André JOYAL - New variations on the notion of topos
Why MathOverflow recommends it
MathOverflow users cite it as the source for Joyal's definitions of locus and paratopos, and for his claim that families of pointed sets are equivalent to copresheaves on the walking arrow with a section.
Recommended by 0 people across 0 different months, 2016 to 2023.
What it is
A one-hour lecture at IHES by category theorist Andre Joyal introducing two new variations on the notion of topos: loci and para-topos. Examples include pointed spaces and spectra as loci, and small weak n-categories as a para-topos.
What people said
Cites user84563 · 2016-12 · MathOverflow · on “Families of Pointed Sets”
Question At around minute 55 in this lecture [this video] by Andre Joyal, he claims that the category $Fam(Set_{*})$ of families of pointed sets is equivalent to the category $[I_{\sharp},Set]$ of copresheaves on the walking-arrow-equipped with a section $I_{\sharp}$ (background is given below). In the video he constructs a functor $F:[I_{\sharp},Set]\rightarrow Fam(Set_{*})$ . I can see how this is essentially surjective, but at the moment cannot see how it is fully faithful (this is probably quite trivial). Can someone either demonstrate the full and faithfulness of $F$ or construct the weak inverse of $F$ ? Background Given a category $C$ , $Fam(C)$ is the category whose objects are pairs $(I,
Cites Karthik Yegnesh · 2017-01 · MathOverflow · on “The "Family" functor for infinity-categories”
Background Given a category $C$, one can consider the category $Fam(C)$ of set-indexed familiies of objects in $C$. Formally, the objects are pairs $(X,F)$ in which $X$ is a set and $F:X\rightarrow C$ is a functor and a map $(X,F)\rightarrow (Y,G)$ is pair $(f,g)$ where $f:X\rightarrow Y$ is a function and $g:F\Rightarrow G\circ f$ is a natural transformation. The same construction works for $\infty$-categories: the objects of $Fam(C)$ for $C$ an $\infty$-category are pairs $(X,F)$ where $X$ is an $\infty$-groupoid and $F:X\rightarrow C$ is a functor and the morphisms are the same. A locus (resp. $\infty$- locus ) is a pointed locally presentable category such that $Fam(C)$ is a topos (resp.
Cites Marc Hoyois · 2018-02 · MathOverflow · on “Generalizations of tangent $\infty$-topos”
This is rarely true. For example, the axioms for ∞-topoi imply that, if $T_S\mathbf H$ is an ∞-topos, then the fibers of $p_S$ must have van Kampen pushouts [LINK] (more generally van Kampen weakly contractible colimits). In particular, the fibers cannot be $n$-categories for any finite $n$, which rules out essentially algebraic ∞-theories that are $n$-categories. This also fails for most Lawvere theories: $A_\infty$-spaces, simplicial commutative rings, ... For a concrete example, $\mathbb Z[x,y]=\mathbb Z[x]\otimes_{\mathbb Z}\mathbb Z[y]$ is not a van Kampen pushout of simplicial commutative rings, since both $\mathbb Z$ and $\mathbb Z[xy]$ have the same intersections with $\mathbb Z[x]$
Cites Jonas Frey · 2023-05 · MathOverflow · on “Why is $\rm{Cat}$ a Cartesian-closed category?”
Not an answer, but Joyal proposed the name "paratopos" for reflective subcats of presheaf cats where the reflector only preserves finite products (rather than finite limits as for toposes). As pointed out above, this preservation of finite products by the reflector is equivalent to the subcategory being an exponential ideal. Here's a video where Joyal makes this suggestion: [this video]