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Dec
12
comment Contradictory Orientations of Faces in Simplicial Complexes
Have you not seen the construction of simplicial homology?
Dec
12
comment Module of differentials in the functorial approach to schemes and quasi-coherent modules
Is your formula correct? I would rather guess $\int_A \prod_{x \in X (A)} \mathrm{Der}_R (A, M (x))$, and even then it's not obvious that it's right. Anyway, sometimes things have both a left universal property and a right universal property.
Dec
12
comment Contradictory Orientations of Faces in Simplicial Complexes
Actually, they are supposed to be contradictory. Otherwise you wouldn't get a chain complex.
Dec
12
comment Module of differentials in the functorial approach to schemes and quasi-coherent modules
I couldn't say, I don't do algebraic geometry!
Dec
11
comment Definition of $\mathbb{A}^n_S$ by glueing
Who says $\mathbb{A}^n_{U_\alpha \cap U_\beta} \to \mathbb{A}^n_{U_\alpha}$ isn't an open immersion? It is. Anyway, just take an open affine cover of $U_\alpha \cap U_\beta$.
Dec
11
comment If $f= \mathrm{ker}\,g$, then $g = \mathrm{coker}\,f$?
Of course not. Cokernels are epimorphic.
Dec
10
comment History of the term “anodyne” in homotopy theory
I believe the phrase is due to Gabriel and Zisman.
Dec
10
comment what is a path that cover all of $S^n$?
It simply means that the image of the path is all of $S^n$.
Dec
10
comment Geometric Homotopy as Chain Homotopy
So what are you asking, geometric homotopies as chain homotopies or the other way around?
Dec
9
comment Sequences or 'chains' of adjoint functors
I also mention that there are arbitrarily long finite strings.
Dec
9
comment Uniqueness of the long exact sequence in homology
I doubt you could get away with that. Actually, I doubt you could avoid looking at the construction of spectral sequences. (The "proof" with spectral sequences is almost surely circular. You need the long exact sequence in homology in order to construct the spectral sequence of a filtered complex.)
Dec
8
comment About certain regular epimorphisms in a Grothendieck Topos
In fact, if you can prove it for $\mathbf{Set}$, then it holds in any Grothendieck topos, because your question only involves finite limits and colimits. Also, every epimorphism is already regular.
Dec
7
comment How to use bijection of two sets to define a bijection of their respective differences?
No. For instance, let $B$ be a countably infinite proper subset of a countably infinite set $A$.
Dec
7
comment Category with pullbacks but not equalizers
At a guess, I would say $\mathbf{LH}$, the category of topological spaces and local homeomorphisms.
Dec
7
comment Intersection of localizations of an integral domain
Well, $A \subseteq \bigcap_\mathfrak{p} A_\mathfrak{p} \subseteq \bigcap_\mathfrak{m} A_\mathfrak{m}$, so if $\bigcap_\mathfrak{m} A_\mathfrak{m} = A$ then certainly $\bigcap_\mathfrak{p} A_\mathfrak{p} = A$. This is just elementary set theory.
Dec
7
comment Does an interpretation of a structure by itself induce a bijection on the automorphism group of the structure?
Yes, but if you read the part where he actually defines the functor he talks about signatures.
Dec
7
comment Does an interpretation of a structure by itself induce a bijection on the automorphism group of the structure?
Your terminology is not entirely precise. One does not have an interpretation of one structure in another but rather of one signature in another, subject to conditions etc.
Dec
6
comment What does $S^z$ mean for each $z\in\mathbb{C}$?
Frankly, no, none of this makes any sense. Not everything is begging to be generalised.
Dec
6
comment No injection $Ord \to A$
So you are being asked to construct the Hartogs number?
Dec
6
comment Colimits in full subcategory (of all monics) of arrow category
Not really. Just read category theory in general, I suppose.