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Stream: deprecated: topos theory

Topic: concrete sheaves variant

Reid Barton (Sep 25 2020 at 13:27):

In a formalization context I encountered the following variation on the notion of a concrete sheaf.
Let $C$ be a concrete site (or something similar, I'm not too concerned with exact conditions yet) and write $R : \mathrm{Set} \to \mathrm{Sh}(C)$ for the codiscrete sheaf functor (so $(RX)(K) = \mathrm{Hom}_{\mathrm{Set}}(|K|, X)$ for $K \in C$ where $|K| = \mathrm{Hom}_C(*, K)$ is the "underlying set" of $K$). I'd like to consider the category whose

• objects are pairs $(X, \mathcal{X})$ where $X \in \mathrm{Set}$ and $\mathcal{X}$ is a subsheaf of $RX$
  (equivalently, a pair of $X \in \mathrm{Set}$ and a monomorphism $\mathcal{X} \to RX$);

• morphisms from $(X, \mathcal{X})$ to $(Y, \mathcal{Y})$ are maps $X \to Y$ such that the induced map $RX \to RY$ carries $\mathcal{X}$ into $\mathcal{Y}$
  (equivalently, a map $X \to Y$ together with a commutative square ... whose top map $\mathcal{X} \to \mathcal{Y}$ is uniquely determined, when it exists, by the bottom map $RX \to RY$).

Compared to the usual category of concrete sheaves, the difference is supposed to be that we don't require the map $\mathcal{X}($) \to RX() = X to be an isomorphism, only a monomorphism. So we can also describe an objects as a concrete sheaf $\mathcal{X}$ plus a possibly bigger set $X$ containing $\mathcal{X}(*)$.

Does anyone know a name for this category?

Reid Barton (Sep 25 2020 at 13:29):

I think it can probably also be described as the category of concrete sheaves on a site $C^+$ obtained by adjoining a new terminal object to $C$. At least it seems to work for the case $C = *$, in which case the category I describe is the category of monomorphisms in the category of sets.

Reid Barton (Sep 25 2020 at 13:31):

A more down-to-earth way to describe the category is as the category whose objects are sets $X$ equipped with certain functions $|K| \to X$ which are "good", satisfying a few axioms, and whose morphisms are functions that preserve the "good" maps. Notably we do not require every map out of $*$ to be good, though.

Reid Barton (Sep 25 2020 at 23:55):

In the paper "Quasitoposes, Quasiadhesive Categories and Artin Glueing" by Johnstone, Lack and Sobociński I found the notation $C \mathbin{/{\mkern-6mu}/} T$ for the full subcategory of the comma category on the monomorphisms, and they show it is a quasitopos when $T : D \to C$ is a functor between quasitopoi that preserves pullbacks (Theorem 16).

Reid Barton (Sep 27 2020 at 00:09):

I also found this specific construction in https://arxiv.org/abs/math/0612727 under the name "quasispaces".