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5

This question is not really about category theory itself (though category theory is the first subject in which the issue you are running into cannot be easily swept under the rug). 1. and 2. could be equally well asked of set theory and basic algebra "In what way does the collection of all sets consist of sets?" "What are collections of sets actually?" ...


5

"Ordered by inclusion" means "$A\le B$ if only if A is a subset of B". For example, the set, U, of all vectors of the form (a, b, 3a+ 2b) is a subspace of $R^3$ so is a subset so "$U\le R^3$". And the set, V, of all vectors of the form (a, 3a, 9a) is a subspace of U: $V\le U$.


4

The set of even numbers is in fact not definable as a subset of $\omega$, considered as a structure in the language of set theory. Here's one quick way to prove it using a little model theory. Let me write $<$ for the $\in$ relation, since on $\omega$ this relation is the usual strict total ordering of natural numbers. By the compactness theorem, there ...


4

Yes, inclusion as in $\subseteq$. A small example may clarify things. Take the usual three-dimensional vector space $\mathbb{R}^3$. Geometrically speaking, there are four types of subspaces of $\mathbb{R}^3$. (i) The space consisting of the zero vector only. (ii) One-dimensional subspaces, which can be identified with lines through the origin. (iii) Two-...


3

Most of your questions have been answered well by user247327, so let me just answer your last question. The poset of all subspaces of a vector space $V$ is not linearly ordered as long as $\dim V>1$. For instance, if $v$ and $w$ are two linearly independent vectors, then $\operatorname{span}(v)$ and $\operatorname{span}(w)$ are two subspaces of $V$, ...


2

The length of the sequence $(F_\eta)_{\eta < \lambda}$ is just $\lambda$. Maybe it's nitpicky, but I'd prefer to call this a "$\lambda$-sequence" and reserve the word "sequence" for the case where $\lambda = \omega$ (the set of natural numbers).


1

In fact, what happens if we only define a "subset" of $M$ as $S=\{x\in M|l_R(x)\neq 0\}$? You can certainly define this set, and it could be considered a set of "torsion elements" inside $M$, but the set does not have as many nice properties in general. As in the example of $\mathbb Z/6\mathbb Z$ given above, $2$ and $3$ are torsion but $3-2=1$ is not, so ...


1

Thanks to Zhen Lin for his comment, here is what I believe is the correct interpretation of the words and phrases having been unclear to me. i) The pullback of the morphism $f_A:A \to C$ along the morphism $f_B:B \to C$ is the morphism $p_B:P \to B$ where $\langle P, p_A, p_B \rangle$ is the pullback, i.e. $f_A \circ p_A = f_B \circ p_B$ where $P$ is ...


1

To say that a category "consists of the data" means that in order to specify a category, one has to tell their reader exactly the data that lies inside the definition. For instance, a monoid consists of the following data: A set $M$; a binary operation $\circ:M \times M \to M$ that is associative, i.e., $\circ(g,\circ(h,k)) = \circ(\circ(g,h),k)$; and a ...


1

The idea is, if $f$ is differentiable at $a$, then "$f$ is approximately affine (linear plus a constant) close to $a$". The intuitive notion of "close to" generally translates as "in some open set". Note, incidentally, that the limit condition $$ \lim_{h \to 0} \frac{\|\epsilon(h)\|}{\|h\|} = 0 $$ implicitly assumes $\epsilon(h)$ is defined in some open ...



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