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Philippe Gaucher
Philippe Gaucher
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I am reading the paper http://www.numdam.org/article/CTGDC_2001__42_1_51_0.pdf fixing the implication $(ii)\Rightarrow (i)$ of Theorem 1.39 of Adamek-Rosicky's book. The correct statement is: if $\mathcal{K}$ is a reflective subcategory of a LFP category $\mathcal{L}$ closed under filtered colimits and such that the theory of the embedding $\mathcal{K}\subset \mathcal{L}$ is a quotient, then $\mathcal{K}$ is a small $\omega$-orthogonality class of $\mathcal{L}$.

I can "prove" that the theory of the embedding $\mathcal{K}\subset\mathcal{L}$ is always a quotient, which is obviously wrong. Where am I wrong in what follows ?

Let $r:\mathcal{L} \to \mathcal{K}$ be the left adjoint of the inclusion functor $\mathcal{K}\subset \mathcal{L}$$i:\mathcal{K}\subset \mathcal{L}$. I am using the terminology of https://ncatlab.org/nlab/show/Gabriel-Ulmer+duality: $\text{Lex}$ is the category of finitely complete (essentially) small categories together with the finite limit-preserving functors. The theory of the embedding $\mathcal{K}\subset \mathcal{L}$ is the $\text{Lex}$-functor $r^{op}:\mathcal{L}_{fp}^{op} \to \mathcal{K}_{fp}^{op}$. To prove that it is a quotient, we have to prove two things: 1) that every object $B$ of $\mathcal{K}_{fp}^{op}$ is isomorphic to an object $r^{op}A$: indeed, $r^{op}i^{op}B\cong B$ since $\mathcal{K}$ is a full subcategory of $\mathcal{L}$; 2) that every map $b:X\to r^{op}A$ factors as a composite $X\to r^{op}A' \to r^{op}A$ where the left-hand map is an isomorphism and the right-hand map is $r^{op}f$ for some $f:A'\to A$, which seems to be a consequence of $\mathcal{K}^{op}_{fp}(X,r^{op}A) \cong \mathcal{K}(rA,X) \cong \mathcal{L}(A,A') \cong \mathcal{L}^{op}_{fp}(A',A)$ with $A'=iX$.

I am reading the paper http://www.numdam.org/article/CTGDC_2001__42_1_51_0.pdf fixing the implication $(ii)\Rightarrow (i)$ of Theorem 1.39 of Adamek-Rosicky's book. The correct statement is: if $\mathcal{K}$ is a reflective subcategory of a LFP category $\mathcal{L}$ closed under filtered colimits and such that the theory of the embedding $\mathcal{K}\subset \mathcal{L}$ is a quotient, then $\mathcal{K}$ is a small $\omega$-orthogonality class of $\mathcal{L}$.

I can "prove" that the theory of the embedding $\mathcal{K}\subset\mathcal{L}$ is always a quotient, which is obviously wrong. Where am I wrong in what follows ?

Let $r:\mathcal{L} \to \mathcal{K}$ be the left adjoint of the inclusion functor $\mathcal{K}\subset \mathcal{L}$. I am using the terminology of https://ncatlab.org/nlab/show/Gabriel-Ulmer+duality: $\text{Lex}$ is the category of finitely complete (essentially) small categories together with the finite limit-preserving functors. The theory of the embedding $\mathcal{K}\subset \mathcal{L}$ is the $\text{Lex}$-functor $r^{op}:\mathcal{L}_{fp}^{op} \to \mathcal{K}_{fp}^{op}$. To prove that it is a quotient, we have to prove two things: 1) that every object $B$ of $\mathcal{K}_{fp}^{op}$ is isomorphic to an object $r^{op}A$: indeed, $r^{op}i^{op}B\cong B$ since $\mathcal{K}$ is a full subcategory of $\mathcal{L}$; 2) that every map $b:X\to r^{op}A$ factors as a composite $X\to r^{op}A' \to r^{op}A$ where the left-hand map is an isomorphism and the right-hand map is $r^{op}f$ for some $f:A'\to A$, which seems to be a consequence of $\mathcal{K}^{op}_{fp}(X,r^{op}A) \cong \mathcal{K}(rA,X) \cong \mathcal{L}(A,A') \cong \mathcal{L}^{op}_{fp}(A',A)$ with $A'=iX$.

I am reading the paper http://www.numdam.org/article/CTGDC_2001__42_1_51_0.pdf fixing the implication $(ii)\Rightarrow (i)$ of Theorem 1.39 of Adamek-Rosicky's book. The correct statement is: if $\mathcal{K}$ is a reflective subcategory of a LFP category $\mathcal{L}$ closed under filtered colimits and such that the theory of the embedding $\mathcal{K}\subset \mathcal{L}$ is a quotient, then $\mathcal{K}$ is a small $\omega$-orthogonality class of $\mathcal{L}$.

I can "prove" that the theory of the embedding $\mathcal{K}\subset\mathcal{L}$ is always a quotient, which is obviously wrong. Where am I wrong in what follows ?

Let $r:\mathcal{L} \to \mathcal{K}$ be the left adjoint of the inclusion functor $i:\mathcal{K}\subset \mathcal{L}$. I am using the terminology of https://ncatlab.org/nlab/show/Gabriel-Ulmer+duality: $\text{Lex}$ is the category of finitely complete (essentially) small categories together with the finite limit-preserving functors. The theory of the embedding $\mathcal{K}\subset \mathcal{L}$ is the $\text{Lex}$-functor $r^{op}:\mathcal{L}_{fp}^{op} \to \mathcal{K}_{fp}^{op}$. To prove that it is a quotient, we have to prove two things: 1) that every object $B$ of $\mathcal{K}_{fp}^{op}$ is isomorphic to an object $r^{op}A$: indeed, $r^{op}i^{op}B\cong B$ since $\mathcal{K}$ is a full subcategory of $\mathcal{L}$; 2) that every map $b:X\to r^{op}A$ factors as a composite $X\to r^{op}A' \to r^{op}A$ where the left-hand map is an isomorphism and the right-hand map is $r^{op}f$ for some $f:A'\to A$, which seems to be a consequence of $\mathcal{K}^{op}_{fp}(X,r^{op}A) \cong \mathcal{K}(rA,X) \cong \mathcal{L}(A,A') \cong \mathcal{L}^{op}_{fp}(A',A)$ with $A'=iX$.

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Philippe Gaucher
Philippe Gaucher
  • 3.9k
  • 20
  • 30

About small $\omega$-orthogonality classes and Gabriel-Ulmer duality

I am reading the paper http://www.numdam.org/article/CTGDC_2001__42_1_51_0.pdf fixing the implication $(ii)\Rightarrow (i)$ of Theorem 1.39 of Adamek-Rosicky's book. The correct statement is: if $\mathcal{K}$ is a reflective subcategory of a LFP category $\mathcal{L}$ closed under filtered colimits and such that the theory of the embedding $\mathcal{K}\subset \mathcal{L}$ is a quotient, then $\mathcal{K}$ is a small $\omega$-orthogonality class of $\mathcal{L}$.

I can "prove" that the theory of the embedding $\mathcal{K}\subset\mathcal{L}$ is always a quotient, which is obviously wrong. Where am I wrong in what follows ?

Let $r:\mathcal{L} \to \mathcal{K}$ be the left adjoint of the inclusion functor $\mathcal{K}\subset \mathcal{L}$. I am using the terminology of https://ncatlab.org/nlab/show/Gabriel-Ulmer+duality: $\text{Lex}$ is the category of finitely complete (essentially) small categories together with the finite limit-preserving functors. The theory of the embedding $\mathcal{K}\subset \mathcal{L}$ is the $\text{Lex}$-functor $r^{op}:\mathcal{L}_{fp}^{op} \to \mathcal{K}_{fp}^{op}$. To prove that it is a quotient, we have to prove two things: 1) that every object $B$ of $\mathcal{K}_{fp}^{op}$ is isomorphic to an object $r^{op}A$: indeed, $r^{op}i^{op}B\cong B$ since $\mathcal{K}$ is a full subcategory of $\mathcal{L}$; 2) that every map $b:X\to r^{op}A$ factors as a composite $X\to r^{op}A' \to r^{op}A$ where the left-hand map is an isomorphism and the right-hand map is $r^{op}f$ for some $f:A'\to A$, which seems to be a consequence of $\mathcal{K}^{op}_{fp}(X,r^{op}A) \cong \mathcal{K}(rA,X) \cong \mathcal{L}(A,A') \cong \mathcal{L}^{op}_{fp}(A',A)$ with $A'=iX$.