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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90 views

Subgroup which “becomes normal in” a particular category

Is there a name for a subgroup $H < G$ which is not necessarily normal, but such that, for any $f : G \rightarrow Y$ where $Y$ belongs to a particular category (e.g. finite groups, linear groups, ...
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110 views

Associative Product Bifunctor

Let $\mathcal{C}$ be a category with finite products. Under the assumption of the Axiom of Global Choice it is posible to define the functor $$\mathbf{Prod}\colon \mathcal{C} \times ...
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108 views

A natural homomorphism of dual modules which is not a monomorphism?

I've been mixing up my reading with a little category theory. When thinking about dual modules, this idea popped up. Suppose $M$ and $N$ are modules over some commutative ring $A$. Suppose $M^\vee$ ...
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90 views

Any amalgam of a set of generators is a generator, in the category of $S$-sets.

Let $S$ be a monoid and denote by $Act-S$ the category of $S$-sets. I am having a problem understanding a step in the proof of the fact that if $\left\{G_i\right\}_{i \in I}$ is a set of generators ...
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147 views

Left-derived functors

Let $F:\mathcal{A}\to\mathcal{B}$ be a covariant right-exact functor between two abelian categories. Suppose $\mathcal{A}$ has enough projectives. Then we define the left derived functors of $F$ by ...
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82 views

Image of projective objects.

Let $A$ and $B$ be two abelian categories. Assume that there exist a functor $F$ between them which is exact, full and essencially surjective. If $x$ is a projective object in $A$, then $F(x)$ is a ...
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162 views

Universal property - inverse limit in the category of sets

Let $\mathscr{J}$ be a small category and $\mathscr{C}$ the category of sets. Suppose we have a functor $F:\mathscr{J} \to \mathscr{C}$ (so that there is an $A_i \in \mathscr{C}$ for each $i \in ...
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328 views

Mac Lane and Eilenberg's motivations for category theory

I'm looking to understand the conceptual process that brought Eilenberg and Mac Lane in developing the basic concepts of category theory. I quote Mac Lane's book "Category theory for working ...
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73 views

Making a Family of Problems into a Category and defining the Morphisms

This question relates to my previous question found here: Defining Category of Problems Let $\left\{\Pi_i \right\}_{i \in I}$ be a family of problems. Let the solution $u_i$ of $\Pi_i$ lie in some ...
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57 views

Choosing a specific model of a Lawvere Theory

Models for Lawvere theories are functors that preserve products. I'm a little confused about this detail: suppose you have a Lawvere theory for groups. If you have a specific group in mind, is there ...
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104 views

Field reductions. part three

Follow-up to Field reductions. part two For a field $K$, an element $a \in K$, and $K(\setminus a)$ the set of field reductions (maximal subfields of $K$ not containing $a$) are there any interesting ...
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39 views

Category of Presheaves on a small category $C$ is locally cartesian closed

I'm trying to fill in the details of the proof and need the following result: $Set^{C^{op}}/P\simeq Set^{D^{op}}$, where $D$ is the category of elements of $P$. The objects are pairs $(x,C)$ with $C ∈ ...
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44 views

The Categories $\mathrm{Set}\times\mathrm{Set}$ and $\mathrm{Set}+\mathrm{Set}$

I'm thinking about the following problem. In $\mathrm{Cat}$ I can form the product $\mathrm{Set}\times\mathrm{Set}$. Elements are tuples, say $(A,X)$. I think that inner products and coproducts are ...
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30 views

Elementary embeddings, elementary substructures,category of sets

I would like to characterize elementary embeddings AND elementary substructures in the category of sets and functions, Set. Not only characterize, but also justify this characterization.
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40 views

coalgebra/algebra of the identity endfunctor

Let $\mathbb{C}$ be a small/locally small category and let $T:\mathbb{C} \to \mathbb{C}$ be an endofunctor. One can then have $T$-algebras and $T$-coalgebras in the usual way: for $X,Y \in ...
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32 views

How many ways can we enrich a category if we fix certain objects and so on?

I am concerned with proposition 10.1.4 here http://www.math.harvard.edu/~eriehl/cathtpy.pdf , pg. 118. Essentially, what is asserted that if we have a closed $\mathcal{V}$-module, then there is a ...
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25 views

Commutative monoidal category free over singleton? Useful in proving coherence?

The proof of coherence in monoidal categories in CWM is based on the existence of a monoidal category free over a singleton. Denoting this category by ...
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51 views

Constructing right adjoint to pullback in a locally cartesian closed category

I am trying to finish the proof here. I haven't been able to convince myself that the implication $h\cdot u=f^{*}p\Rightarrow h^{f}\cdot v=s\cdot p$ is correct. My diagram chase leads to $f^{f}\cdot ...
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29 views

Simple question on proving naturality of an adjunction

Say you want to prove that there exists an adjunction between the functors $F:\mathcal{C}\to\mathcal{D}$, $G:\mathcal{D}\to\mathcal{C}$ with $F$ left adjoint to $G$, and suppose that the bijection on ...
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38 views

2-category as a 2-monad?

It is well known that a category is the same as a monad in the 2-category of spans. So I am wondering if there is a similar statement hold for higher categories: can a bicategory be given as a weak ...
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34 views

Categorization of Mono, Epis in a Category

Let $C$ be a category. Then the following implications on variants for monos hold: Iso $\implies$ SplitMono $\implies$ RegMono $\implies$ StrongMono $\implies$ ExtMono $\implies$ Mono, And dually ...
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50 views

Showing the existence of limits.

Suppose I have an adjunction $\mathcal{C}\overset{R}{\underset{I}\leftrightarrows}\mathcal{D}$, where $R\dashv I$, and $I$ is full and faithful. Now let $F:\mathcal{A}\rightarrow\mathcal{C}$ be any ...
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48 views

Question about accessibility of category of free abelian groups.

I've read, that the accessibility of the category of all free abelian groups is independent on basic set theory (say ZFC). What is the reason for that? And how can I interpret this result? Does it ...
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47 views

Can this category be identified as the category of graphs?

Let $\mathbb I$ be a category with exactly $2$ objects and $4$ arrows. The $2$ arrows that are not identities are parallel and their (common) domain and (common) codomain are distinct. Looking at the ...
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51 views

How does one characterise the reals without points?

The Real number line is defined upto unique isomorphism in the category $Fld$ as a Dedekind complete ordered field. One can view the Reals geometrically with its usual topology. In pointless ...
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38 views

Categorical formulation of linear transformations between vector spaces

My question regards the formulation of linear transformations between vector spaces as morphisms in an appropriate category. I know that any biproduct category admits a calculus of matrix. What I'm ...
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59 views

Showing a subcategory of $\mathbf{Top}$ is Cartesian-Closed

We start with some preliminary definitions (necessary because there is not much literature on this): a test map is a continuous function $\varphi:V\rightarrow X$ where $V$ is an open subspace of ...
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36 views

A field extension $K \subseteq F$ is finitely-generated if and only if there exists a ring epimorphism $K[t_1 \cdots t_n] \twoheadrightarrow F$

Does this make sense as an alternative definition for a finitely-generated field extension?: A field extension $K \subseteq F$ is finitely-generated if and only if there exists a ring epimorphism ...
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40 views

Sheaves: pretopology versus comma pretopology

I'm reading about sheaves on sites and I have a question about a particular example in these notes: http://www.math.harvard.edu/~nasko/documents/stacks.pdf http://homepage.sns.it/vistoli/descent.pdf ...
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27 views

Coherence theorem for symmetric monoidal categories

What's the formal statement for the coherence theorem for symmetric monoidal categories? I've seen there's some notion of permutation around, but I can't get my head around the thing that "all ...
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26 views

Natural Transformation Conditions

What conditions i have to satisfy for showing natural transformation? Just to show that the diagram is commutative or something else? I need to show that the composition of two natural transformation ...
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31 views

Indecomposable groups vs. indecomposable objects

An object $X$ in a category $\cal C$ with an initial object is called indecomposable if $X$ is not the initial object and $X$ is not isomorphic to a coproduct of two noninitial objects. A group $G$ ...
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87 views

Almost continuous and almost cocontinuous functors

I would like to consider Hom-like functors $H:\mathcal{A}\rightarrow\mathcal{B}$ defined between abelian categories or, more specifically, between module categories, in the following sense: $H$ is ...
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21 views

Reference: Topology on Ind-object

In some articles I've recently seen authors mention that pro-finite groups or pro-finite algebras possess a topology, but they do not explicitly describe it. I was wondering how is the topology ...
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24 views

Does the representation ring functor preserve limits?

If I have a diagram of groups $\{H_J\}$ and let $G$ be the limit of that diagram, how well does the representation ring functor "preserve the limit", IE: If I have $\lim_{J} H_J = G$ is it true that ...
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27 views

If $m = g \circ f$ in a dagger category and $m$ is an isometry, is it possible that $f$ fails to be an isometry?

Question. Suppose $m : A \rightarrow B$ is an isometry in a dagger category (by which I mean that $m^\dagger \circ m=\mathrm{id}_A$), and that we're given arrows $f : A \rightarrow Y$ and $g : Y ...
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19 views

Explicit fibrant replacement

Do you know an explicit fibrant replacement in the injective model structure on a functor category (I'm essentially interested in the case of presheaves of groupoids) ? Best
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42 views

How much does a theory tell about categorical properties of its models.

Let take a first order theory $T$ in a language $\mathcal L$. One can form the category $\mathrm{Mod}(T)$ whose objects are the models of $T$ (those $\mathcal L$-structures $\mathfrak M$ such that ...
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41 views

The category V-Cat enriched over V

Consider the category of finite ordinals $ \Delta $ and its objects $ 0, 1, 2, \ldots $. In the category of small categories, we have that $ Cat(0, \mathfrak{X} ) $ is the set of objects. The set $ ...
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51 views

Is it possible to consider this property of “being nice” as a homotopy property?

Consider a Model (Quillen) Category $ M $ (possibly with all oobjects being fibrant or cofibrant (or both)). I'm wondering if the following property is a homotopy property: "Let $ p $ be a morphism ...
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48 views

Where is the mistake? (derived functors )

Assume $pd(M) =n \leq \infty$ for a left $R$-module. I then have to show there exists a free module $F$ such that $Ext_{R}^{n}(M,F) \neq 0 $. I have tried these steps and obtained a contradiction: ...
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26 views

Relations in points fibre are relations in base category

Let $\mathcal E $ be a category with finite limits and let consider the points fibre $\text {Pt}_Y $ over an object $ Y $ and the projection $\pi:\text {Pt}_Y \to \mathcal E$. The question is: ...
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43 views

Is there a connection between contravariant functors and the axiom of choice?

Given that both can be seen as talking about reversing arrows between two objects: Is there a connection between contravariant functors and the axiom of choice? I'm initially motivated geometrical ...
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36 views

Sufficient conditions for Kan extension's to exist and be flat/coflat and exist

I'm very new to category theory and a little confused about how to proceed with the following problem any help is welcome! Suppose $F:A \to B$ is a flat and coflat functor, that is; it preserves ...
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28 views

Category modeling in quantitative Linguistics

I'm currently doing some studies on a new quantitative unit in Linguistics, the so called Motifs. A Motif is a ascending (or descending) sequence of quantitative linguistic properties. For a better ...
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52 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 ...
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58 views

Easy characterization of Cohomology in an Abelian Category

It should be quite an easy question and probably there's also a certain degree of intrinsic silliness in it, but still... Let $\mathcal{C}$ be an abelian category and let $C(\mathcal{C})$ be the ...
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31 views

A question about pre-additive category

Let $C$ be a pre-additive category with a zero object $O$. Suppose that every morphism in $C$ has a kernel and a cokernel and that every monomorphism in $C$ is a kernel of some morphism. Prove that ...
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96 views

Mistake in contradiction argument to show that sheafification commutes with cokernel

The sheafification functor is a left-adjoint to the forgetful functor. Hence it commutes with colimits. The cokernel is a colimit. Hence the cokernel of a sheafified morphism is the same as the ...
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39 views

Rel instead of Set in a concrete category

Concrete category is a pair $( \mathcal{C}; U)$ where $\mathcal{C}$ is a category and $U$ is a faithful functor $\mathcal{C} \rightarrow \mathbf{Set}$. But how to name a a pair $( \mathcal{C}; U)$ ...