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May
16
comment When is $T$-Alg monoidal closed?
I don't see any reason why the algebras would be closed if the base isn't closed. After all, the hom module between two modules is a subset of the hom set... Not to mention, the identity monad is the nicest possible monad there is, so if its algebras are closed, then the base is already closed!
May
16
comment When is $T$-Alg monoidal closed?
Yes, that is the theorem of Kock. Why are you seeking different conditions?
May
16
comment When is $T$-Alg monoidal closed?
It is not necessary to assume that the coequalisers commute with $T$, but some condition about existence of coequalisers and preservation of epimorphisms seems to be required.
May
16
comment Functoriality of the Fundamental group
@Drew As is clear from the construction of $B G$ I give, I am only discussing discrete groups.
May
15
comment How to directly show that Figure 8 injective immersion is not a monomorphism
I prove that no such maps exist in my second paragraph. I believe your understanding is incorrect: if I recall correctly, the regular monomorphisms are embeddings.
May
15
answered How to directly show that Figure 8 injective immersion is not a monomorphism
May
15
answered Functoriality of the Fundamental group
May
15
comment Functoriality of the Fundamental group
Aren't you basically asking whether there is a choice of Eilenberg–MacLane spaces $K(G, 1)$ that is functorial in $G$? The answer is yes.
May
15
comment Plus construction of a presheave factors every sheaf-valued morphism.
Something like that. Doing things internally in $\mathcal{S}$ is a bit too restrictive, however; it would be like working only with small categories.
May
15
comment Plus construction of a presheave factors every sheaf-valued morphism.
Yes, but that's not what I mean. There is a notion of $\mathcal{S}$-topos, where $\mathcal{S}$ is another topos, which one thinks of as being a topos "based on" $\mathcal{S}$, and there is a surprising amount of topos theory that carries over to this relative context including, yes, the machinery of Grothendieck topologies. The whole of Part B of Sketches of an elephant is devoted to relative topos theory.
May
15
comment Plus construction of a presheave factors every sheaf-valued morphism.
General advice: if you're doing topos theory, you can almost always do everything intuitionistically; in particular, you should not need to appeal to the law of excluded middle. So don't do case analysis!
May
15
answered Category of adjunctions inducing a particular monad
May
14
comment Do Boolean rings always have a unit element?
My definition of ring includes a $1$, and my definition of boolean algebra also includes a $1$.
May
14
answered Commuting square of functors
May
12
comment Why does every countable limit ordinal have cofinality $\omega$?
If there is a cofinal $\alpha$-sequence in $\beta$ and a cofinal $\beta$-sequence in $\gamma$, then there is a cofinal $\alpha$-sequence in $\gamma$. Therefore cofinalities are initial ordinals.
May
12
answered Viewing groups as objects of the concrete category $\mathsf{Grp}$
May
11
answered Commutativity of a sheaf of groups from an epimorphism
May
11
answered Space modelled on ring
May
11
comment Space modelled on ring
Functors are not the best way to think about schemes if you're looking for spaces modelled on rings. Rather, a scheme is a locally ringed space that is locally isomorphic to the spectrum of a ring. There is a functorial way of looking at schemes but I find it much more artificial.
May
11
comment Space modelled on ring
Do you mean a scheme?