Articles of limits colimits

Graphic intuition for generalizing to weighted limits

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 $\mathsf{Nat}(\Delta\mathbf 1,\mathsf D(-,F-)):\mathsf{D}^\text{op}\longrightarrow \mathsf{Set}$. […]

Can a nonzero module be cocompact?

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 […]

Modules finitely generated and of finite type (categorical meaning)

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 […]

Topology on $R((t))$, why is it always the same?

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 […]

Inverse image sheaf and éspace étalé

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 $V$ […]

Meaning of pullback

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 $f$ […]

Concrete description of (co)limits in elementary toposes via internal language?

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 […]

Category-theoretic limit related to topological limit?

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 ( […]

In (relatively) simple words: What is an inverse limit?

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 […]

Compact subset in colimit of spaces

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 […]