Add total/cototal category properties (WIP)#254
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I have some rough ideas on some of the others: on Hausdorff spaces and semigroups, I think I should be able to use an idea similar to the one for Cat to keep control over the images of constant maps. For CMon, I think the "subdirectly irreducible" property might have to do with limiting the number of maps to it - though I'm not yet at all sure how to translate that into a contradiction. And on locally ringed spaces, I have a vague idea that I might be able to define a functor whose L(T) would have a number of maps from Spec k which grows faster than possible for any single locally ringed space. Anyway, no rush on reviewing this - I was just working on this off and on over the past week, and wanted to get the progress so far pushed before resuming work on the quasitopos PR. |
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| - property: finitary algebraic | ||
| proof: Take the algebraic theory of a commutative ring. | ||
| proof: Take the algebraic theory of a commutative $R$-algebra. |
| @@ -0,0 +1,11 @@ | |||
| id: cototal | |||
| relation: is | |||
| description: 'A category $\C$ is <i>cototal</i> when its dual category is total, i.e. it is locally essentially small and the contravariant Yoneda embedding $y : \C \to [\C^{\op}, \Set]$ has a left adjoint.' | |||
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| description: 'A category $\C$ is <i>cototal</i> when its dual category is total, i.e. it is locally essentially small and the contravariant Yoneda embedding $y : \C \to [\C^{\op}, \Set]$ has a left adjoint.' | |
| description: 'A category $\C$ is <i>cototal</i> when its dual category is total, i.e. it is locally essentially small and the covariant Yoneda embedding $y : \C \to [\C^{\op}, \Set]$ has a left adjoint.' |
(not sure)
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Actually, the variance is correct on both, it's the notation that's the wrong way around. Whoops...
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| id: total | |||
| relation: is | |||
| description: 'A category $\C$ is <i>total</i> when it is locally essentially small and the covariant Yoneda embedding $y : \C^{\op} \to [\C, \Set]$ has a left adjoint. (For a concrete example of how this left adjoint might look, see <a href="/content/Grp_total_explicit_proof">here</a>.)' | |||
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| description: 'A category $\C$ is <i>total</i> when it is locally essentially small and the covariant Yoneda embedding $y : \C^{\op} \to [\C, \Set]$ has a left adjoint. (For a concrete example of how this left adjoint might look, see <a href="/content/Grp_total_explicit_proof">here</a>.)' | |
| description: 'A category $\C$ is <i>total</i> when it is locally essentially small and the contravariant Yoneda embedding $y : \C^{\op} \to [\C, \Set]$ has a left adjoint. (For a concrete example of how this left adjoint might look, see <a href="/content/Grp_total_explicit_proof">here</a>.)' |
(not sure)
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2 participants
All categories decided for "total" property.
Unknown categories for "cototal" property:
category of commutative monoids
category of Hausdorff spaces
category of locally ringed spaces
category of semigroups