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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Day convolution intuition

In the nLab, Day convolution is introduced as a generalisation of convolution of complex-valued functions, but I'm wondering how exactly to understand this. I can (just about) parse the definitions, ...
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Monomorphism in the category of schemes

Let $(f, f^{\#}): X \rightarrow Y$ be a map of schemes. The stacks project gives a criterion for $f$ to be a monomorphism (see lemma 25.23.6): if (a): $f$ is a monomorphism in the category of ...
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88 views

Is category theory ambiguous? or it just is the case for beginners? [on hold]

First of all, I have to say that I'm not going to offend anyone/anything here; I just need some clarification/studying tips about category theory. I'm totally new in category theory and this happens ...
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92 views

What's an Isomorphism?

I'm familiar with the definition (inverses and bijections, preserving operations) in the context of groups and vector spaces, the hoeomorphism of topological spaces, and have some feeling for the ...
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45 views

subobject classifier for partial orders

Does the category of partial orders have a subobject classifier? (Edit: No, see Eric's answer.) If not, what is a category which is "close" to the category of partial orders (e.g. it should consists ...
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49 views

Why $C(X,Y)$ ,namely the morphisms between $X$ and $Y$, is assumed to be a set rather than a class?

I understand that we introduce the notion of class to bypass the paradox of the "set of all sets". However, shouldn't $C(X,Y)$ considered to be the set of all morphisms between $X$ and $Y$, thus not a ...
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34 views

Metrics and Measures on a Category of Cats : a cauchy complete category of categories

Is there a suitable way to restrict the functors between objects (and I suppose the objects themselves) of a category of categories such that we have a cauchy complete category. Can we find a ...
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14 views

Modular tensor category and the pivotal or shperical condition

I have two related questions: 1) Is a modular tensor category always pivotal? 2) Is a modular tensor category always spherical?
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26 views

Categorically deducding measurability of sections

Two lemmas which are often proved in elementary measure theory courses are that sections of measurable sets are measurable, and sections of measurable functions are measurable. Note $E_x= \left\{y\in ...
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20 views

On the definition of double categories?

I'm trying to understand double categories but I'm having a hard time. A preliminary definition is: Definition. Let $\mathscr{C}$ be a category. We say $\mathscr{I}$ is an internal category to ...
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32 views

Adjoint pair of functors and cogenerator elements

Let $F:\mathcal{A} \rightarrow \mathcal{B}$ and $G:\mathcal{B} \rightarrow \mathcal{A}$ be additive functors between abelian categories, such that $(F,G)$ is an adjoint pair. If $B \in \mathcal{B}$ is ...
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universal property of product: must any map satisfying it be a morphism

I am thinking about the universal property of products: Let $X$ and $Y$ be objects of a category $D$. The product of $X$ and $Y$ is an object $X \times Y$ together with two morphisms $\pi_1 : X ...
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61 views

How do we view natural transformations as functions

1.The definition asserts that natural transformation is a map of two functors. However, from the definition, given tow functors $F,G:C,D$, we associate every element $x$ in $Obj(C)$ a morphism $F(x) ...
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66 views

What is so special about categories that lead people to use them to “formalize math”?

There are countless interesting structures - lists, trees, maps, graphs. Yet, categories - which, if I understand, is just a graph with some constraints on its shape - are apparently special somehow, ...
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79 views

The magic of the morphisms

Given a set $X$. Let $S\subseteq X$ and consider $(X,S)$ as a very simple mathematical structure, lets call it a spotted set. Given two spotted sets, then a morphism $\alpha ...
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28 views

A complete category of categories and embeddings

I have a theory that you should be able to construct a category of categories and embeddings that is complete. Someone already pointed out that if the functors are full and faithful, your cat is not ...
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32 views

Is $\mathsf{nCob}$ bicomplete?

Let $\mathsf{nCob}$ be the category of $n$-cobordisms, whose objects are $(n-1)$-dimensional closed manifolds and morphisms are bordisms. Is this category bicomplete, or even finitely bicomplete? ...
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25 views

Is this category complete or Cauchy complete?

Define a category as follows: the objects are categories and the arrows are embeddings (full and faithful functors). Is this category Cauchy complete or complete?
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71 views

What functors are these?

The category of "just arrow" categories is equivalent to Cat as we see here. If we have a "just arrow" category $C$, I think we can also have the set of equations, $EQ_C$, over the (compositions of) ...
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170 views

Is the category of monoids cartesian closed? Why?

Is the category of monoids cartesian closed? Why? I read Steve Awodey's "Category Theory", and could not solve the exercise in chapter 6, stated above. Here I speak of the "category of monoids" ...
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59 views

Calculating (co)limits of ringed spaces in $\mathbf{Top}$

Let $\mathbf{Top}$ be the category of topological spaces, $\mathbf{RS}$ the category of ringed spaces and $\mathbf{LRS}$ the category of locally ringed spaces. There are forgetful functors $$ ...
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41 views

Tor amplitude of dual complex

Let $E^\bullet$ be a perfect complex of $R$-modules (where $R$ comm. ring). So $E^\bullet$ is quasi-isomorphic to a bounded complex of finitely generated projective R-modules. Now $E^\bullet$ has ...
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45 views

Understanding the Gluing axiom of the Structure Sheaf on $Spec(R)$

Let $X = Spec(R)$ be an affine scheme for some commutative ring $R$. The structure sheaf $\mathscr{O}_{X}$ is a contravariant functor (I think) $\text{Open}(X) \leadsto \text{Ring}$ from the category ...
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33 views

How do dependent products in category theory relate to type theory?

I feel like I understand the construction of dependent product types relatively well, it makes sense to me how the introduction and elimination rules work together to create the concept of a function ...
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49 views

Can we define structures like groups or monoids in the context of pure category theory?

In a category $\mathcal C$ with terminal object $1$ and objects $A$, $B$, $C$ we have $\quad$ $A\times (B \times C) \cong (A\times B) \times C$; $\quad$ $1 \times A \cong A \cong A\times 1$; ...
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Correct definition of model category

When answering this question, In a model category, is the full subcategory of fibrant objects a reflective subcategory? I realized that I wasn't even sure what the correct definition of a model ...
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Is every monomorphism an injection?

We say a morphism is a monomorphism if $fg=fh$ implies $g=h$. So if $f$ is a monomorphism, is it necessarily an injection? i.e. $f(x)=f(y)$ implies $x=y$. My approach is to consider a specific ...
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1answer
46 views

Turning Preordered Sets into Preordered Monoids (Constructing Preordered Monoids from Preordered Sets)

Question: Referring to the Wikipedia article on Adjoint Functors in Section 2 (Motivation), they talk about "turning rngs into rings" (can be rephrased as "constructing rings from rngs"). I do not ...
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33 views

Is there a reasonable Grothendieck topology on the category of modules over a ring?

How about over a field (i.e. f.d vector spaces)? Can these categories be considered as a site?
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95 views

Does $\operatorname{Spec}$ preserve pushouts?

The spectrum-functor $$ \operatorname{Spec}: \mathbf{cRng}^{op}\to \mathbf{Set} $$ sends a (commutative unital) ring $R$ to the set $\operatorname{Spec}(R)=\{\mathfrak{p}\mid \mathfrak{p} \mbox{ is a ...
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50 views

motivation for the direct limit [closed]

I know just the very basics on Category Theory and that's why I'm going to ask a stupid question. I'm trying to get an intuition for direct limits for my course on Commutative Algebra. All the books ...
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42 views

Morphisms of a category with one object, which is a group

I've read two articles(1,2), but still have a question about structure of group, say $G$, as an one-object category. Let's call this category $C$. I understand that morphisms of $G$, which is ...
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69 views

limits' definition

Please, can somebody help me? I was given the following definition of the LIMIT: Let $I$ be a small category and $F:I \to C$ a covariant functor ( where $C$ is a category), $K \in Ob(C)$, ...
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When does the first cohomology group commute with inverse limit?

Let $M_i,i\in\mathbb{N}$ be an inverse system of continous, discrete G-modules and let $M=\varprojlim M_i$. Under what conditions on $M$ and $M_i$ do we have $\varprojlim H^1(G, M_i) \cong H^1(G, M)$? ...
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30 views

cokernel in the pointed set category $Set.$ [closed]

Please, can someone give me an example of cokernel in the pointed set category? Thanks!
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1answer
33 views

Quasicoherent sheaf on the functor of points is the same as on the scheme itself

I've seen the definition of a quasicoherent sheaf $\mathcal{F}$ on an arbitrary functor $$ X : CRing \to Sets $$ as a specification of an $R$-module $\mathcal{F}(x)$ for each $R$-point $x \in X(R)$ ...
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Is I-adic completion a ring epimorphism?

Let $R$ be a commutative ring and let $I \subset R$ be an ideal. For any $n \ge 1$, the ring homomorphism $R \rightarrow R/I^n$ is surjective, hence an epimorphism in the category of rings. What about ...
2
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2answers
36 views

Defintion of the $\mathrm{hom}$ functor in category theory

I am going through some notes 'Physics, Topology, Logic and Computation: A Rosetta Stone', on category theory. We first define the opposite category: Given a category $C$, we define the opposite ...
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Frobenius-Perron dimension on a fusion category

Let $C$ be a fusion category with simple objects $V_i\in I$. The fusion rule is $V_i\otimes V_j \cong N_{i,j}^k V_k$. The Frobenius-Perron dimension of a simple object $V_i$, $\mathrm{FPdim}(i)$, is ...
4
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1answer
43 views

Tensor of cocomplete categories

Let $C$, $D$ and $E$ be cocomplete categories. Is there a construction $C \otimes D$ such that there is a correspondence between functors $C \otimes D \to E$ preserving colimits and functors $C ...
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32 views

an example of direct product in Ab

Can someone help me with this? Are there direct products in the category $Ab$ (the category of the abelian groups)? If yes, please, can you give me an example? Thank you?
2
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1answer
52 views

“Coforgetful” functors?

Let $F : \mathbf{Sets} \to \mathbf{Cat}$ be the free functor that takes the elements of a set to the objects of a discrete category. Does it has a left adjoint? Awodey (2010 p.249 ex.8) says it does, ...
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1answer
1k views

Which is the most powerful language, set theory or category theory? [closed]

As far as I know, mathematics is written based on a language which can be for example set theory or category theory. My concern is about the power of these languages. How can we realize which language ...
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2answers
80 views

Is there any comprehensive definition of the opposite category $\mathcal C^{op}$ of a category $\mathcal C$?

In the definition of the opposite category $\mathcal C^{op}$ of a category $\mathcal C$, it is said that : objects remain the same and arrows' directions are changed (that is, and arrow $f:A \to B$ in ...
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63 views

Order-Preserving Bijection $f:A\to A^*$?

Let $A$ be a well-quasi-ordered infnite set. Does there exist an order-preserving bijection $f:A\to A^*$, where $A^*$ is the free monoid over $A$ under the subword ordering? Would this subword ...
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1answer
47 views

Exactness of a right adjoint functor

Let $F: \mathcal{A} \longrightarrow \mathcal{B}$, $G: \mathcal{B} \longrightarrow \mathcal{A}$ be two additive functors between abelian categories, such that $(F, G)$ is an adjoint pair. I want to ...
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Category Theory Zoo

There are a few very useful websites when it comes to either finding a specific object with certain properties (and maybe lacking other properties) or finding out which properties a certain object ...
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83 views

Name for categories with a certain property on coproducts

Is there a name for categories with the following property: The category has zero morphisms, coproducts, and for each family $(X_i)_{i \in I}$ of objects the natural map $$\hom(Y,\bigoplus_{i \in I} ...
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728 views

Example of a category without product

A question was raised in our class about the non-existence of product in a category. The two examples that came up in the discussion was the category of smooth manifolds with boundary and the category ...
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30 views

Is the braid category biclosed and bicomplete?

Let $\mathcal{B}$ be the braid category, as in Categories for the Working Mathematician §XI.4 p.262 (objects are natural numbers and morphisms are the braids $n\to n$). Then this can be given a ...