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Results tagged with adjoint-functors
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user 11640
6
votes
Accepted
Simple technical adjunction question
No, it is not always true. For instance, suppose $G$ is the forgetful functor from monoids to sets. Then $F$ is the free monoid functor, and the two maps in question are
$$((x_{1,1}, \ldots, x_{1,n_1} …
- 15.9k
4
votes
Accepted
When is a finitary functor induced by Ind (co)continuous
Allow me to generalise to $\kappa$-accessible categories for infinite regular cardinals $\kappa$. Your guess for (2) is correct: if $F$ preserves $\kappa$-small colimits then $\tilde{F}$ preserves col …
- 15.9k
20
votes
Accepted
Is there a monad on Set whose algebras are Tychonoff spaces?
No. In fact any full subcategory of $\mathbf{Top}$ that contains all the discrete spaces cannot be monadic over $\mathbf{Set}$ unless it contains only discrete spaces. Indeed, for any such subcategory …
- 15.9k
4
votes
Accepted
A syntactic characterisation of morphisms of algebraic theories whose induced algebraic func...
I thought about this a bit when trying to understand $\lambda$-rings – the same example that Wraith offers – and I think the answer is basically what is remarked in the cited lecture notes: it is nece …
- 15.9k