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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47 views

Do counit-unit adjunctions have the expected universal properties?

Let a unit-counit adjunction between the categories $C$ and $D$ be defined as a pair of functors $F: D\rightarrow C$ and $G: C\rightarrow D$ with natural transformations $\varepsilon : FG \rightarrow ...
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64 views

What, if anything, does higher category theory have to say about situations where each subcategory induces a sub-$2$-category?

Let $\mathbf{Top}$ denote the $2$-category of topological spaces, continuous mappings, and homotopies between them. Let $\mathbf{C}$ denote a wide subcategory of $\mathbf{Top}$. Then we get a wide ...
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25 views

closed monoidal posets

Let $X$ be a set, regarded as discrete category. if $X$ has structure of closed monoidal category $(X,\cdot,e)$, then it is easy to show that $X$ is a group: since all the morphisms are identities, ...
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49 views

Differential topology and/or geometry from the viewpoint of Cauchy completions

According to an nLab article that I don't really understand, if we take the "Cauchy completion" of the category $\mathbf{C}$ whose objects are open subsets of $\mathbb{R}^n$ and whose morphisms are ...
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79 views

What is this categorical notion called?

If we take a category $\mathcal{C}$ of objects with functions as morphisms and restrict the morphisms to injections (monomorphisms?), then this defines a partially ordered set of isomorphism classes. ...
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196 views

Can Isomorphism between categories be defined only if both categories are small?

In wikipedia page, "Isomorphism of categories ", A functor F : C → D yields an isomorphism of categories if and only if it is bijective on objects and on morphism sets. I have heard that ...
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32 views

Galois Connection Between Posets

I have the next doubt about this problem: In a Galois Connection between posets, show that the subset $\{p\mid p=RLp\}$ of $P$ is equal $\{p\mid p=Rq \; for\; some \; q\}$ and give a bijection from ...
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1answer
32 views

What does 'coherent isomorphism' mean in the sense of pseudofunctors?

From what I've been able to find, psuedofunctors are not-quite-functors, in the sense that they preserve the identity morphism and composition of morphisms only up to coherent isomorphism, and not 'on ...
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1answer
40 views

Adjunction of two functors. If the right adjoint functor is linear, then the left adjoint functor is also linear.

Let $R$ be a Ring, $M$ be the category of R modules, and $(L,R, \tau)$ an adjunction from $M$ to $M$ such that $R$ is linear $(F(f+g)=F(f)+F(g))$. I want to show that $L$ is also linear. I would just ...
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97 views

$\mathcal{V}$-naturality in enriched category theory

Let $\mathcal{V}$ be a monoidal category, in section 1.2 of "Basic concepts of enriched category theory" (http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf) Max Kelly introduces the terms ...
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76 views

What is it called when every object of a category is the quotient object of some free object?

For example, every group is the quotient of a free group. Is there a name for this property in a general category, or does this property necessarily follow from the universal property of free objects? ...
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58 views

Categorifiying Implication [closed]

How is implication, e.g., $X\implies Y$, represented categorically? How can we show/prove implication using category theory? Update: to clarify, the "implication" I am referring to is one that refers ...
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29 views

Is it ever convenient to mod out reparameterizations in a category of spaces?

Often we choose nice categories of spaces by choosing full subcategories of larger categories of spaces. Is it ever convenient to instead look at subcategories of these supercategories by modding out ...
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47 views

Turnstile (\vdash) in adjoint functor and type theory.

Is common symbol $\vdash$ an abuse of notation or there is a deep sacred connection between $$\Gamma \vdash \lambda(a:A).a:\Pi(a:A).A$$ which is preorder and $$G \vdash F \quad \mbox{($F$ is left ...
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1answer
54 views

$X,Y$ be connected ; $f:X\to Y$ be a continuous function which is right-cancellative w.r.t. continuous maps on connected spaces ; is $f$ surjective? [closed]

Let $X,Y$ be connected topological spaces and $f:X\to Y$ be a continuous function such that for any connected space $Z$ and any continuous functions $g_1,g_2:Y \to Z$ , $g_1 \circ f=g_2 \circ f ...
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1answer
40 views

Comparing Free Monoids and Kleene Closures (Stars)

These are going to be a straight-to-the-point questions: What is the difference between a free monoid and a Kleene Closure (Star) when generated by the set $A=\{1\}$? Let $A^*$ be the free monoid ...
2
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1answer
65 views

What Is Associativity in Composition

Link to Original Question I never thought of this until recently when I began learning about categories. In the past, with function composition, I understand associativity as follows. Suppose that ...
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2answers
209 views

What Is a Morphism?

I know that there are several posts on the same question. They all ask for examples for morphisms that are not functions. So, morphisms are more general than functions; they are the arrows ...
2
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1answer
53 views

Covariant and Contravariant Functors

In Category Theory, we have covariant and contravariant functors. Mathematically I know the difference between the two - I can picture that. What I would like to know why these concepts are named ...
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3answers
125 views

Natural transformations in Awodey's Category Theory Exercise 7.11.8

I've been having troubles trying to make sense of the last part of Exercise 7.11.8 in Awodey's Category Theory book (p.182). The exercise asks us to Show that a functor category ...
4
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1answer
77 views

Applications of the Lawvere Fixed Point Theorem for Sets

I'm not familiar with the general theorem for closed, cartesian categories (as I'm not familiar with closed, cartesian categories), but I am aware of this version of the fixed point theorem for sets: ...
3
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1answer
33 views

Internal semantics of a category based on a fixed topos.

I'm not certain as to how I should formulate this question; it might be considered a soft question. I am interested in finding a general way to take a category $\mathbb{C}$ and an (elementary) topos ...
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0answers
26 views

Derived functors on 2 term exact sequence

Suppose we have a not-short exact sequence 0 -> A -> B -> 0 in some abelian category. Now let us apply right exact functor F: F(A) -> F(B) -> 0. So could I consider a left derived functor H to obtain ...
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1answer
55 views

How to state the Axioms of Category Theory in Predicate Logic?

I have been trying to formally state the axioms of category in predicate logic. It seems that I will need one-place predicates for objects and arrows, two-place predicates for heads and tails of ...
2
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1answer
63 views

On the join of simplicial sets as a dependent product

Prop. 3.5 of Joyal notes in quasicategories gives a description of $X\star Y$ as $i_*(X,Y)$, where $i^*\dashv i_*$ is the adjunction $$ i^*\colon \mathbf{sSet}/\Delta[1] \leftrightarrows ...
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2answers
37 views

Checking commutativity of a diagram of modules over some ring and what the commutativity of the diagram implies.

Suppose that you have the following diagram of modules over some ring: These are my questions: (1) To prove that the diagram is commutative, we needs to prove that $gf=kh$, $wf=rv$, $zh=uv$, ...
3
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1answer
50 views

Category of modules

For given a ring $R$, we can define the category of left $R$-modules. In fact, objects are all left $R$-modules and morphisms are $R$-module homomorphisms. NOw my question is: If we do not fix $R$, ...
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2answers
95 views

Applying the Yoneda-Lemma to prove the existence of Tensor-products

In class the professor said when he came to prove the existence of the tensor-product for $A$-modules ($A$ any ring) that the existence and properties of the tensor-product would be one-liners having ...
3
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1answer
67 views

Hom / tensor adjunction for $O_X$ modules?

Does the hom-tensor adjunction hold for $O_X$ modules also? With sheaf hom and sheaf tensor product, the statement would consist of a natural transformation $Hom_O (M \otimes_O N, K)\cong_{nat} ...
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1answer
46 views

Isomorphic kernels imply pullback?

In Hilton/Stammbach's A Course in Homological Algebra, they are treating the Ext functor, and they give the following lemma: [][2 He implies (but doesn't say) that the same is not true if we ...
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1answer
34 views

Applications of small object argument outside model categories.

Are there applications of the small object argument outside of its original application of constructing the factorisation for a model category?
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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), ...
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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
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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 ...
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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
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1answer
41 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 ...
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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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125 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
44 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
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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 ...
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1answer
168 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 ...
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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 ...
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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 ...
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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 $\mathrm{Hom}_S(Y,M)=\mathrm{Hom}_{(Sch/S)'}(\mathcal{Y},F)$. ...
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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 ...