One of the ways to define a limit of a functor $F:\mathsf C\longrightarrow\mathsf D$ is a representation of $\mathsf{Nat}(\Delta-,F)$. Along the journey of generalization to the enriched setting, one notes there’s a bijection natural in $d$: $$\mathsf{Nat}(\Delta d,F)\cong \mathsf{Nat}(\Delta\mathbf 1,\mathsf D(d,F-)).$$ Thus a limit may be equivalently defined as representing of $\mathsf{Nat}(\Delta\mathbf 1,\mathsf D(-,F-)):\mathsf{D}^\text{op}\longrightarrow \mathsf{Set}$. […]

Let $R$ be a ring and $M$ be an $R$-module. Say that $M$ is cocompact if the functor $\operatorname{Hom}(-,M)$ turns filtered limits into filtered colimits. (This is just saying that $M$ is compact as an object of $R\text{-Mod}^{op}$.) My question is: Is it possible for a nonzero module $M$ to be cocompact? I suspect the […]

An object $C$ in an additive category admitting all filtered direct limits $\mathcal{C}$ is called “of finite type” if the canonical map $$\underrightarrow{\lim} Hom_{\mathcal{C}}(C,F(i))\to Hom_{\mathcal{C}}(C,\underrightarrow{\lim}F)$$ is injective for every $I$ directed poset for every functor $F:I\to \mathcal{C}$ In the case $\mathcal{C}$=Mod-R prove that this definition is equivalent to the definition of “finitely generated” The exercise […]

Let $R$ be a topological ring (commutative with $1$) and let $R((t))$ the ring of Laurent power series. So, it is the ring containing the formal power series: $$\sum_{i\ge m} a_i t^i,\quad m\in\mathbb Z,\;a\in R$$ It is easy to see that $R((t))$ can be obtained as an ind-pro objects in the following way: $$R((t))=\varinjlim_{m\in\mathbb Z}\left […]

Let $f\colon X \rightarrow Y$ be a continuous map of topological spaces. Let $\mathcal{F}$ be a sheaf of abelian groups on $Y$. The inverse image sheaf $f^{-1}(\mathcal{F})$ is the sheaf associated to the presheaf which assigns $\operatorname{colim}_{f(U) \subset V} \mathcal{F}(V)$ for every open subset $U$ of $X$, where $V$ runs through every open subset $V$ […]

I was wondering if the following two meanings of pullback are related and how: In terms of Precomposition with a function: a function $f$ of a variable $y$, where $y$ itself is a function of another variable $x$, may be written as a function of $x$. Then $f(y(x)) \equiv g(x)$ is the pullback of $f$ […]

In the category of sets, limits and colimits can be concrete described respectively as subobjects of products and quotients of coproducts. It seems like these descriptions make sense in any elementary topos. I was wondering whether they actually describe the corresponding limits and colimits, and how to see this. Clarification. I’m asking about concrete element-based […]

Is there any connection between category-theoretic term ‘limit’ (=universal cone) over diagram, and topological term ‘limit point’ of a sequence, function, net…? To be more precise, is there a category-theoretic setting of some non-trivial topological space such that these different concepts of term ‘limit’ somehow relate? This question came to me after I saw ( […]

I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of the guys in my office, and despite a very shady explanation he ended up muttering that “you usually […]

I found at the beginning of tom Dieck’s Book the following (non proved) result Suppose $X$ is the colimit of the sequence $$ X_1 \subset X_2 \subset X_3 \subset \cdots $$ Suppose points in $X_i$ are closed. Then each compact subset $K$ of $X$ is contained in some $X_k$ Now I really don’t know how […]

Intereting Posts

If $(|G|, |H|) > 1$, does it follow that $\operatorname{Aut}(G \times H) \neq \operatorname{Aut}(G) \times \operatorname{Aut}(H)$?
cyclic three variable inequality
Pullback and Pushforward Isomorphism of Sheaves
Why this polynomial represents this figure?
Only 12 polynomials exist with given properties
The binomial formula and the value of 0^0
Prove an integral inequality $|\int\limits_0^1f(x)dx|\leq\frac{1-a+b}{4}M$
Structure theorem of finite rings
Understanding the definition of Cauchy sequence
Suppose $(s_n)$ converges and that $s_n \geq a$ for all but finitely many terms, show $\lim s_n \geq a$
relative size of most factors of semiprimes, close?
Question regarding differentiation
On terms “Orientation” & “Oriented” in different mathematical areas?
Is there a geometric meaning to the outer product of two vectors?
$\dim(\ker\varphi\cap\ker\psi)=n-2$ proof