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$\newcommand{\Hom}{\operatorname{Hom}}$Suppose we want to define the category of arrows of $S$. Below are two forms of doing it. Definition 1: If $D$ is of the following type: $\bullet \to \bullet$, …

Considering both the ubiquity of Kan extensions in category theory (as MacLane stated, 'The notion of Kan extensions subsumes all the other fundamental concepts of category theory.'), its early introd …

Wikipedia seems to have an answer "The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories in 1965. The more general concept of bicategory (or weak 2-cat …

These days I am wandering on a wild adventure in an incredible but intimidating land. Fortunately, I could find a guide to some animals of this land Phenomena of gerbes But someone said to me that thi …

There is an article where you can find some ideas about this. Towards a new geometry of forms In: The notion of space in Grothendieck: from schemes to a geometry of forms, John Alexander Cruz Morale …

There is a (seemingly simple) statement in the literature on sheaf theory, namely, If $E$ is the site of opens of a topological space $X$, the notion of sheaf over $X$ coincides with that of sheaf of …

I would like to enter the world of derivators. We can find a little history here and there about the limitations of triangulated categories and the motivation to enhance them, but also to compute homo …

We often find in Grothendieck terminology the words variable and fixed (or absolute). For example in SGA 4 studies variable topological spaces, groups, and categories as examples of morphisms of topos …

Take a look at the paper of Barwick, Glasman and Haine (https://arxiv.org/pdf/1807.03281.pdf). In particular the section "Exodromy for schemes & the Reconstruction Theorem".

Maybe you would like to see the thesis of O. Leroy, Groupoïde fondamental et théorème de Van Kampen en théorie des topos, available from https://plmbox.math.cnrs.fr/f/7ffa366379144dd4bacc/?dl=1 — the …

It is not clear to me the role of the domain and target in the definition of prederivators. For instance, the classical references put the domain as $\mathit{Dia}$, others as $\mathit{Cat}$ itself. So …

In "Toward a Galoisian interpretation of homotopy theory" (2000), B. Toën wrote: Pour expliquer notre point de vue sur la notion de champs rappelons une construction (non conventionnelle) du topos de …