Here, I will share with you some of the most important factors as far as the right time to launch your email campaign is concerned. Let us have a look at it in detail.

Day of week: According to a survey done it has been observed that the best days to send your email marketing campaigns are Tuesdays, Wednesdays and Thursdays. Mondays are not considered to be good as after a weekend more B2B Email List in boxes are inundated and the chances are high that your prospects may easily miss your email. Fridays are also difficult to pitch as most of them are busy wrapping up the week. Saturdays and Sundays are not suitable as most business people don't work on weekends.

Time of day: It is tough and difficult to determine the most optimal time due to different time zones and secondly, emails might go out over a period of few minutes or to an hour depending on the size of your email list. Although, mornings and at the end of the day are the best times when you can target B2B Email List, as in this time zone most people are not in meetings and most likely catching up with their email. It is better to take into consideration your past experience and keep a record of a particular time zone when you see the most opens. If you do - then it is better to follow the same route.

Time of year: Summers are always slow with a lot of people and other such time include long weekends. New Years week, Christmas and Thanks Giving are such times when people are not at work and hence it is recommended to avoid these for launching your email campaigns B2B Email List.

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We are glad to announce the following event:

For more information and the link to registration, please visit the event website.

We look forward to seeing many of you there!

Following in the footsteps of late Professor Charles Ehresmann (1966), who suggested that category theory--in light of its unifying mathematical concepts and constructions--should be taught as early as linear algebra, I made a case for learning and teaching category theory beginning with pre-university in our Science eLetter:

Universal yearning for understanding

https://science.sciencemag.org/content/372/6537/36/tab-e-letters

Your time permitting, please critique.

I look forward to your corrections of any mistakes I might have made in my characterization of category theory.

Thanking you,

posina

Ehresmann, C. (1966) Trends toward unity in mathematics, Cahiers de Topologie et Géométrie Différentielle Catégoriques 8: 1-7. http://www.numdam.org/article/CTGDC_1966__8__A1_0.pdf

]]>There are many examples of this type. For example, let $\partial_1 : B^\to \to B$ be the fundamental fibration of a category $B$ which has pullbacks, the right adjoint takes any object $I$ in $B$ to the identity on $I$, making $B$ a reflexive subcategory of $B^\to$. $\partial_1$ preserves connected limits, hence preserves pullbacks. There are some other trivial examples of this kind: if $C$ has a terminal object, then the projection fibration $ \pi_B : B \times C$ has a right adjoint, sending each $I$ in $B$ to $(I,1)$ in $B \times C$, making $B$ a reflexive subcategory. The projection obviously preserves pullbacks. Another perhaps more interesting case is the fibration of polynomial functors $\mathbf{Poly}$ onto $\mathbf{Set}$: the evaluation on the singleton set $ -(1) : \mathbf{Poly} \to \mathbf{Set}$ is a fibration, and it has both a left and right adjoint, both of which are fully faithful. The right adjoint of it sends each set $A$ to the constant functor on $A$, and it makes $\mathbf{Set}$ a reflexive subcategory of $\mathbf{Poly}$. All these fibration structures are supposed to be described by the general case in the previous paragraph.

The case where it might get interesting for topos theory is that it makes every geometric inclusion of sheaf category into the corresponding presheaf category fibrational. For any site $(\mathcal C,J)$, the sheafification $a : \to \mathbf{Sh}(\mathcal C,J)$ is actually a fibration according to our previous discussion, with fibres of a sheaf $\mathcal F$ being the collection of all presheaves whose sheafification is isomorphic to $\mathcal F$, and $\mathcal F$ can be viewed as a canonical colimit of this fibre, with the unit of the sheafification adjunction being the cocone morphisms. By the way, since a quasi-topos is a reflexive subcategory of a presheaf category whose localisation preserves products, from this connection with fibration it might also be interesting to consider a reflexive subcategory of a presheaf category whose localisation preserves pullbacks.

If I'm not taking these examples wrong, I'm wondering whether this connection goes any further and whether or not it is interesting or has more interesting things to say. Due to my relatively limited background, I also do not know whether these observation has been made or remained folklore or not. If there are literatures on these, I'm also happy to know where I can find more of these related things.

Merci beaucoup!

]]>https://agrothendieck.github.io/

You can now access the recordings of Grothendieck's lectures on topoi in 1973 (Algebraic geometry and algebraic groups).

See the section 1970-1974!

The page is very much work in progress.

Enjoy!

]]>I am glad to announce the first version of our joint work with Riccardo Zanfa on relative toposes:

Relative topos theory via stacks

Here is the abstract:

*We introduce new foundations for relative topos theory based on stacks. One of the central results in our theory is an adjunction between the category of toposes over the topos of sheaves on a given site (C,J) and that of C-indexed categories. This represents a wide generalization of the classical adjunction between presheaves on a topological space and bundles over it, and allows one to interpret several constructions on sheaves and stacks in a geometrical way; in particular, it leads to fibrational descriptions of direct and inverse images of sheaves and stacks, as well as to a geometric understanding of the sheafification process. It also naturally allows one to regard any Grothendieck topos as a 'petit' topos associated with a 'gros' topos, thereby providing an answer to a problem posed by Grothendieck in the seventies. Another key ingredient in our theory is a notion of relative site, which allows one to represent arbitrary geometric morphisms towards a fixed base topos of sheaves on a site as structure morphisms induced by relative sites over that site.*

We shall progressively release expanded versions of this text contaning new developments in the directions sketched in the introduction.

This work has been recently presented at the conference *Toposes online*:

The following video focuses on the specialization of the fundamental adjunction in the setting of presheaves (or discrete fibrations):

]]>The slides and videos of talks and courses given at the recent event *Toposes online* are now available from the conference website.

Thanks again to all the speakers at this event, who have allowed us to set up a very rich and varied programme, and to IHES for making the videos available on its YouTube channel!

Feel free to engage in discussions on the content of talks in the relevant section of the Around Toposes forum. Looking forward to seeing many of you there!

]]>It is suspected that Exodromy is related to this note, see the *0.10 Acknowledgements.*

See more at Thèmes pour une harmonie.

]]>We seek to understand topoi as a certain class of stacks on the big site Sp of locales with étale covers (if you restrict to topoi with enough points, you can take sober topological spaces), that are of the form X\mapsto \text{Core(Geom(Sh(}X),\mathcal{E})) for a topos \mathcal{E} . Here Core denotes the maximal subgroupoid of a category.

The first insight (which was essentially mentioned by Prof. Caramello in her talk) is that by weak left Kan extending the functor \text{Sh}:Sp\to \text{Core}_{(2,1)} Topos along the Yoneda embedding y:Sp\to Stacks(Sp), we get a (2,1)-adjunction ( \text{Core}_{(2,1)} is the maximal sub-(2,1)-category):

Lan_y\text{Sh}:\text{Core}_{(2,1)} Topos\to Stacks(Sp)

and

y:Stacks(Sp)\to \text{Core}_{(2,1)} Topos

defined by

X\mapsto \text{Core(Geom(Sh(}X),\mathcal{E})) .

The idea now is to characterize topoi as the* quotient stacks* of localic groupoids: We claim a stack \mathcal{X} on Sp is in the essential image of y iff:

- The diagonal \Delta:\mathcal{X}\to \mathcal{X}\times \mathcal{X} is representable i.e. for any representable functor yX for a locale X and map f:yX\to \mathcal{X}\times \mathcal{X}, the pullback along \Delta is again representable.

This corresponds to the fact, that the diagonal morphism of a topos \mathcal{E} is a localic geometric morphism. Thus, the base change of the diagonal along any map \text{Sh}(X)\to \mathcal{E}\times \mathcal{E} will be a localic geometric morphism with target \text{Sh}(X) and hence a localic topos itself. - There exists a locale X and a map p:yX\to\mathcal{X} with the following property: For any locale Y and map f:yY\to\mathcal{X}, the base-change (which is representable by a locale Y' which follows from (i)) \bar{p}: yY'\to yY corresponds to an open surjection of locales Y'\to Y. This corresponds to the fact, that any topos admits an open surjective geometric morphism from a localic one.

From this data, we can reconstruct a localic groupoid and hence a topos.

This presentation would be more convenient than the localic groupoids themselves, because they tend to be rather big and complicated. For the same reason, algebraic stacks instead of "smooth groupoids in algebraic spaces" are used in algebraic geometry.

My question is now, whether this is a well-known characterization. And if not, whether this is actually correct and if somebody wants to help me with working out the details.

]]>The definition of solid Abelian group in the notes uses signed measures. If you use non-negative measures, do you usefully get a similar notion of solid Abelian monoid?

(I’m interested in understanding how far the notions for condensed sets transfer to point-free topological spaces. For them we have some decent notions of measure and integration, but you have to be much more careful about whether reals are Dedekind or 1-sided, and that makes subtraction delicate.

By the way, Ming Ng’s talk this afternoon includes suggestions that the scaling factor p from Ostrowski’s theorem in the Archimedean case is best considered as an upper real, ie with the topology of upper semicontinuity.)

Steve.

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