Skip to main content
deleted 390 characters in body; Post Made Community Wiki
Source Link
Tim Campion
  • 64k
  • 13
  • 143
  • 384

Theorem 4.7 in the BRICS version of Lack and Sobicisnki's Adhesive categories shows that any adhesive category has the "effective unions" (in Barr's terminology) of the question. However, if you look at the proof, you will find that the full strength of adhesivity is not used. Unwinding the proof, we need substantially less:

Theorem: (Lack and Sobicinski): Let $\mathcal C$ be a category. Suppose that

  1. $\mathcal C$ has pullbacks along monomorphisms;
  2. $\mathcal C$ has pushouts of monomorphisms along monomorphisms, and the resulting pushout squares consist of monomorphisms;
  3. Whenever $A \xrightarrow f C \xleftarrow g B$ is a jointly epimorphic pair of monomorphisms in $\mathcal C$, and $C' \xrightarrow h C$ is any map, then the base change $A' \xrightarrow{f'} C' \xleftarrow{g'} B'$ is a jointly epimorphic pair of morphisms.

Then $\mathcal C$ has effective unions.

Notes:

  • (3) holds if $\mathcal C$ has binary coproducts and pullback-stable epimorphisms.

  • In particular, this theorem implies both that topoi have effective unions and that abelian categories have effective unions.

  • To be explicit: following Barr, I say here that $\mathcal C$ has effective unions if the subobject posets in $\mathcal C$ have binary joins, which are given by $A \cup B = AB = A \amalg_{A \cap B} B$ as in the question.

    I think this still doesn't cover the abelian case.

Theorem 4.7 in the BRICS version of Lack and Sobicisnki's Adhesive categories shows that any adhesive category has the "effective unions" (in Barr's terminology) of the question. However, if you look at the proof, you will find that the full strength of adhesivity is not used. Unwinding the proof, we need substantially less:

Theorem: (Lack and Sobicinski): Let $\mathcal C$ be a category. Suppose that

  1. $\mathcal C$ has pullbacks along monomorphisms;
  2. $\mathcal C$ has pushouts of monomorphisms along monomorphisms, and the resulting pushout squares consist of monomorphisms;
  3. Whenever $A \xrightarrow f C \xleftarrow g B$ is a jointly epimorphic pair of monomorphisms in $\mathcal C$, and $C' \xrightarrow h C$ is any map, then the base change $A' \xrightarrow{f'} C' \xleftarrow{g'} B'$ is a jointly epimorphic pair of morphisms.

Then $\mathcal C$ has effective unions.

Notes:

  • (3) holds if $\mathcal C$ has binary coproducts and pullback-stable epimorphisms.

  • In particular, this theorem implies both that topoi have effective unions and that abelian categories have effective unions.

  • To be explicit: following Barr, I say here that $\mathcal C$ has effective unions if the subobject posets in $\mathcal C$ have binary joins, which are given by $A \cup B = AB = A \amalg_{A \cap B} B$ as in the question.

Theorem 4.7 in the BRICS version of Lack and Sobicisnki's Adhesive categories shows that any adhesive category has the "effective unions" (in Barr's terminology) of the question. However, if you look at the proof, you will find that the full strength of adhesivity is not used. Unwinding the proof, we need substantially less:

Theorem: (Lack and Sobicinski): Let $\mathcal C$ be a category. Suppose that

  1. $\mathcal C$ has pullbacks along monomorphisms;
  2. $\mathcal C$ has pushouts of monomorphisms along monomorphisms, and the resulting pushout squares consist of monomorphisms;
  3. Whenever $A \xrightarrow f C \xleftarrow g B$ is a jointly epimorphic pair of monomorphisms in $\mathcal C$, and $C' \xrightarrow h C$ is any map, then the base change $A' \xrightarrow{f'} C' \xleftarrow{g'} B'$ is a jointly epimorphic pair of morphisms.

Then $\mathcal C$ has effective unions.

Notes:

  • I think this still doesn't cover the abelian case.
added 765 characters in body
Source Link
Tim Campion
  • 64k
  • 13
  • 143
  • 384

According to Theorem 174.7 in the BRICS version of Lack and Sobicisnki's Lack and Sobicinski, Adhesive Categories, this holds inAdhesive categories shows that any adhesive category. According to Corollary 18 has the "effective unions" (in Barr's terminology) of the same paperquestion. However, this impliesif you look at the proof, you will find that the subobject latticefull strength of any object in an adhesive categoryadhesivity is distributive. Abelian categories generally don't have distributive subobject lattices, so they are generally not adhesiveused. So this criterion of being adhesive still doesn't coverUnwinding the abelian case.proof, we need substantially less:

I don't understand the structure of Lack and Sobicinski's paper though, and in particular I don't know where the proofs of Theorem 17Theorem: (Lack and Corollary 18 appear.Sobicinski): Let $\mathcal C$ be a category. Suppose that

  1. $\mathcal C$ has pullbacks along monomorphisms;
  2. $\mathcal C$ has pushouts of monomorphisms along monomorphisms, and the resulting pushout squares consist of monomorphisms;
  3. Whenever $A \xrightarrow f C \xleftarrow g B$ is a jointly epimorphic pair of monomorphisms in $\mathcal C$, and $C' \xrightarrow h C$ is any map, then the base change $A' \xrightarrow{f'} C' \xleftarrow{g'} B'$ is a jointly epimorphic pair of morphisms.

Then $\mathcal C$ has effective unions.

Notes:

  • (3) holds if $\mathcal C$ has binary coproducts and pullback-stable epimorphisms.

  • In particular, this theorem implies both that topoi have effective unions and that abelian categories have effective unions.

  • To be explicit: following Barr, I say here that $\mathcal C$ has effective unions if the subobject posets in $\mathcal C$ have binary joins, which are given by $A \cup B = AB = A \amalg_{A \cap B} B$ as in the question.

According to Theorem 17 of Lack and Sobicinski, Adhesive Categories, this holds in any adhesive category. According to Corollary 18 of the same paper, this implies that the subobject lattice of any object in an adhesive category is distributive. Abelian categories generally don't have distributive subobject lattices, so they are generally not adhesive. So this criterion of being adhesive still doesn't cover the abelian case.

I don't understand the structure of Lack and Sobicinski's paper though, and in particular I don't know where the proofs of Theorem 17 and Corollary 18 appear...

Theorem 4.7 in the BRICS version of Lack and Sobicisnki's Adhesive categories shows that any adhesive category has the "effective unions" (in Barr's terminology) of the question. However, if you look at the proof, you will find that the full strength of adhesivity is not used. Unwinding the proof, we need substantially less:

Theorem: (Lack and Sobicinski): Let $\mathcal C$ be a category. Suppose that

  1. $\mathcal C$ has pullbacks along monomorphisms;
  2. $\mathcal C$ has pushouts of monomorphisms along monomorphisms, and the resulting pushout squares consist of monomorphisms;
  3. Whenever $A \xrightarrow f C \xleftarrow g B$ is a jointly epimorphic pair of monomorphisms in $\mathcal C$, and $C' \xrightarrow h C$ is any map, then the base change $A' \xrightarrow{f'} C' \xleftarrow{g'} B'$ is a jointly epimorphic pair of morphisms.

Then $\mathcal C$ has effective unions.

Notes:

  • (3) holds if $\mathcal C$ has binary coproducts and pullback-stable epimorphisms.

  • In particular, this theorem implies both that topoi have effective unions and that abelian categories have effective unions.

  • To be explicit: following Barr, I say here that $\mathcal C$ has effective unions if the subobject posets in $\mathcal C$ have binary joins, which are given by $A \cup B = AB = A \amalg_{A \cap B} B$ as in the question.

added 164 characters in body
Source Link
Tim Campion
  • 64k
  • 13
  • 143
  • 384

According to Theorem 17 of Lack and Sobicinski, Adhesive Categories, this holds in any adhesive category. According to Corollary 18 of the same paper, this implies that the subobject lattice of any object in an adhesive category is distributive. Abelian categories generally don't have distributive subobject lattices, so they are generally not adhesive. So this criterion of being adhesive still doesn't cover the abelian case.

I don't understand the structure of Lack and Sobicinski's paper though, and in particular I don't know where the proofs of Theorem 17 and Corollary 18 appear...

According to Theorem 17 of Lack and Sobicinski, Adhesive Categories, this holds in any adhesive category. According to Corollary 18 of the same paper, this implies that the subobject lattice of any object in an adhesive category is distributive. Abelian categories generally don't have distributive subobject lattices, so they are generally not adhesive. So this criterion of being adhesive still doesn't cover the abelian case.

According to Theorem 17 of Lack and Sobicinski, Adhesive Categories, this holds in any adhesive category. According to Corollary 18 of the same paper, this implies that the subobject lattice of any object in an adhesive category is distributive. Abelian categories generally don't have distributive subobject lattices, so they are generally not adhesive. So this criterion of being adhesive still doesn't cover the abelian case.

I don't understand the structure of Lack and Sobicinski's paper though, and in particular I don't know where the proofs of Theorem 17 and Corollary 18 appear...

Source Link
Tim Campion
  • 64k
  • 13
  • 143
  • 384
Loading