Spahn mates (changes)

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K(a,b′)(⋄ lx,y□ l)≅K(b,a′)(x□ r,⋄ ry) K(a,b')(\diamond_l x,y \Box_l) \cong K(b,a')(x \Box_r,\diamond_r y)
a →x a′ □ l↓ λ⇓ ↓⋄ l b →y b′↦b →□ r a →x a′ →1 a′ 1↓ ϵ⇓ □ l↓ λ⇓ ↓⋄ l ⇓η′ ↓1 b →1 b →y b′ →⋄ r a′ \array{ a & \overset{x}{\to} & a' \\ \mathllap{\Box_l} \downarrow & \mathllap{\lambda} \Downarrow & \downarrow \mathrlap{\diamond_l} \\ b & \underset{y}{\to} & b' } \;\;\;\;\; \mapsto \;\;\;\;\; \array{ b & \overset{\Box_r}{\to} & a & \overset{x}{\to} & a' & \overset{1}{\to} & a' \\ \mathllap{1} \downarrow & \mathllap{\epsilon} \Downarrow & \mathllap{\Box_l} \downarrow & \mathllap{\lambda} \Downarrow & \downarrow \mathrlap{\diamond_l} & \Downarrow \mathrlap{\eta'} & \downarrow \mathrlap{1} \\ b & \underset{1}{\to} & b & \underset{y}{\to} & b' & \underset{\diamond_r}{\to} & a' }

and

b →y b′ □ r↓ ⇑ ↓⋄ r a →x a′↦a →□ l b →y b′ →1 b′ 1↓ η⇑ ↓ □ r⇑ ↓⋄ r ⇑ϵ′ ↓1 a →1 a →x a′ →⋄ l b′ \array{ b & \overset{y}{\to} & b' \\ \mathllap{\Box_r} \downarrow & \Uparrow & \downarrow \mathrlap{\diamond_r} \\ a & \underset{x}{\to} & a' } \;\;\;\;\; \mapsto \;\;\;\;\; \array{ a & \overset{\Box_l}{\to} & b & \overset{y}{\to} & b' & \overset{1}{\to} & b' \\ \mathllap{1} \downarrow & \mathllap{\eta} \Uparrow &\downarrow & \mathllap{\Box_r} \Uparrow & \downarrow \mathrlap{\diamond_r} & \Uparrow \mathrlap{\epsilon'} & \downarrow \mathrlap{1} \\ a & \underset{1}{\to} & a & \underset{x}{\to} & a' & \underset{\diamond_l}{\to} & b' }

That this is a bijection follows from the triangle identities?. The 2-cells λ\lambda and □ r\Box_r are called mates (or sometimes conjugates) with respect to the adjunctions □ l⊣□ r\Box_l \dashv \Box_r and ⋄ l⊣⋄ r\diamond_l \dashv \diamond_r (and to the 1-cells xx and yy).

Properties

Strict 2-functors preserve adjurnctions and pasting diagrams, so that i\Box_l F:K→JF \colon K \to J is a 2-\Box_l\Box_rnctor and i\Box_l λ\lambda and □ r\Box_r are mates wrt □ l⊣□ r\Box_l \dashv \Box_r and ⋄ l⊣⋄ r\diamond_l \dashv \diamond_r in KK, then FλF \lambda and F□ rF \Box_r are mates wrt F□ l⊣F□ rF \Box_l \dashv F \Box_r and F⋄ l⊣F⋄ rF \diamond_l \dashv F \diamond_r in JJ.

I\Box_l α:F⇒G\alpha \colon F \Rightarrow G is a 2-nat\Box_rral trans\Box_lormation?, then the nat\Box_rrality identities α b∘F□ l=G□ l∘α a\alpha_b \circ F \Box_l = G \Box_l \circ \alpha_a and α a∘F□ r=G□ r∘α b\alpha_a \circ F \Box_r = G \Box_r \circ \alpha_b are mates wrt F□ l⊣F□ rF \Box_l \dashv F \Box_r and G□ l⊣G□ rG \Box_l \dashv G \Box_r.

Created on December 23, 2012 at 22:41:02. See the history of this page for a list of all contributions to it.

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