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Oct
31
comment How to construct a contractible space but not locally path connected?
Actually the space that I'm pretty sure that if you intersect the space above with the closed unitary ball of $\mathbb R^2$ you should find a contractible space non locally path connected which is a cone. The space $X$ should be the intersection of space in the answer with the unit circle.
Oct
31
answered How to construct a contractible space but not locally path connected?
Oct
31
answered Prove that the centralizer subgroup is normal in the normalizer subgroup
Oct
31
comment Is this an accurate portrayal of what category theorists with an interest in the foundations are actually trying to achieve?
Is not opposed, just different. We can deal with foundation in two different ways, the first one I would call applied foundation (like in applied maths) and the second one as theoretical foundation. The first one is interested in codifying all maths concept in the language of a theory and proving the existence of some construction inside the foundational theory. The second is interested in the comparison of different theories which can be used as foundations. The problem is that this second one require a meta-theory i.e. a foundational theory to be developed.
Oct
31
comment Is this an accurate portrayal of what category theorists with an interest in the foundations are actually trying to achieve?
@user18921 The point is that a foundational theory is a theory in which we develop all the mathematics. Once we choose a topos whose theory is our foundational theory all the mathematics we are considering live inside that topos. Comparing different theories (i.e. toposes) is still a large part of logic but that's not foundations.
Oct
29
answered Is this an accurate portrayal of what category theorists with an interest in the foundations are actually trying to achieve?
Oct
29
comment Do adjoint functors really define monads?
@student I've edited the answer. Let me know if now it's more clear.
Oct
29
revised Do adjoint functors really define monads?
improved formatting
Oct
29
comment Do adjoint functors really define monads?
Now I see probably the answer is too long and confusing, I'll try to shorten it a little bit.
Oct
29
comment Do adjoint functors really define monads?
@student Well if you are referring to the last proof then that's just the proof of the exercise in Weibel's book, i.e. that for two generic adjoint functor the equality involving the $\epsilon$ should holds. Instead I'm also convinced that the pair of functor that you have indicated are not adjoint, at least not with the counit you've indicated.
Oct
28
comment Do adjoint functors really define monads?
@student after some long and accurate (I hope) calculations I've finally found the problem. As apologize for being a little messy before I've also added the proof that $\epsilon \circ LR \epsilon = \epsilon \circ \epsilon_{LR}$ for $\epsilon$ a counit of an adjunction.
Oct
28
revised Do adjoint functors really define monads?
added 1078 characters in body
Oct
28
revised Do adjoint functors really define monads?
added 1040 characters in body
Oct
27
comment Do adjoint functors really define monads?
I don't think you were wrong. It should be a pair of adjoint functor, and I'm pretty much sure that your mapping is exactly the counit.
Oct
27
comment Do adjoint functors really define monads?
@student I've edited the answer, I hope I made more clear why the equation should hold in this case. If you're interested in seeing a proof of the general result for the comonad induced by the adjoint pair of functor I'll ask you to be a little patient, since I've to write it down. Usually to prove that $LR,\mu,\eta$ induce a monad I've always followed Maclane's direct approach :)
Oct
27
revised Do adjoint functors really define monads?
added 89 characters in body
Oct
27
answered Free abelian group has a subgroup of index n
Oct
27
answered Are concrete categories whose arrows don't preserve all the structure ever interesting?
Oct
27
comment Do adjoint functors really define monads?
@student I think I've found your mistake take a look and let me know if it convinces you :)
Oct
27
revised Do adjoint functors really define monads?
corretected the answer