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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94 views

What does it mean for a category to be “tensored over” another category?

What does it mean for a category to be "tensored over" another category? I was reading "Stable model categories are categories of modules" by Schwede and Shipley ...
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27 views

Universal Properties and Equalizer

Good evening. At the moment I am looking into category theory and at the moment I am trying to proof the Universal Property of the kernel as described here. I use the definition that the kernel of ...
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1answer
42 views

Functorizing a choice of sections

Take $\mathcal{C}$ and $T$ to be categories (if it helps, assume $T$ is a poset with a minimal element and $\mathcal{C}$ is cartesian closed). Take a functor $P\colon T\to \mathcal{C}$ where the image ...
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17 views

Inverse limit of small categories

It is well known that category $\mathcal{Cat}$ of small categories has all small limits and colimits. In particular it has all iverse limits. I am wondering if there is an explicit constraction of an ...
3
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1answer
70 views

(co)limits in the category of diffeological spaces vs. category of smooth manifolds

I am wondering which (co)limits that exist in the category of smooth manifolds are preserved by the inclusion into the category of diffeological spaces? Are there any results that allow us to ...
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1answer
46 views

N-Tuples or N-functions in category theory

When I'm writing out categories of my Haskell programs, I often get stuck whilst trying to describe morphisms that involve functions that involve more than one argument, such as 2-tuple construction. ...
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1answer
32 views

Predecessor on the final coalgebra (the extended natural numbers): difference of notation

In the Wikipedia article, the predecessor $f$ on the final coalgebra is not defined at 0, it is only defined at $n+1$ and $\infty$. In the $n$Lab article, $\operatorname{pred}(0) = *$, but $*$ is ...
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1answer
57 views

Isn't this a non-surjective epimorphism on the category of sets?

I am trying to prove that a morphism in the category of sets is epic iff it is a surjective function. Recall that for objects $A,B,C$, $f \in \hom(A,B)$ is epic when $g_1 \circ f = g_2 \circ f ...
5
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55 views

Prove that the isomorphism between vector spaces and their duals is not natural [duplicate]

In preparation for an introductory talk on category theory, I recently spent some time thinking about natural transformations. The first example, or maybe the second, that everyone gives to motivate ...
2
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1answer
41 views

Bijection in the Yoneda Lemma

To prove the Yoneda Lemma one defines a bijection between $[\mathcal{A}^{op},\mathbf{Set}](\hom(-,A),X)$ and $X(A)$ and shows that this bijection is natural in $A$ and $X$. In my textbook this ...
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1answer
20 views

Power set functors preserve monicness

This link discusses power set functors. Proposition 5.7 If $f$ is a epimorphism then so is $\exists_f$. Proposition 5.8 If $f$ is an monomorphism then so is $\forall_f$. A little ...
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65 views

In an abelian category is it true that $\ker f \cong \ker (\operatorname{coker} (\ker f))$?

In an abelian category is it true that $\ker f \cong \ker (\operatorname{coker} (\ker f))$? I am teaching myself category and was playing with the definitions of kernel and cokernel and think I ...
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61 views

Set of axioms for finite subset of Natural Numbers

I would like to get a set of Peano like first-order axioms for a finite subset of natural numbers $N'$ such that $0 \leq N' \leq Max$, with $Max$ denoting the upper-bound. (So my signature might be ...
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3answers
126 views

Name of some category with two objects

Does the category that consists of two objects and exactly one non-identity morphism that connects both objects have a specific name?
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2answers
177 views

Identify the universal property of kernels

I'm reading Mac Lane's "Categories for the Working Mathematician". I found the following sentence in page 59 of it: Similarly, the kernel of a homomorphism (in $\mathbf{Ab}$, $\mathbf{Grp}$, ...
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39 views

Colimits in $Ch_R$, help with a step of the proof

I want to prove that the category of chain complexes of R-modules admits small colimits. I was told to try proving that the chain complex defined degree wise as the colimit of the modules of the same ...
3
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1answer
78 views

What's up with this endofunctor $\mathbf{Aff}_k \rightarrow \mathbf{Aff}_k$?

(Work over a fixed field $k$.) The nLab offers a list of definitions of the concept "affine space". Here's two of them: An affine space is a set $A$ together with a vector space $V$ and an ...
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1answer
75 views

The definition of the $false$ truth value

In "Topoi: The Categorial Analysis of Logic" by R. Goldblatt the $false: 1 \to \Omega$ truth value is defined as the characteristic arrow of the arrow $0_1: 0 \to 1$. This definition requires that ...
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26 views

functors with a morphism lifting property

By analogy to the familiar situation in homotopy theory (i.e., (unique) path lifting in covering spaces), it is natural to consider the following. Let $P:C\to D$ be a functor. Say that $P$ has ...
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46 views

Generalisation of posets

The notion of (set) monoid can be generalised to that of monoid object in a monoidal category. Can the notion of poset be generalised in a similar fashion? How?
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1answer
27 views

Does an injection of finitely generated abelian groups always induce a surjection via $Hom(-,U(1))$?

I was recently interested in the following conjecture, which at first sight seemed pretty elementary. Conjecture: Let $i: A \hookrightarrow B$ be an injection into a finitely generated abelian group. ...
3
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1answer
58 views

Why are functors exact if they preserve all short exact sequences?

If a functor $F\colon \mathcal C → \mathcal D$ of abelian categories preserves short exact sequences, why is it exact? I know the argument is supposed to be that you can split up long exact sequences ...
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3answers
532 views

Is Stokes' Theorem natural in the sense of category theory?

Stokes' Theorem asserts that for a compactly-supported differential form $\omega$ of degree $n-1$ on a smooth oriented $n$-dimensional manifold $M$ we have the marvellous equation $$\int_M d\omega = ...
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1answer
29 views

Natural Isomorphism Between Two Desciptions of Tangent Space

Let $M$ be a smooth manifold. Let $(T_pM)_{alg}$ denotes the "algebraist's" tangent space at $p\in M$, that is the tangent space via derivations, and $(T_pM)_{kin}$ denote the "kinematic" or ...
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1answer
58 views

(Hopefully) Simple question about the exterior algebra functor

I have some (hopefully super) basic questions about the exterior algebra functor $$ \wedge:R\text{-Mod}\rightarrow R\text{-Alg}. $$ As I (think I) understand it, if one considers it as a functor ...
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1answer
26 views

Is there a characterization of coverings in subcanonical pretopologies?

Let $\mathcal C$ be a category. A sieve for $\mathcal C$ is called strictly universally epimorphic if it is one of the covering sieves for the canonical topology on $\mathcal C$. SGA4 gives the ...
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1answer
37 views

find a coproduct in a cauchy complete category

In an additive Cauchy complete category $\mathcal C$, i.e. all idempotents in $\mathcal C$ splits. let $f: X \rightarrow Y$ and $g: Y \rightarrow X$ such that $g \circ f = id_X$. Is there an object ...
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1answer
44 views

Does the bicategory of bimodules really have left Kan extensions?

Let $\mathsf{Bimod}$ denote the bicategory whose $0$-cells are rings $A$, a $1$-cell $A \longrightarrow B$ is an $(A,B)$-bimodule and a $2$-cell is a bimodule map. The composition of $1$-cells is ...
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1answer
73 views

Distributor? Distributive analog of commutator and associator?

Motivation: "the commutator gives an indication of the extent to which a certain binary operation fails to be commutative" (http://en.wikipedia.org/wiki/Commutator). For example (courtesy of ...
2
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1answer
60 views

A Characterization of Categories with a Conservative Forgetful Functor to SET

Examples of categories over $\bf{Set}$ such that the forgetful functor is conservative include the "algebraic" categories of groups, rings, modules, monoids, etc., but does not include the ...
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83 views

Examples of categories which appear naturally without objects

Regarding the morphisms-only-definition of a category (which is equivalent to the usual one dealing with objects and morphisms), I would like to ask: Which examples of categories in practice appear ...
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1answer
49 views

Reference request: Derived category of category with sufficiently many injectives

I'm studying derived categories and have encountered problem with references I have. Namely, proof of the following theorem: Theorem: Let $\mathcal A$ be Abelian category and $\mathcal I$ full ...
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24 views

Proof of Quillen's lemma about minimal fibrations

In Hovey's book $\textit{Model categories}$ (available online here http://www.math.rochester.edu/people/faculty/doug/otherpapers/hovey--model-cats.pdf) one finds the classical result that any ...
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59 views

$\mathbb Z$ is not a dense generator in $\mathsf{Ab}$

Why is $\mathbb{Z}$ not a dense generator in $\mathsf{Ab}$, the category of abelian groups? This is exercise 4.8.6 in Borceux's Handbook of Categorical Algebra. There is a hint which says to consider ...
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1answer
37 views

Showing that the moduli stack of triangles is equivalent to the quotient stack $[\tilde{T}/S_3]$

$\require{AMScd}$ A good reference for this question is found in the first chapter in the unfinished book Algebraic Stacks by Behrend, Conrad, Edidin, Fulton, Fantechi, Göttsche, and Kresch. ...
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175 views

The free monoid functor is fully faithful?

For every set $A$, there is a free monoid $A^*$ and a function $i_A : A \rightarrow A^*$, such that for all monoids $Z$ and functions $j : A \rightarrow Z$, there is a unique monoid morphism $j^* : ...
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0answers
34 views

A monomorphism in the category of compact Hausdorff spaces is regular

Let $f \colon X \rightarrow Y$ be a monomorphism of compact Hausdorff spaces. This is just a continuous injection. I am trying to show that $f$ is regular, i.e. it is an equaliser. My first thought ...
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1answer
36 views

Right adjoint of a triangle functor is also unique

In general, right adjoint of a functor is unique. In triangulated categories, this is also true. My question is why the natural isomorphism between two right adjoints is compatible with the triangle ...
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27 views

Subobject classifier in $Sets^{Q}$

Let Q be the linearly ordered set of rational numbers considered as a category while $R^{+}$ is the set of reals with $\infty$. In $Sets^{Q}$,prove that the subobject classifier $\Omega$ has ...
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56 views

Topological insight on an algebraic theory.

I recently read about the marvelous bar construction. As far as I understand, it is something of a free resolution in an abstract setting. I'm wondering if it can be used to get topological insights ...
2
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1answer
29 views

Coherence result for (braided) monoidal functors

Is there any coherence result for (braided) monoidal functors? (like Mac Lane's coherence theorem for monoidal categories) What I have in mind is a theorem like the following: "Let F be a (braided) ...
6
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1answer
70 views

Motivation for the definition of an infinitesimal object

An infinitesimal object $D$ in a Cartesian closed category $\mathsf{C}$ is one for which the internal Hom functor $$(-)^D: \mathsf{C} \to \mathsf{C}$$ has a right adjoint. I am wondering what is the ...
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1answer
51 views

Examples of certain types of toposes

I'm looking for examples of (non-degenerate) categories $\mathcal{C}$ such that both $\mathcal{C}$ and $\mathcal{C}^{op}$ are toposes (assuming that such categories even exist). On a related note, ...
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1answer
34 views

Is Rel a topos?

Is the category Rel of sets and relations a topos? I've done a few Google searches about this question but I haven't found any answers either way. And I can't recall any answers either way in any of ...
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1answer
118 views

commutes homology

I am trying to prove the following: Let $R,A$ be rings and $\mathrm T:$$\mathscr M_R$ $\to $$\mathscr A_R$ such that $\mathscr M_R$ is category of left R modules and $\mathscr A_R$ is category of ...
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Adjoint functor to an R-algebra only “remembering” itself as a ring

I have been wondering this question while trying to comprehend adjoint functors and the various definitions. If you let $$F:\mathbf {R\text - Alg}\to \mathbf {Ring}$$ be the functor that sends ...
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44 views

Category Group has unique identity morphism

Let $\mathcal C$ be a category which as finite products and a terminal element $\ast$. A monoid is a pair $(G,e,m)$, where $m: G \times G \rightarrow G$ and $e: \ast \rightarrow G$ are morphisms ...
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23 views

Filtered colimits are exact in abelian categories

It is well known that filtered colimits commute with finite limits in $\mathsf{Set}$, and hence in every algebraic category - $R\mathsf{Mod}$ in particular. Unless I'm wrong, from the Mitchell ...
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1answer
35 views

Examples of thick subcategory

I'd like to know several examples of thick/Serre subcategory of an abelian category, I have no one in mind now. Help me please!
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58 views

Direct products, direct sums and coproducts in category of groups

I have couple questions about terms I mentioned in the title. Why we don't define direct sum of non-abelian groups (subset of direct products which consists of elements with almost every component ...