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May
14
answered What is a branch point?
May
14
comment Question on the definition of a locally presentable category
Given a fully faithful functor $\mathcal{C} \to \mathbf{Psh} (\mathcal{K})$ with a left adjoint, if $\mathcal{C}$ is accessible, then the functor is also accessible. This is an easy exercise.
May
14
comment Question on the definition of a locally presentable category
You do not need to change $K$. The inclusion is automatically accessible in that case.
May
14
comment Question on the definition of a locally presentable category
They are generators but they are not necessarily presentable.
May
13
comment Naturality in linear algebra
Well, as you say, "natural predicate" is an imprecise notion. If you define it the way you do, then it becomes a precise notion, but then what more is there to ask?
May
13
awarded  algebraic-geometry
May
12
comment What are the best topics to learn for a first (and second) course in Category Theory?
Quasicategories might well refer to something like non-locally-small categories (cf Joy of Cats) rather than simplicial sets with inner horn fillers.
May
12
answered Separated scheme stable under base extension.
May
12
comment Volume 3 of Johnstone's “Sketches of an Elephant”
@DavidRoberts I had heard that there is a draft version of the SDG chapter but I haven't seen it myself.
May
11
comment Are there any nontrivial examples of contradictions arising in non-foundational or applied math due to naive set theory?
Suppose you discovered a "natural-seeming" inconsistency in, say, Peano arithmetic. Then the only way to recover would be to reject some "natural-seeming" principle – most likely unrestricted induction. But would you really be prepared to continue doing mathematics after you give up unrestricted induction?
May
11
comment Abelian subcategory generated by a full subcategory.
It now occurs to me that free "strict" abelian categories are automatically free abelian categories as well: you can always turn any abelian category into a strict one, and any "strictly" exact functor is in particular exact, so every functor out of the generating subcategory extends to an exact functor.
May
11
comment Abelian subcategory generated by a full subcategory.
The fact that every monomorphism and epimorphism is normal is equivalent to $\operatorname{coker} \ker \cong \ker \operatorname{coker}$, which can be handled by defining an inverse to the canonical comparison $\operatorname{coker} \ker \to \ker \operatorname{coker}$. As for the existence and preservation of limits, that is automatic when we have an essentially algebraic theory.
May
11
comment Abelian subcategory generated by a full subcategory.
@QiaochuYuan For kernels, we introduce a map $\operatorname{ker} : \operatorname{mor} \mathcal{C} \to \operatorname{mor} \mathcal{C}$ and a partially defined map $c_\mathrm{ker} : \operatorname{mor} \mathcal{C} \times \operatorname{mor} \mathcal{C} \to \operatorname{mor} \mathcal{C}$ such that $c_\mathrm{ker} (g, f)$ is defined if and only if $\operatorname{codom} f = \operatorname{dom} g$ and $g \circ f = 0$, in which case $\operatorname{dom} c_\mathrm{ker} (g, f) = \operatorname{dom} f$ and $\operatorname{codom} c_\mathrm{ker} (g, f) = \operatorname{dom} \ker f$ etc.
May
11
comment Abelian subcategory generated by a full subcategory.
Hmmm. Actually, come to think of it, the 2-category of small abelian categories and exact functors to the 2-category of small categories has 2-colimits for filtered diagrams, 2-cotensors, small products, inserters, and equifiers, and these are all preserved by the forgetful 2-functor to the 2-category of small categories (or small $\mathbf{Ab}$-categories), so the only obstruction to the existence of a left 2-adjoint should be local presentability...
May
11
comment Abelian subcategory generated by a full subcategory.
@QiaochuYuan I don't have an explicit construction, but it is clear because the "strict" abelian categories I described are models for an essentially algebraic theory. The existence of the left adjoint follows from general principles, or if you like, the accessible adjoint functor theorem.
May
11
comment Abelian subcategory generated by a full subcategory.
I think there is a free abelian category. Certainly, the forgetful functor from the category of small abelian categories with chosen zero objects, biproducts, kernels, and cokernels and functors that strictly preserve this structure to the category of small categories (or even small $\mathbf{Ab}$-categories) has a left adjoint; but it's not so clear to me whether you can do the same thing with the obvious 2-categories.
May
11
comment Are there any nontrivial examples of contradictions arising in non-foundational or applied math due to naive set theory?
@TrevorWilson The whole enterprise of mathematical foundations only exists because there is the intuition that there is no "natural-seeming" way to prove the inconsistency of naïve set theory. I think if there were, we would have long ago given up on rigour.
May
11
comment Textbooks on higher category theory
Lurie's book is a textbook, but as you say, it not a textbook on $\infty$-categories. Leinster's book is not a textbook on $\infty$-categories in the same sense that CWM is a textbook on categories – you won't even find a higher Yoneda lemma in there. The same goes for the book of Cheng and Lauda.
May
11
revised Some questions about écarts
deleted 8 characters in body; edited title
May
10
comment What is the Eilenberg-Moore category of this diagonal-like monad?
Yes, of course.