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Mar
7
comment Can we simultaneously freely adjoin both limits and colimits to a category?
This is described in Joyal's paper to some degree, but it's not very explicit.
Mar
7
comment Can we simultaneously freely adjoin both limits and colimits to a category?
It does not seem to be available online.
Mar
7
comment Why is $Set$ not equivalent to $Set^{op}$?
The second paragraph is the explanation. Try dualising the two properties in question.
Mar
6
comment In the category $\mathsf{Set}$ regular epis are stable under pullback
In fact, regular epimorphisms in $\mathbf{Set}$ = surjective maps. If you work concretely you will immediately see that the pullback of a surjective map is a surjective map.
Mar
5
comment Size issues in 2-categories
For me, $\mathbf{Cat}$ is the category of small categories.
Mar
5
comment Size issues in 2-categories
The obvious analogue is to have the hom-categories be small.
Mar
5
comment Why would the category of topological spaces be a balanced category (i.e. monic epimorphisms are isomorphisms)?
@RobArthan A common misconception. Epimorphisms in $\mathbf{Top}$ are surjective.
Mar
4
comment What is the most general category in which exist short exact sequences?
There are all kinds of gadgets: regular categories, homological categories, semi-abelian categories...
Mar
3
comment Relation between two notions of $BG$
I think the difference roughly corresponds to the difference between general sheaves and locally constant sheaves.
Mar
2
comment Intuition for microlinearity / infinitesimal linearity?
Linear independence is not guaranteed. After all, $n$ can be arbitrarily large.
Feb
29
comment Category of finitely presented $R$-algebras cartesian closed?
Indeed. A more likely candidate would be something like the opposite of the category of finite $R$-algebras, perhaps.
Feb
29
comment Category of finitely presented $R$-algebras cartesian closed?
If anything, it seems more likely that the opposite of the category of "very small" $R$-algebras is cartesian closed. For instance, the formal dual of $R [x] / (x^2)$ is exponentiable, if I recall correctly.
Feb
28
comment $1/\det$ is a polynomial function?
Well, $Y \det X - 1 = 0$, so the function you want is just $Y$. No?
Feb
28
comment Intuition for microlinearity / infinitesimal linearity?
Well, you have to check that it does not depend on $z$, for one thing, and then you have to check all the equations...
Feb
28
comment Forgetful functor preserves coequalizers.
An equivalence relation in the category of models is a congruence (in the sense of universal algebra) and it is easy to check the quotient of an algebra by a congruence is again an algebra.
Feb
28
comment Image of presheaf morphism
The image of a presheaf morphism is the obvious thing: do it componentwise.
Feb
27
comment Geometric meaning of $D_k(n)=\left\{ (x_1,\dots ,x_n)\in R^n\mid \prod_{i=1}^{k+1}x_{\ell _i}=0 \; \forall \ell_i \right\}$
Sure. That part is obvious.
Feb
26
comment Is this property of continuous maps equivalent to properness?
Please read the question. My definition of properness includes a closedness condition.
Feb
24
comment Why is the category of n-categries cocomplete?
It seems to me that any reasonable definition of strict $n$-category will be formalisable by a finite limit sketch. It is a fact that the category of models for a finite limit sketch is locally finitely presentable – in particular, cocomplete.
Feb
23
comment Tensor of cocomplete categories
The accessible adjoint functor theorem is described in Adamek/Rosicky and also Borceux. The second thing, I don't think is described in textbooks.