Questions tagged [limits-colimits]
For questions about categorical limits and colimits, including questions about (co)limits of general diagrams, questions about specific special kinds of (co)limits such as (co)products or (co)equalizers, and questions about generalizations such as weighted (co)limits and (co)ends.
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Writing an enriched graph as a colimit
I am working with enriched directed graphs (aka, directed graphs/quivers such that the edges are objects in a category V). I can write every graph as a filtered colimit of finite graphs, and I can ...
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Are finite colimits closed under finite colimits?
Let $C$ be a cocomplete category and $S$ a set of objects of $C$. We may assume, if need be, that the objects of $S$ are compact. Consider $S'$ the class of objects spanned by finite colimits of ...
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Coequalizer that is not absolute
A coequalizer is called absolute when it is preserved by each functor. Could somebody give me an example of a coequalizer that is not absolute (with proof) ? If possible it would be great if the ...
- 1,659
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Is the category of graded modules over a graded-commutative ring an AB5 category?
Is the category of $\mathbb{Z}$-graded modules over a graded-commutative ring an AB5 category? It is abelian, the subobjects of each object form a set, and it admits arbitrary coproducts. But I don't ...
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Prove that the forgetful functor $U: \textbf{Ab} \to \textbf{Set}$ preserves all filtered colimits
I realize that my question is exactly the same as this post here. However, I tried finding the book that was mentioned, Borceux's Handbook of Category Theory I, but my efforts to find the book here ...
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1
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Reference for the alternative construction of the free cocompletion
Given any small category $\mathcal{C}$, the Yoneda embedding $y:\mathcal{C} \to \mathbf{Set}^{\mathcal{C}^{\operatorname{op}}}$ is well-known to represent the free cocompletion of $\mathcal{C}$. That ...
- 10.1k
2
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1
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Generalizing pullbacks
A cospan is two morphisms having the same codomain. A pullback is a limit of a cospan. If instead of having two morphisms, we have a set of morphisms having the same codomain, is there a name ...
- 308
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$\mathrm{Ext}$ and direct limit
Let $R$ be a commutative Noetherian ring. Then, for $R$-modules $\{X_i\}_i$ and $Y$, do we always have $$\mathrm{Ext}^n_R(\varinjlim X_i,Y) \cong \varprojlim \mathrm{Ext}^n_R( X_i,Y)$$ for all $n\geq ...
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Cartesian products in categories of subobjects
Let $\mathcal{C}$ be a category and $A$ be an object of $\mathcal{C}$. If the inclusion functor from the category $Sub_{\mathcal{C}}(A)$ of subobjects of $A$ (objects are monomorphisms of $\mathcal{C}$...
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Properties of the Projective Limit (in $\textbf{Set}$)
Let $I$ be an index set with a preorder $\leq$ and let $(G_i)_{i \in I}$ be a family of sets. Furthermore, for all $i,j \in I$ with $i \leq j$ let $f_{ij} \colon G_i \longrightarrow G_j $ be maps such ...
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On kernels and stalks of sheaves.
Suppose we're given sheaves $F,G$ on a space $X$ and a morphism of sheaves (of abelian groups) $\phi:F\to G$.
I want to prove two things :
the presheaf $\ker \phi$, defined by $(\ker\phi)(U):=\ker(\...
- 1,048
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47
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Representable ind-objects
Let $C$ and $I$ be categories and $I$ is filtered. Let $F$ be an inductive system indexed by $I$ in $C$. Then we have the ind-object
$$X\to \varinjlim\limits_{i\in I} \mathrm{Hom}_C(X, F(i)),$$
which ...
- 4,123
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2-pullbacks in 2-functor 2-categories
Let $\mathcal{C}$ and $\mathcal{D}$ be 2-categories, and let $[\mathcal{C}, \mathcal{D}]$ be the 2-category of 2-functors from $\mathcal{C}$ to $\mathcal{D}$. If $\mathcal{D}$ has 2-pullbacks, does it ...
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Dual Concept of a Well-Powered Category
I was studying the SAFT theorem (Special Adjoint Functor Theorem), using Leinster's Basic Category Theory and I had the homework to dualize it. So far I have the following,
Consider a category $\...
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$F : I → C$ is a functor from a filtered category with any colimit. $ι : J → I$ is the embedding with $J$ cofinal. Then colim $F \cong$ colim($F ◦ ι$)
Let $I$ be a filtered category. We say a full subcategory $J$ is cofinal if for
every object $A \in I$, there is an object $B \in J$ so that $\text{Hom}(A, B) \neq \emptyset$. Let $F : I \to \mathcal ...
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