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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questions about extremal epimorphisms in category theory

let $K$ be a category with equalizers, show that every extremal epimorphism is epic. for composable morphisms $f: A \rightarrow B $ and $g: B \rightarrow C$ in $K$, show that if $gf$ is an extremal ...
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1answer
153 views

show the following conditions are equivalent for a category $ C $

I need to show the following conditions are equivalent for a category $ C $ (a) $ C $ has binary products, equalizers, and a terminal object; (b) $ C $ has pullbacks and a terminal object; (c) $ C ...
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2answers
105 views

Nice notation for projection maps

Let $X\times Y$ be a product of two object of a category, and consider the natural projections $$ X\times Y \to X \quad\text{ and }\quad X\times Y \to Y. $$ Usually I denote them by $\pi_X$ and ...
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2answers
99 views

Problem on initial and final objects in category theory.

Let $C$ be a category and let $F:C^{op}\rightarrow Set$ be the functor which takes any $X$ in $C$ to a (fixed) singleton set (and morphisms to the only possible map). If $F$ is representable, an ...
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1answer
86 views

a problem on functor

Let $I$ be a small category and suppose $F:C\rightarrow Set$ and $G:I\rightarrow C$ are functors. (i) How to construct a morphism $\alpha:F(\varprojlim_{i}G(i))\rightarrow \varprojlim_{i}F(G(i))$ ...
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1answer
27 views

Can I test the property to be an epimorphism of sheaves on a sort of 'generalized covers' of representables?

Let $C$ be a Grothendieck-site with finite limits (perhaps I have to assume that the topology it defines is subcanonical, but it would be nice if I wouldn't have to) and let $T=Shv(C)$ denote the ...
2
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1answer
67 views

Can the obvious “product” of complete atomistic Boolean algebras be realized as a categorial product?

Let $X$ and $Y$ denote sets, and $\eta_X,\eta_Y : X,Y \rightarrow X+Y$ denote the natural injections to the disjoint union. Then intuitively, the "product" of the Boolean algebras $2^X$ and $2^Y$ ...
6
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0answers
102 views

On locally small category

Maybe this is a very trivial question for those who are familiar with Set theory. Let $C$ be a category, and $\hat{C}$ be its presheaves. I hear that if both $C$ and $\hat{C}$ are locally small, ...
3
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1answer
450 views

lambda calculus and category theory

I am not particularly knowledgeable in either lambda calculus or category theory, but I am starting to learn Haskell so I would like to ask: are there connections between category theory and lambda ...
2
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3answers
124 views

$0$-th exterior power, empty product of modules and their tensor product

$\def\finiteprod#1#2{#1_{1}\times#1_{2}\times\dots\times#1_{#2}}$In Lang's algebra he defines the tensor product as a universal object in the category of multilinear maps from $\finiteprod En$ where ...
5
votes
1answer
214 views

Foundational theories, their uses, interactions and comparisons?

Until now, I heard that there are some theories for building mathematical objects (or at least it is what it seems to my poor knowledge). Some of these are: Set theory; Logic; Category theory; Type ...
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1answer
58 views

Help on how to read 2-categories diagrams

I've seen 2-categories diagrams, where there are 2-cells and 1-cells morphisms, but I do not understand how to read them. Apparently it seems to be some kind of rule that says that you have to follow ...
0
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2answers
70 views

Natural Transformation and Isomorphism

How can we see that a natural transformation $\alpha:F\to G$ between functors $F,G:C\to D$ is a natural isomorphism iff for each object $c$ in the category $C$, $\alpha_c$ is an isomorphism in $D.$
2
votes
1answer
52 views

Question on an induced map into the pullback of epimorphisms

Let \begin{eqnarray} W&\to& Z\\ \downarrow &&\downarrow\\ X&\to&Y \end{eqnarray} be a pullback diagram in a category $C$ where all the involved morphisms are epimorphisms. Let ...
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0answers
36 views

Do colimits preserve free group actions?

Suppose $C$ is a small category, $M$ is a category and $G$ is a discrete group. Let $X$ be a $G$-object in $M^C$ and suppose that the action of $G$ on $X$ is object-wise free. Under what conditions is ...
16
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1answer
336 views

What is a concrete example of why one wants to have a *derived category* in algebraic geometry?

My question asks for a concrete (and hopefully easy) example, why one wants to derive things in algebraic geometry. I heard, that a resolution of an object by free ones behaves much better than the ...
3
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2answers
109 views

For what categories do category algebras exist?

The monoid algebra $R[M]$, for a commutative ring $R$ and a monoid $M$, can be described as the free $R$-algebra on $M$. We think of $R[M]$ as the set of finite formal sums of elements of $M$ with ...
0
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1answer
70 views

Monics and monomorphisms are the same as kernels in the additive category of R-modules.

How can we show that in an additive category monics and monomorphisms are the same as kernels? Actually, I can show that a kernel is a monic and a monomorphism but I could not show that "a monic is a ...
4
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1answer
82 views

Linear structure on the category of formal groups

Let $R$ be a commutative ring. If $R$ is a $\mathbb{Q}$-algebra, then the category of formal groups over $R$ (or the category of formal group laws) carries the structure of an $R$-linear category; ...
11
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1answer
239 views

Applications of TQFTs beyond physics

I'm giving a talk at a postgrad seminar on the topic of topological quantum field theories (TQFTs) with a mixed audience of pure and applied mathematicians. As such, I'd like to be able to offer some ...
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0answers
39 views

Existence of colimit-dense subset

Let $C$ be a locally presentable category and let $I$ be a full subcategory of $C$ closed under colimits. Does $I$ have a colimit-dense subset of objects? (I hope that this Q doesn't depend on the ...
3
votes
1answer
63 views

Help with exercise on Reports of the Midwest Category Seminar IV

At the end of the LMN 137, "Reports of the Midwest Category Seminar IV", there is a list of exercises. "5. Considering a left-adjoint as male and a right adjoint as female, give the correct term for ...
4
votes
1answer
212 views

Lawvere theories: an equivalence.

I'm having trouble understanding Lawvere theories (as defined below). Definition: A Lawvere Theory is a category $\mathcal{L}$ with finite products and with a distinguished object $A$ such that ...
5
votes
1answer
64 views

Exact sequence in a category with zero morphisms

Let $C$ be a category with zero morphisms (equivalently, $\mathsf{Set}_*$-enriched), for example it could be a linear category. Then we can talk about kernels and cokernels of morphisms in $C$. I ...
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1answer
97 views

problem on products in category theory

Let $C$ be the category of torsion abelian groups. (1) How do you prove that products are representable in $C$? (2) Could you also please give me an example where the product in $C$ is not isomorphic ...
3
votes
2answers
129 views

Should axioms be viewed as part of the signature?

I included category theory in the tags in order to get feedback from the categorial logic community. It goes without saying that this isn't really category theory. A semigroup can be defined as a ...
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votes
2answers
243 views

What is a (the?) good starting point for learning the modern “higher” mathematics?

As many of you know, category theorists are currently doing, among other things, a great job in advertising their modern developments. And I must say, this works for me - in particular, I find myself ...
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1answer
45 views

Why is the unit of a compact closed category coordinate-independent?

Compact closed categories are equipped with a unit and counit $$\eta_A: I \to A^* \otimes A$$ $$\varepsilon_A : A^*\otimes A \to I$$ For a particular compact closed category, say FdVect it is ...
2
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1answer
65 views

Can functor carry over a monoidal structure?

Suppose we have two categories $C$ and $D$ and a functor $F: C \rightarrow D$, furthermore suppose that the functor $F$ is an equivalence of categories, if the category $C$ is a monoidal category, can ...
0
votes
1answer
47 views

Definition for the action of a category on a set.

I'm trying to understand the definition of the action of a category on a set which is given in nLab, more particularly the first one. If one has a functor $\rho: C \to Set$, one takes the set S as the ...
3
votes
3answers
412 views

Confused about the definition of a group as a groupoid with one object.

A groupoid is defined to be a category where every morphism is an isomorphism. So sometimes a group is said to just be a groupoid with one object. When I try to make sense of this, I denote the ...
6
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1answer
65 views

$M\times N$ Doesn’t Have a Module Structure

In Keith Conrad's notes (page 4) is written: For two $R-$modules $M$ and $N$ , $M\oplus N$ and $M\times N$ are the same sets, but $M\oplus N$ is an $R-$module and $M\times N$ doesn’t have a ...
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2answers
81 views

When should we take direct limit and when should we take inverse limit?

We know that we can take direct limit for a direct system and inverse limit for an inverse system. For example, when can defined the stalk of a presheaf $\mathcal{F}$ on a topological space $X$ at a ...
3
votes
1answer
73 views

Property of finitely presented objects

Let $\mathcal{C}$ be a Grothendieck category. Take a short exact sequence in this category: $$ 0 \to X \to Y \to Z \to 0 $$ where the object $Y$ is finitely presented. Is true that $Z$ is finitely ...
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5answers
228 views

Deeper studies in Category Theory: suggestions and references.

I discovered Category Theory one year and a half ago and I got addicted. I studied some of the basic concepts of this branch (limits, adjunctions etc.), including some sightseeing into abelian ...
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1answer
110 views

Name for a category

Is there any name or notation for this category? Let $U$ be a set. By "function" I will mean a function $U\rightarrow U$. objects are functions; morphisms from a function $A$ to a function $B$ are ...
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1answer
70 views

What properties no natural isomorphisms between functors preserve

If $F,G: C \rightarrow D$ are functors between regular categories and $F \Rightarrow G$ is a natural isomorphism, is it true that if $F$ is faithful {resp. full) then $G$ is full? I believe this is ...
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2answers
96 views

How can we formalize $\mathrm{Subset}$, the “category” of all subsets?

Suppose $X$ is an object of $\mathrm{Set}$. Then we can define a category of subsets of $X$ in the usual way; objects are equivalence classes of monomorphisms $f : A \rightarrow X,$ etc. Now this ...
3
votes
1answer
88 views

On the definition of the direct sum in vector spaces

We say that if $V_1 , V_2, \ldots, V_n$ are vector subspaces, the sum is direct if and only if the morphism $u$ from $V_1 \times \cdots \times V_n$ to $V_1 + \cdots + V_n$ which maps $(x_1, \ldots, ...
1
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0answers
70 views

colimit on presheafs

Given a category $C$ with limits and colimits. Let $F: D \to C$ be a connected diagram in $C$. I want to show that if we just take the iterated pushouts then I will eventually get the colimit. For ...
5
votes
1answer
133 views

Examples of canonical projections that are not epimorphisms and canonical injections that are not [duplicate]

Although in $\mathsf{Set}$, canonical projections from a product are surjective and canonical injections to a coproduct are in fact injections, there seems to be nothing forcing this to be the case ...
4
votes
2answers
152 views

Adjunctions in Category Theory

My question is how can we see that if $F$ and $F'$ are both left adjoints of $G$, there is natural isomorphism between $F$ and $F'$? So how can we show that adjoints are unique upto isomorphism?
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1answer
75 views

What is homotopy in $(\infty,1)$-categories (as weak Kan complexes)

One of the peculiar (and somewhat appealing) features of quasi-categories is that many properties from ordinary category theory characterized equality are characterized by some form of homotopy ...
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2answers
94 views

Notation for “parallel” morphisms in a diagram

Suppose $f\colon A\to B$ and $g\colon A\to B$ are possibly-distinct morphisms. How do I stick them both in a diagram (along with, e.g., their (co)equalizer) without suggesting that they are equal?
4
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1answer
105 views

What are presentable categories?

Locally presentable categories are categories which have enough presentable objects. Jacob Lurie drops the word "locally" in his work, and I would like to follow this terminology. On the other hand, ...
1
vote
1answer
126 views

Fibered coproducts in $\mathsf{Set}$

Following my not-entirely-successful attempt to define fibered products in $\mathsf{Set}$, I will attempt to define the fibered coproducts: Let $A,B,C$ be sets, and let $\alpha\colon C\to A$ and ...
3
votes
1answer
86 views

Retrieve affine schemes by adjunction

In an introductory course about schemes, I've seen the adjunction $$ {\mathbf{LocRngSpace}}^{\mathrm{op}} \overset{\mathrm{Spec}}{\underset{\Gamma}{\leftrightarrows}} \mathbf{Ring}, $$ where ...
3
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0answers
43 views

Are these categories toposes?

Apart from usual examples of toposes, I'd like to know if some of the following categories and some of their subcategories are known to be toposes : the category $\text{Heyt}$ of Heyting algebras ...
3
votes
3answers
135 views

Fibered products in $\mathsf {Set}$

I'm just starting to work through Aluffi, and one question asks the reader to define fibered products (and coproducts) in the category of sets and functions. I'm just looking to check that I have the ...
2
votes
0answers
33 views

Endofunctor as a presheaf

This looks pretty obvious, but an endofunctor $F : Set \to Set$ seems to be a presheaf over $Set^{op}$. Is there any useful fact that can be learnt from this view of endofunctors?