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 ...

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7
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1answer
101 views

Why the octahedral axiom?

My question is about the octahedral axiom (OA) in the definition of a triangulated category. For what I can understand so far (cf. Huybrechts, Fourier-Mukai in algebraic gometry, Definition 1.32), ...
1
vote
0answers
17 views

Weaker sufficient conditions to reflect monos and epis than faithfulness?

I am looking for a weaker condition than faithfulness to test when a functor might reflect both monomorphisms and epimorphisms. It seems that faithfulness is somewhat overkill for this; I don't really ...
5
votes
1answer
67 views

Homotopy cardinality of the category of categories

The category of finite sets has homotopy cardinality $e$, because $$ |{\bf FinSet}|=\sum_{n=0}^{\infty}\frac{1}{\left|\operatorname{Aut}\ [n]\right|}=\sum_{n=0}^{\infty}\frac{1}{n!}. $$ What is the ...
2
votes
2answers
55 views

Why aren't spaces contravariant functors?

The functor of points approach to algebraic geometry often starts with the definition of a k-space as an object in the functor category $\mathsf{Sp}_k=\mathrm{Fun}(\mathsf{Comm}_k,\mathsf{Set})$. That ...
4
votes
1answer
40 views

How do I prove that canonical monomorphisms of a coproduct in the category of pointed spaces are topological embeddings?

Let $\{(X_i,p_i)\}$ be a family of pointed spaces and $(\coprod X_i,j_i)$ be a coproduct of $\{(X_i,p_i)\}$ in the category of pointed spaces. I have proven that canonical monomorphisms $j_i$'s are ...
11
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0answers
94 views

Sheaves of categories

I recently read an answer on this MO post explaining that one reason people are interested in higher category theory is to make reasonable sense of something like a "sheaf of categories" on a ...
8
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0answers
124 views

When can stalks be glued to recover a sheaf?

Let $\mathcal{F}$ be a sheaf over some topological space. The stalks are $\mathcal{F}_x= \underset{{x\in U}}{ \underrightarrow{\lim}} \mathcal{F}(U)$. Is there a special name for a sheaf that ...
1
vote
1answer
43 views

universal properties of dependent types

What is the universal property of dependent product / dependent sum? (I want to see a diagrams) They are must be different from usual ones, aren't they? (i'm trying to understand category theory ...
3
votes
1answer
28 views

How do you break up an exact sequence of any length to a “succession of short exact sequences”?

Note that if $\text{Hom}_R(D,-)$ functor takes short exact sequences to short exact sequences then it takes exact sequences of any length to exact sequences since any exact sequence can be broken ...
0
votes
1answer
33 views

Why are concretizable categories locally small?

I have seen it mentioned in a few places that concrete (or concretizable) categories are locally small, but never seen any proof. Is it particularly trivial? If not, does anybody have some reference ...
2
votes
0answers
56 views

Sheafification and monomorphisms.

I was showing that a monomorphism of sheaves induces a monomorphism in stalks. I used the classical fact about filtered colimits but I was wondering that if the inclusion is adjoint to sheafification ...
1
vote
1answer
50 views

Coproducts and products are same in any preadditive category

Here is the proof that coproducts and products are same in any preadditive category from the Stack project. I have few questions regarding the above proof. I don't understand what do they mean by ...
4
votes
2answers
85 views

Assume a homomorphism of groups gives a full and faithful functor on reps. Was it surjective?

Let $\phi: H \to G$ be a finite group homomorphism. Then there is a functor on representations $\operatorname{Rep}(\phi): \operatorname{Rep}(G) \to \operatorname{Rep}(H)$ given by precomposition with ...
1
vote
0answers
45 views

Why is every object in a locally presentable category small

The definition I am working with is the following, a category $\mathcal{C}$ with all small colmits is called locally presentable if it has a set of small objects $S\subset Obj(\mathcal{C})$ every ...
8
votes
1answer
167 views

What Yoneda tells us about algebraic geometry

I am currently learning about relative algebraic geometry, and I'm just trying to walk myself through some of the foundations and motivating examples before moving on to the proper stuff (symmetric ...
0
votes
1answer
38 views

In an abelian category, every map $f:B \to C$ factors as $B \xrightarrow{e} \text{im}(f) \xrightarrow{m} C$

In an abelian category, every map $f:B \to C$ factors as $B \xrightarrow{e} \text{im}(f) \xrightarrow{m} C$ with $m = \ker(\text{coker}f)$ monic and $e $ epi. How can $\text{im}(f)$ be defined in ...
3
votes
0answers
38 views

Is every topos equivalent to a full subtopos of U-small objects in another topos?

Let $\mathcal{E}$ be a topos, and $\mathcal{U}$ an $\mathcal{E}$-universe (as discussed on this nLab page). Let $\mathcal{E}_\mathcal{U}$ be the full subcategory of $\mathcal{U}$-small objects in ...
0
votes
0answers
25 views

Corepresents and yoneda lemma

This question comes from G ́omez's notes on Algebraic stacks. I think because of the yoneda lemma, it's true that $M$ represents $F$ iff $\mathrm{Hom}_S(Y,M)=\mathrm{Hom}_{(Sch/S)'}(\mathcal{Y},F)$. ...
1
vote
1answer
29 views

The elements in the bounded derived category of a hereditary category

I am looking for a proof for the following statement. Let $\mathcal A$ be a hereditary category, and $D^b(\mathcal A)$ be its bounded derived category. Then for any $M \in D^b(\mathcal A)$, we ...
0
votes
0answers
20 views

When does having a coseparating set imply having a single coseparator?

Under what conditions on a category is it the case that having a coseparating set implies having a single coseparator? The rest of this text is motivation for the above question. The General Adjoint ...
0
votes
1answer
29 views

Specific formulation of Yoneda lemma with comma category

In a problem sheet there is a statement of the co-Yoneda lemma as follows (the whole question is included for context): My question is, what would be the formulation of the usual Yoneda lemma ...
4
votes
1answer
57 views

Definition of the category of group representations

One usually considers the category of complex linear group representations for a fixed group $G$. It is defined as the category whose objects are group morphisms $G \rightarrow GL(V)$ where $V$ is a ...
1
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0answers
37 views

Set-theoretical problems of regarding Grothendieck topologies as presheaves

I apologise for the vagueness of this question - it's just something I was idly wondering about. Let $(\mathcal{C},J)$ be a site with $\mathcal{C}$ small. The axioms for the Grothendieck topology $J$ ...
8
votes
2answers
275 views

Axiom of Choice needed to “categorify” the cardinals?

I was playing around in $\mathsf{Set},$ trying to reduce it modulo isomorphisms to make a category $\mathsf{Card},$ letting the objects of $\mathsf{Card}$ be the isomorphism classes of $\mathsf{Set}$ ...
0
votes
2answers
35 views

$F \times G$ as defined is indeed a product in $A^I$, $ \ A $ an additive category.

Trying to prove: $F \times G$ is a product in $A^I$. It has been defined as $(F\times G)(i) = F(i) \times G(i)$ and $(F\times G)(f) = $ unique map $F(i) \times G(i) \to F(j) \times G(j)$ described in ...
4
votes
1answer
43 views

Calculating $\operatorname{Ext}$ in special cases.

Is there a set of "methods" for calculating $\operatorname{Ext}$ in some special cases? For instance, I would be interested in calculating $\operatorname{Ext}_{\mathbb{Z}}^n (\mathbb{Z}/4\mathbb{Z}, ...
1
vote
0answers
40 views

Optimal object in category? Metric, objective function on category objects. Optimization over category?

Is there notion about optimal object in category (that can be found by some algorithms, or - more importantly - that can be constructed (if unknown) by some algorithms), about metric and objective ...
0
votes
1answer
18 views

Showing that $F \times G \in A^I$ respects composition in $I$, where $A$ is additive cat.

If $i \xrightarrow{g} j \xrightarrow{f} k$ in $I$, then $F(i) \xrightarrow{F(g)} F(j) \xrightarrow{F(f)} F(k) \\ G(i) \xrightarrow{G(g)} G(j) \xrightarrow{G(f)} G(k)$ in $A$. And since $A$ is ...
2
votes
1answer
118 views

Morphisms! Time to set the record straight

In my limited mathematical reading, I often come across authors that declare functions as isomorphisms, homomorphisms, or homeomorphisms (or any other variety of morphism). Although I've found ...
0
votes
1answer
31 views

How can we define $(F \times G) (f)$ in a general category, where $F, G$ functors, $f$ a map.

Let $F,G$ be two functors in $A^I$, with $A$ an additive category. Since the product $a \times b$ exists in $A$ for any objects $a, b$, define $(F \times G)(j) \equiv F(j) \times G(j), \ j \in I$. ...
1
vote
1answer
24 views

$f0 = 0$ in an Ab-cat with a zero object?

$0 \equiv B \to 0 \to C$ in $\text{Hom}(B,C)$. I want to show that for any map $f: C \to D$ in an Ab-category, $f0 =( 0: B \to D)$. $f0 = e \neq 0 \implies f0 - e = 0$ gives me nothing. Thank you! ...
1
vote
1answer
45 views

Inverse limits over cofinal subsets

OK, I am looking for no "hands waiving" proof that inverse limit over a cofinal index subset is isomorphic to the inverse limit over its superset. For instance: Given an (Abelian) category $C$ with ...
-1
votes
1answer
45 views

$\{\alpha_i : A_i \to C_i \}$ family of maps $\implies \exists ! \alpha : \prod_i A_i \to \prod_i C_i$ such that $\pi_i^C \alpha \to \alpha_i \pi_i^A$

Suppose that $\prod_i A_i, \prod_i C_i$ exist in a category, and that there is a family of maps $\{\alpha_i : A_i \to C_i\}$. There exists a unique $\alpha : \prod_i A_i \to \prod_i C_i$ such that ...
1
vote
1answer
76 views

When they say $\text{Hom}(A,B) \approx \text{Hom}(C,D)$ in category theory, what do they mean?

For instance in Weibel. Do they mean that the two hom sets are bijective or something in addition to that? Show that $\text{Hom}_{\mathcal{C}}(A, \prod_i C_i) \approx \prod_i ...
3
votes
4answers
63 views

In a category with a $0$ object any two kernels of a morphism $f$ are isomorphic “in an evident sense”

This is from Weibel. A kernel of a morphism $f : B \to C$ is a morphism $i : A \to B$ such that $fi = 0 = (A \to 0 \to C)$ and for any other $e: A' \to B$ such that $fe = 0$, then $e$ factors ...
2
votes
0answers
87 views

$E_{\infty}$ algebra in characteristic zero

Let $A^{\bullet}$ be a cosimplicial commutative algebra over a field $\Bbbk$. Denote with $N(A)^{\bullet}$ the conormalized Moore complex. Since $A^{\bullet}$ is equipped with a product, the ...
5
votes
2answers
106 views

Description of generated Grothendieck topology

Let $C$ be a small category, and let $\tau$ be a set of sieves in $C$. Assume that $\tau$ contains all the maximal sieves, and is stable under pullbacks. How to describe the Grothendieck topology ...
3
votes
0answers
37 views

Why does a skew-linear form on kG determine a triangular structure on k[G]?

I'm trying to understand braidings on finite group representations. They are the same as quasitriangular structures on the group algebra $k[G]$. The original reference seems to be ...
2
votes
1answer
35 views

Can we always define a congruence category?

In Awodey's Category Theory the congruence category is defined as follows... We have a congruence ~ on a category $C$. Then $C^\tilde{}$ is defined as: $(C^\tilde{})_0=C_0$ $(C^\tilde{})_1=\{ ...
7
votes
1answer
93 views

Which spaces can be used as “test spaces” for the Stone-Čech compactification?

Stone-Čech compactification $\beta X$ of a completely regular space $X$ is defined by the following property: Let $X$ be a completely regular space. Let $i \colon X \hookrightarrow \beta X$ be an ...
2
votes
1answer
29 views

Exponential in the category Relation

Is it possible to define exponential and currying in the category Relation? If not, what is the reason that we cannot?
6
votes
1answer
100 views

Finding a space with $X \cong X+2$ and $X \not\cong X+1$.

Question. Is there a topological space $X$ with $X \cong X+2$ and $X \not\cong X+1$? Here, $X+n$ denotes the disjoint union (i.e. coproduct) of $X$ with $n$ isolated points. This question is similar ...
2
votes
1answer
66 views

Triple Products are Isomorphic

I am currently working through Awodey's Introduction to Category Theory, and I'm learning how to move around complicated diagrams. I want to show that $A\times(B\times C)\cong(A\times B)\times C$; ...
1
vote
1answer
49 views

An adjoint with unit the identity, Does it imply counit identity?

I have the next doubt: If $S:A\rightarrow C$ is faithful, full and surjective in objects (for each $c\in Obj(C)$ $Sa=c$ for $a\in Obj(A)$) then we have the adjunction $\langle T,S, ...
11
votes
0answers
143 views

How does the internal language of a topos come to be?

There are several books and articles on topos theory which mention the internal language, but I can't manage to see the big picture from any of them. I would like a soft explanation of how the ...
1
vote
1answer
48 views

Concrete categories and the concept of to be free

Let $F$ be free on the set $S$, let $F′$ be free on the set $S′$, and assume that $|S| = |S′|$. Prove that there is an isomorphism $g : F → F′$. By definition we have: $f: S → U(G)$, $i: S → U(F)$ ...
0
votes
0answers
49 views

Example of Faithful, but not Full Functor

I have a couple examples of functors that were given to me in class that are faithful, but not full. However, I'd like an actual proof of these facts in case I have to explain myself on an exam. ...
3
votes
1answer
53 views

Lifting Functors to Adjoints

There are well-understood theorems that give sufficient conditions for a functor $R: D\to C$ to have a left adjoint. For example, $R$ should preserve limits and $D$ should have nice categorical ...
2
votes
1answer
55 views

Cone of an adjunction

I came across this sentence "...let $\varepsilon: GG^\vee \to \operatorname{Id}$ be the counit of adjunction and $Z$ its cone." I thought that cones were constructions on functors. $\varepsilon$, ...
3
votes
0answers
30 views

Projective objects of chain category

I am tempted to think that the projective objects in the chain category $\text{Ch}(\mathcal C)$ for $\mathcal C$ abelian are exactly the complexes $P_\bullet$ for which each $P_i$ is projective. Is ...