[Paper Review] Lawvere-Tierney sheaves, factorization systems, sections and $j$-essential monomorphisms in a topos

Zeinab Khanjanzadeh, Ali Madanshekaf|arXiv (Cornell University)|Mar 17, 2015

Homotopy and Cohomology in Algebraic Topology8 references3 citations

TL;DR

This paper investigates Lawvere-Tierney sheaves, factorization systems, and $j$-essential monomorphisms in a topos, establishing that an arrow $f: X \to B$ is a $j_B$-sheaf if and only if its graph is a section in $\mathcal{E}/B$ and the object of sections $S(f)$ is a $j$-sheaf in $\mathcal{E}$. It further introduces $j$-essential monomorphisms and proves every presheaf in a presheaf topos admits a maximal essential extension.

ABSTRACT

Let $j$ be a Lawvere-Tierney topology (a topology, for short) on an arbitrary topos $\mathcal{E}$, $B$ an object of $\mathcal{E}$, and $j_B = j imes 1_B$ the induced topology on the slice topos $\mathcal{E}/B$. In this manuscript, we analyze some properties of the pullback functor $Π_B:\mathcal{E} ightarrow \mathcal{E}/B$ which have deal with topology. Then for a left cancelable class $\mathcal{M}$ of all $j$-dense monomorphisms in a topos $\mathcal{E}$, we achieve some necessary and sufficient conditions for that $(\mathcal{M} , \mathcal{M}^{\perp})$ is a factorization system in $\mathcal{E}$, which is related to the factorization systems in slice topoi $\mathcal{E}/B,$ where $B$ ranges over the class of objects of $\mathcal{E}$. Among other things, we prove that an arrow $f : X ightarrow B$ in $\mathcal{E}$ is a $j_B$-sheaf whenever the graph of $f$, is a section in $\mathcal{E}/B$ as well as the object of sections $S(f)$ of $f$, is a $j$-sheaf in $\mathcal{E}$. Furthermore, we introduce a class of monomorphisms in $\mathcal{E}$, which we call them $j$-essential. Some equivalent forms of those and some of their properties are presented. Also, we prove that any presheaf in a presheaf topos has a maximal essential extension. Finally, some similarities and differences of the obtained result are discussed if we put a (productive) weak topology $j$, studied by some authors, instead of a topology.

Motivation & Objective

• To characterize $j$-sheaves in a topos via the section object $S(f)$ and the graph of a morphism $f$ in the slice topos $\mathcal{E}/B$.
• To define and study $j$-essential monomorphisms and their properties in relation to injectivity and factorization systems.
• To establish the existence of a maximal essential extension for any presheaf in a presheaf topos.
• To compare results under a Lawvere-Tierney topology versus a productive weak topology, identifying where analogies break down.

Proposed method

• Analyzes the pullback functor $\Pi_B: \mathcal{E} \to \mathcal{E}/B$ and its behavior with respect to $j$-dense monomorphisms and $j_B$-sheaves.
• Introduces $j$-essential monomorphisms as monomorphisms that reflect $j$-dense monomorphisms under pullback, using the slice topos structure.
• Applies Zorn’s Lemma to prove the existence of a maximal $j$-essential extension of a presheaf in $\widehat{\mathcal{C}}$.
• Establishes that $f: X \to B$ is a $j_B$-sheaf iff its graph is a section in $\mathcal{E}/B$ and $S(f)$ is a $j$-sheaf in $\mathcal{E}$, using categorical duality and section objects.
• Compares results under a full Lawvere-Tierney topology and a productive weak topology, noting that the sufficiency of certain factorization conditions fails for weak topologies.
• Uses internal logic and subobject classifier properties to verify closure under composition and pullback for $j$-dense monomorphisms.

Experimental results

Research questions

• RQ1 When is a morphism $f: X \to B$ in a topos $\mathcal{E}$ a $j_B$-sheaf in the slice topos $\mathcal{E}/B$?
• RQ2 Under what conditions is the pair $(\mathcal{M}, \mathcal{M}^\perp)$ a factorization system, where $\mathcal{M}$ is the class of $j$-dense monomorphisms?
• RQ3 What characterizes $j$-essential monomorphisms, and how do they relate to injectivity and essential extensions?
• RQ4 Does every presheaf in a presheaf topos admit a maximal $j$-essential extension, and how is this related to $j$-injectivity?
• RQ5 How do results for Lawvere-Tierney topologies differ when replaced by productive weak topologies, particularly in factorization and monomorphism closure?

Key findings

• An arrow $f: X \to B$ in $\mathcal{E}$ is a $j_B$-sheaf in $\mathcal{E}/B$ if and only if its graph is a section in $\mathcal{E}/B$ and the object of sections $S(f)$ is a $j$-sheaf in $\mathcal{E}$.
• $j$-essential monomorphisms are characterized as monomorphisms that reflect $j$-dense monomorphisms under pullback via $\Pi_B$, and they preserve the $j$-sheaf property.
• Every presheaf $F$ in a presheaf topos $\widehat{\mathcal{C}}$ admits a maximal $j$-essential extension, proven via Zorn’s Lemma applied to the poset of essential extensions.
• A $j$-injective presheaf (i.e., $j$-sheaf) has no proper $j$-essential extension, as such an extension would contradict the injectivity condition via a non-monomorphic retraction.
• The pullback functor $\Pi_B$ reflects $j$-essential monomorphisms, meaning if $\Pi_B(f)$ is $j_B$-essential, then $f$ is $j$-essential.
• For a productive weak topology $j$, many results analogous to those for full topologies hold, but the sufficiency of certain factorization conditions fails, as the converse of dense monomorphism closure does not hold in general.