Reputation
Next tag badge:
93/100 score
27/20 answers
Badges
2 54 128
Impact
~369k people reached

May
15
comment Proving that a category is cartesian closed
It would perhaps be better to divide the proof into two steps: show that your category is isomorphic to the category of presheaves on the category $\mathbb{B} \mathbb{N}$, which has a unique object and is generated by a non-trivial endomorphism, and then show that every category of presheaves on a small category is cartesian closed.
May
15
comment Derived functors - homotopical vs homological approach
The connection between universal $\delta$-functors and Quillen–Verdier derived functors is a bit subtle. I only know how to prove that they coincide when we can explicitly calculate both in terms of acyclic resolutions.
May
15
comment Right adjoint of covariant hom functor
Well, does this functor preserve colimits or not?
May
14
comment In a closed monoidal category, is $[-,-]$ always a bifunctor?
It is well known and considered obvious. If you really want a citeable justification, look at Theorem 3 in [CWM, Ch. IV, §7].
May
14
comment How general $ [X,[Y,Z]] \cong [X \times Y, Z] $ is?
It is not "well known". In fact, it is false unless one interprets $\times$ and "algebraic category" in a very particular way.
May
14
comment If a composition of two maps is smooth, as well as one of the maps, then so is the other.
At minimum $g$ must be surjective: for example, we could take $M = \emptyset$, then $g$ and $f \circ g$ are vacuously smooth but $f$ can be arbitrary.
May
14
comment What is a branch point?
That is because the circle is too big and does in fact wind around $0$.
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
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
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.