oo-topos, but not with respect to the global geometric morphisms $\mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\leftarrow}{\to}} \infty Grd$, but with respect to a relative such morphisms $\mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\leftarrow}{\to}} \mathbf{H}_{red} \stackrel{\overset{\Pi_{inf}}{\to}}{\stackrel{\leftarrow}{\to}} \infty Grd$, where $\mathbf{H}$ is an infinitesimal thickening of $\mathbf{H}_{red}$, which in turn is a finite thickening of the point.

The picture emerging here is still being drawn at structures in an (oo,1)-topos.

I just wish the day had more hours. It’s shame how long I have been elaborating on all this already, and still counting.