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Jul
15
comment Definition of (left) resolution
OK, and did you show that $H_n = 0$ for $n \ne 0$?
Jul
15
comment Algebra structure on dual to coalgebra
Use the unit–counit equation to get rid of the S-shaped bend, and then push past the braiding.
Jul
15
comment A proof that right adjoints preserve limits?
There is something to be extracted here. First of all, the limit cone is encoded into the counit of the adjunction $\Delta \dashv \varprojlim$. Secondly, the proof that adjoints compose also tells you something about the counits. So you just need to unfold these proofs a bit.
Jul
15
comment Canonical arrow between $\varinjlim _C \varprojlim _D F(C,D)\rightarrow \varprojlim_D \varinjlim _CF(C,D)$ in $\mathsf{Set}$
Let me be more explicit: you should change your notation. That might alleviate some confusion.
Jul
14
comment Canonical arrow between $\varinjlim _C \varprojlim _D F(C,D)\rightarrow \varprojlim_D \varinjlim _CF(C,D)$ in $\mathsf{Set}$
Then maybe you should think about it more carefully.
Jul
14
comment Canonical arrow between $\varinjlim _C \varprojlim _D F(C,D)\rightarrow \varprojlim_D \varinjlim _CF(C,D)$ in $\mathsf{Set}$
There is a problem with your notation – $s$ and $p$ both depend on both $C$ and $D$.
Jul
14
revised Canonical arrow between $\varinjlim _C \varprojlim _D F(C,D)\rightarrow \varprojlim_D \varinjlim _CF(C,D)$ in $\mathsf{Set}$
edited body
Jul
14
comment Canonical arrow between $\varinjlim _C \varprojlim _D F(C,D)\rightarrow \varprojlim_D \varinjlim _CF(C,D)$ in $\mathsf{Set}$
Keep unwinding the definitions!
Jul
13
comment Is the category of (pre)sheaves over a singleton isomorphic to the category of sets?
Yes, sheaves of sets on a point are equivalent to sets. For presheaves, well, that depends on what you mean...
Jul
13
comment In what kinds of categories is a monic epi an isomorphism?
A monomorphism that is an extremal epimorphism is an isomorphism, more or less by definition. Extremal epimorphisms seem to be the weakest commonly used notion of epimorphism with this property, and quite conveniently, it seems that every reasonably well-behaved epimorphism is extremal.
Jul
13
comment In what kinds of categories is a monic epi an isomorphism?
This is precisely the definition of a balanced category. Incidentally, your characterisation of epimorphisms in the category of topological spaces is incorrect. (They are exactly the surjective continuous maps.)
Jul
13
comment Simple question in “Sheaves in geometry and logic”
It is a general fact about commutative squares involving binary products of that form. You can deduce it from the pullback pasting lemma, if you like.
Jul
12
comment The Brauer group is a set
Sure. But there are not so many of them.
Jul
7
comment Opposite directions of adjunction between direct and inverse image in $\mathsf{Set}$ and $\mathsf{Sh}(X)$
It's just a question of terminology. Nothing more, nothing less.
Jul
6
comment Inverse image sheaf functor: proving $(f^\ast P)_x=P_{f(x)}$
@Arrow The codomain category would be the (very large) category of (large) categories. It is literally a functor the way I define it.
Jul
6
answered Inverse image sheaf functor: proving $(f^\ast P)_x=P_{f(x)}$
Jul
6
comment Inverse image sheaf functor: proving $(f^\ast P)_x=P_{f(x)}$
The only abstract nonsense proof I am aware of is restricted to sheaves (not presheaves), and it is based on the fact that the composite of two left adjoints is the left adjoint of the composite (of the right adjoints).
Jul
5
awarded  Popular Question
Jul
3
comment Intuitionistic Proof of $(a \Rightarrow b) \Rightarrow (\lnot b \Rightarrow \lnot a)$
@z5h No, the point is that $\lnot a$ is an abbreviation of $a \to \bot$, so if you want to prove $\lnot a$, you should assume $a$ and prove $\bot$; similarly, $\lnot b$ is an abbreviation of $b \to \bot$, so if you want to deduce $\bot$ from $\lnot b$, you should prove $b$.
Jul
3
answered Intuitionistic Proof of $(a \Rightarrow b) \Rightarrow (\lnot b \Rightarrow \lnot a)$