Tagged Questions
Various structures are studied in category theory using properties of objects and morphisms between them. Many constructions are special cases of categorical limits and colimits (e.g. products in various categories). The notions of functor and natural transformations are very important in category ...
0
votes
0answers
16 views
Direct and indirect definitions by recursion
For any given function that we wish to define by recursion, there's usually two different ways of implementing the definition, a "direct" approach and an "indirect" approach. For example, the ...
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votes
1answer
33 views
Constructing short exact sequence from complex
Suppose
$$ A^\bullet \colon \; \cdots \to A^{i-1} \xrightarrow{f^{i-1}} A^{i} \xrightarrow{f^{i}} A^{i+1} \to \cdots
$$
is a complex in an abelian category. I would like to construct an exact sequence
...
3
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0answers
51 views
Can anyone help me get the following isomorphism of rings?
Let $\mathcal{C}$ be an additive category, $\Psi (C)$ the category of all complexes over $\mathcal{C}$ with chain maps, $\mathcal{K} (C)$ the homotopy category of $\Psi (C)$, the derived category of $...
1
vote
1answer
27 views
“Dualizing” a particular Set-valued colimit
Let $F:I\to \mathcal{C}$ be a diagram, and consider the colimit $$\varinjlim_{i\in I}\hom_{\mathcal{C}}(F(i),A),$$ where $A$ is a fixed object of $\mathcal{C}$.
Is it considered "legal" to write $\...
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0answers
37 views
Reference Request: Nucleotides as adjoints [on hold]
If anyone could point me in the direction of someone or some group that has modeled codons using category theory in this manner, I would appreciate it.
3
votes
1answer
54 views
Do Maclane and Moerdijk mean “Coherent logic” when they write “Geometric logic”?
Just want to make sure. The nlab says coherent logic differs from geometric logic in that it only allows finite disjunctions. At the beginning of chapter X, Maclane and Moerdijk seem to define ...
2
votes
1answer
32 views
What does this sign mean in the definition of triangulated categories?
I have two questions about Wikipedia's definition of triangulated categories.
One of the axioms for distinguished triangles (TR 2) says that if $X\overset u\to Y \overset v\to Z \overset w\to X[1]$ ...
3
votes
0answers
56 views
$\text{Hom}_{\Lambda}(\text{Hom}_{\mathcal{C}}(V,M_1), \text{Hom}_{\mathcal{C}}(V,M_2)) \cong \text{Hom}_{\mathcal{C}}(M_1,M_2)$?
Let $\mathcal{C}$ be an additive category, $M$ an object in $\mathcal{C}$, and $add(M)$ the full subcategory of $\mathcal{C}$ consisting of all direct summands of finite sums of copies of $M$. Suppose ...
1
vote
2answers
32 views
Proof that Category Product is a Category
let A and B be categories, and let's define category product as:
$A \times B \;\;$ where:
$(1)$ the objects are pairs $(a, b)$ such that $a \in A$ and $b \in B$.
$(2)$ the arrow work like $(f,g) : (...
5
votes
0answers
57 views
Explicit construction of first derived functors.
I would like to give an explicit description of the first derived functor of familiar functors. Let me start with an example to explain what I need exactly.
Fix a group $G$ and consider the category ...
3
votes
1answer
65 views
Show a category has an initial object
Consider the category wherein:
• Objects are triples $(X,a,\phi)$, where $X$ is a set, $\;a\;$ is an element of $\;X$, and $ \;\phi: X \rightarrow X$ is an endomorphism of $\;X$.
• Morphisms $(X, a, ...
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0answers
63 views
Interesting results of Schemes through Theory of Sheaves
Recently, I started reading scheme theory and in particular all this theory which has been built up by Grothendieck and his "relatives". Whilst I was reading, thought that there must be a reason for ...
1
vote
1answer
21 views
Preserving a 2-adjunction?
Let $C$ and $D$ be small categories and let $F:C\rightleftarrows :D$ be a n adjunction between them. Given a 2-functor $J$, it is well known that $J(F):J(C)\rightleftarrows :J(D)$ is also an ...
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votes
3answers
107 views
Why aren't morphisms defined as a set of relations?
I've been studying category theory and on the books, it's never too clear what a morphism really is. Some say that a morphism could be a function, but there are examples of morphisms which are not ...
4
votes
1answer
92 views
+50
Is this an adjunction?
(Write $\langle S\rangle$ for the submonoid generated by $S$.)
Let $A$ denote a set and $S$ denote a subset of $A^*$ equipped with a distinguished well-ordering. Let $M$ denote a monoid and $f : S \...
