Made explicit the sidedness of the adjoint, ie that we want reflective categories and not coreflective ones

Source Link

edit approved Nov 1, 2022 at 2:25

448
5
26

Consider an elementary class, $K$, of some $\mathcal{L}$-theory, $T$ equipped with the usual $\mathcal{L}$-structure homomorphisms. (Not elementary embeddings, which elementary classes are more frequently equipped with.) Suppose we have $K' \subseteq K$, the elementary class of models of $T'$. When is $K'$ a reflective subcategory of $K$, i.e. when does the inclusion functor $\iota:K'\to K$ admit ana left adjoint functor?

For example, I was considering monoids as a subcategory of groups. Here the theory of groups is given by $T' = T\cup \{\forall x \exists y\;xy=1\}$, where T is the theory of monoids. As far as I can tell though, the similarly structured sentence $\forall x\exists y\; x\sim y$ does not form a reflective subcategory for graphs, so presumably this cannot be characterized in terms of quantifier complexity, i.e. just being $\Pi_n$ or $\Sigma_n$.

Consider an elementary class, $K$, of some $\mathcal{L}$-theory, $T$ equipped with the usual $\mathcal{L}$-structure homomorphisms. (Not elementary embeddings, which elementary classes are more frequently equipped with.) Suppose we have $K' \subseteq K$, the elementary class of models of $T'$. When is $K'$ a reflective subcategory of $K$, i.e. when does the inclusion functor $\iota:K'\to K$ admit an adjoint functor?

For example, I was considering monoids as a subcategory of groups. Here the theory of groups is given by $T' = T\cup \{\forall x \exists y\;xy=1\}$, where T is the theory of monoids. As far as I can tell though, the similarly structured sentence $\forall x\exists y\; x\sim y$ does not form a reflective subcategory for graphs, so presumably this cannot be characterized in terms of quantifier complexity, i.e. just being $\Pi_n$ or $\Sigma_n$.

Consider an elementary class, $K$, of some $\mathcal{L}$-theory, $T$ equipped with the usual $\mathcal{L}$-structure homomorphisms. (Not elementary embeddings, which elementary classes are more frequently equipped with.) Suppose we have $K' \subseteq K$, the elementary class of models of $T'$. When is $K'$ a reflective subcategory of $K$, i.e. when does the inclusion functor $\iota:K'\to K$ admit a left adjoint functor?

For example, I was considering monoids as a subcategory of groups. Here the theory of groups is given by $T' = T\cup \{\forall x \exists y\;xy=1\}$, where T is the theory of monoids. As far as I can tell though, the similarly structured sentence $\forall x\exists y\; x\sim y$ does not form a reflective subcategory for graphs, so presumably this cannot be characterized in terms of quantifier complexity, i.e. just being $\Pi_n$ or $\Sigma_n$.

added 56 characters in body

Source Link

edited Nov 1, 2022 at 2:11

Joel David Hamkins

236.4k
44
777
1.4k

Consider an elementary class, $K$, of some $\mathcal{L}$-theory, $T$ equipped with the usual $\mathcal{L}$-structure homomorphisms. (Not elementary embeddings, which elementary classes are more frequently equipped with.) Suppose we have $K' \subseteq K$, the elementary class of models of $T'$. When is $K'$ a reflective subcategoryreflective subcategory of $K$, i.e. when does the inclusion functor $\iota:K'\to K$ admit an adjoint functor?

For example, I was considering monoids as a subcategory of groups. Here the theory of groups is given by $T' = T\cup \{\forall x \exists y\;xy=1\}$, where T is the theory of monoids. As far as I can tell though, the similarly structured sentence $\forall x\exists y\; x\sim y$ does not form a reflective subcategory for graphs, so presumably this cannot be characterized in terms of quantifier complexity, i.e. just being $\Pi_n$ or $\Sigma_n$.

Consider an elementary class, $K$, of some $\mathcal{L}$-theory, $T$ equipped with the usual $\mathcal{L}$-structure homomorphisms. (Not elementary embeddings, which elementary classes are more frequently equipped with.) Suppose we have $K' \subseteq K$, the elementary class of models of $T'$. When is $K'$ a reflective subcategory of $K$, i.e. when does the inclusion functor $\iota:K'\to K$ admit an adjoint functor?

For example, I was considering monoids as a subcategory of groups. Here the theory of groups is given by $T' = T\cup \{\forall x \exists y\;xy=1\}$, where T is the theory of monoids. As far as I can tell though, the similarly structured sentence $\forall x\exists y\; x\sim y$ does not form a reflective subcategory for graphs, so presumably this cannot be characterized in terms of quantifier complexity, i.e. just being $\Pi_n$ or $\Sigma_n$.

Consider an elementary class, $K$, of some $\mathcal{L}$-theory, $T$ equipped with the usual $\mathcal{L}$-structure homomorphisms. (Not elementary embeddings, which elementary classes are more frequently equipped with.) Suppose we have $K' \subseteq K$, the elementary class of models of $T'$. When is $K'$ a reflective subcategory of $K$, i.e. when does the inclusion functor $\iota:K'\to K$ admit an adjoint functor?

For example, I was considering monoids as a subcategory of groups. Here the theory of groups is given by $T' = T\cup \{\forall x \exists y\;xy=1\}$, where T is the theory of monoids. As far as I can tell though, the similarly structured sentence $\forall x\exists y\; x\sim y$ does not form a reflective subcategory for graphs, so presumably this cannot be characterized in terms of quantifier complexity, i.e. just being $\Pi_n$ or $\Sigma_n$.