New entry representable morphism, in the sense of Grothendieck school. The notion is used at closed immersion of schemes where I just made some changes.

]]>I have added a new paragraph to direct image about direct image functor with compact support $f_! F$. Eventually I would create a separate entry direct image with compact support, but not yet.

]]>A while ago, we had some brief discussion on potential higher structures in categorical probability theory. We now have a definition of a certain abstract categorical structure which captures these phenomena and also many other examples. See the current working document.

Differently from “ordinary” higher category theory, composition in our “compositories” has the property that composing an $n$-morphism with an $m$-morphism along a common $k$-morphism face results in an $(n+m-k)$-morphism. We believe that this kind of composition is a natural structure which arises in many situations. Think, for example, of the nerve of a category, in which sequences of composable morphisms can simply be concatenated. Other examples arise from mathematical structures which can be glued along pairwise intersection, even though the general sheaf condition fails; this applies e.g. to the presheaf of metrics on a set.

Compositories may also provide a potential answer to Urs’ question on hyperstructures and higher spans.

Now the questions are:

- has anyone looked at these kind of structures before?
- is there a simple way to reformulate them in ordinary (higher) categorical terms?
- can you think of other examples?
- would it make sense to develop “compositorial” analogues of category-theoretical concepts like limits, adjunctions, etc.?

Thanks for any feedback!

]]>The nLab entry Cech cohomology claims in its first sentence that Cech cohomology of a site $C$ is the cohomology of the $(\infty,1)$-topos of presheaves on $C$ localized at Cech covers. I’m having trouble reconciling this with Remark 7.2.2.17 of Higher Topos Theory, which claims that the cohomology of this topos is ordinary sheaf cohomology. I used to assume the claim the nLab makes without thinking about it (it’s pretty plausible…), but now I think it may be wrong. Specifically, Eilenberg-Mac Lane objects are truncated and therefore hypercomplete, so they automatically satisfy descent with respect to all hypercovers, and so cohomology of an $(\infty,1)$-topos and of its hypercompletion should always be the same.

So, is the $(\infty,1)$-topos referred to in the nLab page something different than the topos of sheaves defined by Lurie? Or is there no way to view Cech cohomology as the cohomology of an $(\infty,1)$-topos?

]]>The sheafification of a (1-)presheaf on a site is classically constructed in a two-step process $X^{++}$, where $X^+$ consists of matching families in $X$, is always separated, and is a sheaf if $X$ is separated. But the sheafification can also be constructed in a single step by looking at matching families over *hypercovers*. However, the only published reference I can find which mentions this latter fact is *Higher Topos Theory* (section 6.5.3), and it doesn’t really give a proof. Does anyone know of a reference on “good old” 1-sheaves which discusses sheafification via hypercovers?

I created the page Witt Cohomology.

]]>Remake of Street’s Gummersbach paper: Characterization of Bicategories of Stacks (zoranskoda).

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