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I am stuck in trying to interpret a definition in the paper "Categories of continuous functors" by P. Freyd and M. Kelly (click).

They say:

A category $\cal A$ with a proper factorization system $(\mathfrak E, \mathfrak M)$ [i.e. a factorization system where the left class is contained in the class $Epi$ and the right class in the class $Mono$ of monic arrows] has a generator when it has a small full subcategory $\cal G$ such that the family of all morphisms $G\to A$ with domain $G\in\cal G$ is in $\mathfrak E$. If $\cal A$ admits coproducts, then $\cal G$ is a generator iff the canonical arrow $\coprod_{G\to A}G\to A$ lies in $\mathfrak E$ for any $A\in\cal A$.

Proving that the two conditions are equivalent in presence of coproducts is easy: one simply uses the universal property of the coproduct. What is seems incredible to me is that this notion is the right one to capture the notion of generator, or that of separator, in $\cal A$.

In fact, one of the main point of Freyd-Kelly's paper is that the two notions are not equivalent (as they are stated on the nlab or wikipedia, if I remember well): in a finitely complete -or discrete-cocomplete- category a generator separates arrows; with a particular choice of the factorization system, a separator is a generator.

My problem is that if I interpret "the family of all morphisms $G\to A$ with domain $G\in\cal G$ is in $\mathfrak E$" in the unique possible sense, I can't obtain what I expected: "each arrow $*\to X$ is an epi in $\bf Set$" is a blatantly false statement, even if in that case the terminal object separates arrows.

Can you help me? I feel I'm lost in something easy, but I don't see where.

I am stuck in trying to interpret a definition in the paper "Categories of continuous functors" by P. Freyd and M. Kelly (click).

They say:

A category $\cal A$ with a proper factorization system $(\mathfrak E, \mathfrak M)$ [i.e. a factorization system where the left class is contained in the class $Epi$ and the right class in the class $Mono$ of monic arrows] has a generator when it has a small full subcategory $\cal G$ such that the family of all morphisms $G\to A$ with domain $G\in\cal G$ is in $\mathfrak E$. If $\cal A$ admits coproducts, then $\cal G$ is a generator iff the canonical arrow $\coprod_{G\to A}G\to A$ lies in $\mathfrak E$ for any $A\in\cal A$.

Edit: Proving that the two conditions are equivalent in presence of coproducts seemed to be easy but trying to reproduce the argument I noticed that the diagram I used wasn't commutative. I wanted to say that the diagram

gives by lifting property the desired arrow to show that each $G\to A$ lies in $\mathfrak E = {}^\perp\mathfrak M$. 

What is seems incredible to me is that this notion is the right one to capture the notion of generator, or that of separator, in $\cal A$.

In fact, one of the main point of Freyd-Kelly's paper is that the two notions are not equivalent (as they are stated on the nlab or wikipedia, if I remember well): in a finitely complete -or discrete-cocomplete- category a generator separates arrows; with a particular choice of the factorization system, a separator is a generator.

My problem is that if I interpret "the family of all morphisms $G\to A$ with domain $G\in\cal G$ is in $\mathfrak E$" in the unique possible sense, I can't obtain what I expected: "each arrow $*\to X$ is an epi in $\bf Set$" is a blatantly false statement, even if in that case the terminal object separates arrows.

Can you help me? I feel I'm lost in something easy, but I don't see where.

I am stuck in trying to interpret a definition in the paper "Categories of continuous functors" by P. Freyd and M. Kelly (click).

They say:

A category $\cal A$ with a proper factorization system $(\mathfrak E, \mathfrak M)$ [i.e. a factorization system where the left class is contained in the class $Epi$ and the right class in the class $Mono$ of monic arrows] has a generator when it has a small full subcategory $\cal G$ such that the family of all morphisms $G\to A$ with domain $G\in\cal G$ is in $\mathfrak E$. If $\cal A$ admits coproducts, then $\cal G$ is a generator iff the canonical arrow $\coprod_{G\to A}G\to A$ lies in $\mathfrak E$ for any $A\in\cal A$.

Proving that the two conditions are equivalent in presence of coproducts is easy: one simply uses the universal property of the coproduct. What is seems incredible to me is that this notion is the right one to capture the notion of generator, or that of separator, in $\cal A$.

In fact, one of the main point of Freyd-Kelly's paper is that the two notions are not equivalent (as they are stated on the nlab or wikipedia, if I remember well): in a finitely complete -or discrete-cocomplete- category a generator separates arrows; with a particular choice of the factorization system, a separator is a generator.

My problem is that if I interpret "the family of all morphisms $G\to A$ with domain $G\in\cal G$ is in $\mathfrak E$" in ht eunique possible sense, I can't obtain what I expected: "each arrow $*\to X$ is an epi in $\bf Set$" is a blatantly false statement, even if in that case the terminal object separates arrows.

Can you help me? I feel I'm lost in something easy, but I don't see where.

I am stuck in trying to interpret a definition in the paper "Categories of continuous functors" by P. Freyd and M. Kelly (click).

They say:

A category $\cal A$ with a proper factorization system $(\mathfrak E, \mathfrak M)$ [i.e. a factorization system where the left class is contained in the class $Epi$ and the right class in the class $Mono$ of monic arrows] has a generator when it has a small full subcategory $\cal G$ such that the family of all morphisms $G\to A$ with domain $G\in\cal G$ is in $\mathfrak E$. If $\cal A$ admits coproducts, then $\cal G$ is a generator iff the canonical arrow $\coprod_{G\to A}G\to A$ lies in $\mathfrak E$ for any $A\in\cal A$.

Proving that the two conditions are equivalent in presence of coproducts is easy: one simply uses the universal property of the coproduct. What is seems incredible to me is that this notion is the right one to capture the notion of generator, or that of separator, in $\cal A$.

In fact, one of the main point of Freyd-Kelly's paper is that the two notions are not equivalent (as they are stated on the nlab or wikipedia, if I remember well): in a finitely complete -or discrete-cocomplete- category a generator separates arrows; with a particular choice of the factorization system, a separator is a generator.

My problem is that if I interpret "the family of all morphisms $G\to A$ with domain $G\in\cal G$ is in $\mathfrak E$" in the unique possible sense, I can't obtain what I expected: "each arrow $*\to X$ is an epi in $\bf Set$" is a blatantly false statement, even if in that case the terminal object separates arrows.

Can you help me? I feel I'm lost in something easy, but I don't see where.

I am stuck in trying to interpret a definition in the paper "Categories of continuous functors" by P. Freyd and M. Kelly (click).

They say:

A category $\cal A$ with a proper factorization system $(\mathfrak E, \mathfrak M)$ [i.e. a factorization system where the left class is contained in the class $Epi$ and the right class in the class $Mono$ of monic arrows] has a generator when it has a small full subcategory $\cal G$ such that the family of all morphisms $G\to A$ with domain $G\in\cal G$ is in $\mathfrak E$. If $\cal A$ admits coproducts, then $\cal G$ is a generator iff the canonical arrow $\coprod_{G\to A}G\to A$ lies in $\mathfrak E$ for any $A\in\cal A$.

Proving that the two conditions are equivalent in presence of coproducts is easy: one simply uses the universal property of the coproduct. What is seems incredible to me is that this notion is the right one to capture the notion of generator, or that of separator, in $\cal A$.

In fact, one of the main point of Freyd-Kelly's paper is that the two notions are not equivalent (as they are stated on the nlab or wikipedia, if I remember well): in a finitely complete -or discrete-cocomplete- category a generator separates arrows; with a particular choice of the factorization system, a separator is a generator.

My problem is that if I interpret "the family of all morphisms $G\to A$ with domain $G\in\cal G$ is in $\mathfrak E$" in ht eunique possible sense, I can't obtain what I expected: "each arrow $*\to X$ is an epi in $\bf Set$" is a blatantly false statement, even if in that case the terminal object separates arrows. On the other hand, it seems quite obvious that every arrow $X\to *$ is epic in $\bf Sets$.

Can you help me? I feel I'm lost in something easy, but I don't see where.

I am stuck in trying to interpret a definition in the paper "Categories of continuous functors" by P. Freyd and M. Kelly (click).

They say:

A category $\cal A$ with a proper factorization system $(\mathfrak E, \mathfrak M)$ [i.e. a factorization system where the left class is contained in the class $Epi$ and the right class in the class $Mono$ of monic arrows] has a generator when it has a small full subcategory $\cal G$ such that the family of all morphisms $G\to A$ with domain $G\in\cal G$ is in $\mathfrak E$. If $\cal A$ admits coproducts, then $\cal G$ is a generator iff the canonical arrow $\coprod_{G\to A}G\to A$ lies in $\mathfrak E$ for any $A\in\cal A$.

Proving that the two conditions are equivalent in presence of coproducts is easy: one simply uses the universal property of the coproduct. What is seems incredible to me is that this notion is the right one to capture the notion of generator, or that of separator, in $\cal A$.

In fact, one of the main point of Freyd-Kelly's paper is that the two notions are not equivalent (as they are stated on the nlab or wikipedia, if I remember well): in a finitely complete -or discrete-cocomplete- category a generator separates arrows; with a particular choice of the factorization system, a separator is a generator.

My problem is that if I interpret "the family of all morphisms $G\to A$ with domain $G\in\cal G$ is in $\mathfrak E$" in ht eunique possible sense, I can't obtain what I expected: "each arrow $*\to X$ is an epi in $\bf Set$" is a blatantly false statement, even if in that case the terminal object separates arrows.

Can you help me? I feel I'm lost in something easy, but I don't see where.