Timeline for Galois connections
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 2, 2011 at 2:44 | vote | accept | James Propp | ||
| Aug 2, 2011 at 2:43 | vote | accept | James Propp | ||
| Aug 2, 2011 at 2:43 | |||||
| Jul 30, 2011 at 0:45 | comment | added | Todd Trimble | to write down analogous formulas for $R/T$ and $S \backslash R$ so that one has $V$-isomorphisms $\hom(S, R/T) \cong \hom(S \otimes T, R) \cong \hom(T, S\backslash R)$$ between enriched homs. These give you the adjunctions you need to make a complete analogy. | |
| Jul 30, 2011 at 0:42 | comment | added | Todd Trimble | @Qiaochu: it's very general, as you surmise. One generalization considers $V$-enriched profunctors $R: X \times Y \to V$ for say a bicomplete symmetric monoidal closed category $V$. (The classical case discussed above takes $V = \\{0 \leq 1\\}$.) Then you can get a contravariant enriched adjunction between the functor categories $V^X$ and $V^Y$. Briefly, using the monoidal structure on $V$, there's an obvious way of combining two functors $S: X \to V$, $T: Y \to V$ to get a functor $S \otimes T: X \otimes Y \to V$. Then, using completeness and the closed structure of $V$, it is possible [cont] | |
| Jul 30, 2011 at 0:17 | comment | added | Gerhard Paseman | The relation of satisfiability between algebraic structures and identities (universally quantifies equations which are sentences) is an example involving a class and a set that gives rise to a Galois connection of equationally closed theories and classes of structures closed under certain operators. If the original poster takes care to call a class a class, a set a set, and not mix the two, then again there are no worries. Gerhard "Ask Me About System Design" Paseman, 2011.07.29 | |
| Jul 29, 2011 at 23:29 | comment | added | Qiaochu Yuan | How general is this construction? Can I take a functor $X \times Y \to C$ for categories $X, Y, C$ and turn this into a pair of adjoint functors between $X$ and $Y$? If not, what extra conditions do I need? | |
| Jul 29, 2011 at 20:37 | history | answered | Todd Trimble | CC BY-SA 3.0 |