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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Proof of Yoneda Lemma

Can anyone explain to me Yoneda Lemma proof in great details? i.e. they usually say " ... it is easy to see that these morphisms are inverse to each other.." without explanation.
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In a triangulated category with coproducts any idempotent splits

In a triangulated category with coproducts any idempotent splits. Is there a proof of this fact different from that in Neeman, Prop. 1.6.8? In particular I'm looking for one which doesn't use the ...
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134 views

Simple Category Theory/Grps Problem

I'm doing a little homework but I think my brain has ceased to function late at night, was hoping you could help me out with what I am certain is a very simple group/cat theory problem. Similarly to ...
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53 views

$\mathsf{Top}$ with proper maps has products.

In I.M. James' General Topology and Homotopy Theory, he presents proper maps before introducing compact sets, by defining $\phi:X \to Y$ to be proper iff $\phi \times \text{Id}_T$ is closed for all $T ...
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33 views

Maclane's Coherence Theorem: why not just use the functors themselves?

I have a softball question on Maclane's proof. Suppose $B=\left ( B, \square , \alpha ,\rho ,\lambda \right )$ is a monoidal category. Fix $b\in B$. Define $W$, the (monoidal) category of binary ...
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48 views

Relation between being epic and having a “full” image.

Let $g: A \to B$ be a morphism in an abelian category. Is it true that $g$ is epic iff $im(g)=B$? Context: Given a short sequence in an abelian category $0 \to A \to B \to C \to 0$ with maps $f: A ...
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66 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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60 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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28 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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57 views

T-Algebras for a monad

Suppose $R$ is a ring with identity, $G:R$-$Mod\rightarrow Set$ is the forgetful functor and $F:Set\rightarrow R$-$Mod$ its left adjoint. I want to prove that the structure maps for the T-Algebras of ...
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49 views

Does this prove an equivalence of categories?

The definition I am working with (I know there is a stronger notion) states that a functor is an $\textbf{equivalence of categories}$ if it is fully faithful and essentially surjective. I was reading ...
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159 views

Formalizing finitism in category theory

If we assume that finitism can be formalized by primitive recursive arithmetic (PRA), what category could it correspond to? In particular, which sort of a natural numbers object (NNO) may it contain? ...
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56 views

Relations in a finitely complete category with images

I can't use tikz inside MathStackExchange; so my question is on an image below.
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28 views

A category is locally finitely presented if the relative purity and purity coincide

Let $A$ be a locally finitely presented additive category, $X$ an additive subcategory. A sequence $0\rightarrow A_1\rightarrow A_2\rightarrow A_3\rightarrow 0$ in $A$ is pure exact if it is ...
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40 views

When defining a Grothendieck pretopology,can we get away with less than the fibre product axiom?

$\newcommand\restr[2]{{\left.#1\right|_{#2}}}$ I'm fairly new to this whole area, so correct me if there are any technical errors in any of this. The base category for a classical sheaf is the ...
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62 views

Derived version of projection formula

Let $f \colon X \to Y$ be a continuous map of locally compact spaces. Denote by $Sh(X)$, $Sh(Y)$ the categories of sheaves of $k$-vector spaces for some field $k$ and by $D^b(X)$, $D^b(Y)$ their ...
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63 views

Coproducts and pushouts of Boolean algebras and Heyting algebras

I am having trouble of find a reference explaining how to compute coproduts and pushouts in the category of Boolean algebras and in the category of Heyting algebras. To be precise I am looking for ...
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79 views

Higher transformations between natural transformations and so on

In category theory, arrows between categories are functors, arrows between functors are natural transformations, so a natural question is to ask what are arrows between natural transformations ? I've ...
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51 views

What are the projective and the injective objects in the category of spectra?

What are the projective and the injective objects in the category of spectra (of simplicial sets)? Does the category of spectra have enough projectives and injectives? An object $P$ of a ...
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72 views

Does zero-kernel imply monic in Abelian categories?

I'm trying to learn how to perform diagram-chasing in abstract Abelian categories. Instead of an approach with some elements one have to use universal properties somehow in the proof. But I reckon the ...
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41 views

Lost Chevalley Manuscript

In the end notes of chapter 9 in Mac Lane's "Categories for the Working Mathematician", he mentions a lost Chevalley manuscript A systematic treatment of all possible properties of limits was ...
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34 views

Protomodular categories

The axioms for abelian categories are nice and clear. The axioms for protomodular categories - and therefore semi-abelian categories - are beyond me entirely. I'm looking for a breakdown of the ...
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33 views

Derived Category in terms of Torsion Theory?

It is known that there's a bijection between hereditary torsion theories on, and localizations of, a fixed abelian category. Is this bijection natural? How/why not? How can I think of the derived ...
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75 views

Grothendieck Topology on the Category of Elements

We are given a site $(C, J)$ for a small category $C$ and a Grothendieck topology $J$. If $F\in Sh(C, J)$, we take the natural topology $J_F$ on its category of elements $el(F)$ induced by $J$. I am ...
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44 views

Reference for closed categories and monoidal categories

I'm looking for a book that: Defines closed categories separately from monoidal categories, and then proves in detail that the structure induced by a left adjoint to the internal hom is closed ...
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76 views

Image sheaf is the sheafification of the image presheaf

This is an exercise in Vakil's notes on foundations of algebraic geometry. Suppose $\Phi:\mathscr{F}\to\mathscr{G}$ is a morphism of sheaves of abelian groups, show that the image sheaf Im $\Phi$ ...
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52 views

Coproduct of Lie algebras

Fix a commutative ring $k$ and look at the category of Lie algebras over $k$. How do coproducts in that category look like? Notice that what is usually called the "direct sum" of Lie algebras is not ...
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72 views

Connecting morphism in an abelian category

I'm trying to understand how one gets the long exact sequence in homology from a short exact sequence of chain complexes in an arbitrary abelian category. So far I have the commutative diagram ...
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31 views

Does the 2-functor $PsAlg\to \mathfrak{X} $ reflect equivalences?

Consider a $2$-monad $ T: \mathfrak{X}\to \mathfrak{X} $ and consider its 2-category of pseudoalgebras $PsAlg$. There is a forgetful functor $ U: PsAlg\to \mathfrak{X} $. Does this forgetful functor ...
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47 views

What kind of objects are both subobjects and quotients?

Fix an object $B$ in some category. What does the existence of a diagram $A \rightarrowtail B \twoheadrightarrow A$ imply about $A$ and $B$? What if $A \rightarrowtail B \twoheadrightarrow ...
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76 views

Categorical description of Alexander horned sphere?

Looking at the construction of the Alexander horned sphere: Is it possible to describe as some kind of direct limit of spaces? How can one formally see this?
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37 views

Structural / design / meta optimization - is there mathematical theory. Optimization over categories?

There is huge branch of mathematical optimization theory, but it mostly considers the finding optimal parameter values for the predefined structures. There are variational calculus and optimal control ...
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37 views

pseudonatural vs natural

By a general result of steve lack's article, I know that there is a nice adjunction between the 2-category of 2-functors (and 2-natural transformations) and the 2-category of 2-functors and ...
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53 views

Does the five lemma hold true for Lie algebras?

According to wikipedia, the Five Lemma is true in Abelian categories. But the category of Lie algebras is not Abelian. Then is the Five Lemma still true for Lie Algebras?
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108 views

Solutions to questions in “Categories for the Working Mathematician”

Does anyone have solutions for the exercises in "Categories for the Working Mathematician"? I'm working my way through them, and want to check my answers.
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63 views

Monomorphisms in a concrete category

Let $\mathcal{C}$ be a concrete category, i.e., a category which admits a faithful functor $C:\mathcal{C}\rightarrow \mathsf{Set}$. It is certainly not the case that $f$ a mono in $\mathcal{C}$ ...
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43 views

How to prove that a particular (sub-)category has a projective generator.

Suppose that $\mathcal{C}$ is an abelian $k$-linear category ($k$ a field) in which every object is of finite length and every $k$-vector space $\text{Hom}(X,Y)$ is finite dimensional. How does one ...
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62 views

Adjoints with a Parameter

I am tryng to prove a theorem in McLane without looking at his proof and I am stuck on one point. Suppose $F:X\times P\rightarrow A$ is a bifunctor, that for each $p\in P $, the functor ...
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71 views

Yoneda lemma for enriched categories

Let $\mathcal{M}$ be a monoidal category. Is there a generalization of the Yoneda lemma to categories enriched over $\mathcal{M}$? In the specific case I need, $\mathcal{M}$ would be the category ...
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48 views

A question regarding Yoneda's lemma.

Suppose you have two objects $A$ and $A'$ in a category $\mathfrak{C}$, and morphisms $i_C:\text{Mor}(C,A)\to\text{Mor}(C,A')$ for any object $C\in\mathfrak{C}$. Show that the $i_C$ are induced ...
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60 views

Commutative diagrams with antiparallel arrows

A diagram in category theory is said to commute when for all objects $A$ and $B$ in it, every the composite morphism resulting from a possible path from $A$ to $B$ are the identical. Does that mean ...
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34 views

The closure of the graph of a certain composite in a topos.

Let $\mathcal{E}$ be an elementary topos with subobject classifier $\Omega$ and let $j\colon \Omega\to\Omega$ be a Lawvere-Tierney topology on it. Assume that, for an object $C$ of $\mathcal{E}$, each ...
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90 views

Transfinite composition and $\kappa$-categories

I wonder whether the following ideas make sense, and whether something in that direction has been written down somewhere; do you know a reference? The last definition is supposed to be a ...
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0answers
34 views

Locally Cartesian Closed implies Cartesian Closed

We follow Awodey's definition (page 235, Category Theory, 2nd ed.) of locally cartesian closed categories: A category $\mathcal E$ is locally cartesian closed provided $\mathcal E$ has a terminal ...
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125 views

Commuting with kernels implies left exactness in Abelian category

I'm following Vakil's notes - chapter on category theory. One issue that is unclear in the notes is the conclusion that right adjoint functors are left exact. The notes define a left exact functor as ...
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If models of set theory can be construed as categories, can notions of forcing be construed as functors?

Consider the following passage from Blass and Scedrov's paper "Complete topoi representing models of set theory"(Annals of Pure and Applied Logic, vol. 57 (1992),PP. 1-26) where 'set theory' in this ...
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115 views

Must diagrams be commutative?

Given a category C and a function $ \Theta : Mor(\textbf{C}) \times Mor(\textbf{C}) \longrightarrow Mor(\textbf{Rel}) $ and suppose that the relation, with $(r,s)\in\Theta(u,\bar{u})$ and so forth, ...
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51 views

Category of Hilbert Spaces

Is it possible to triangulate the category of Hilbert spaces and bounded linear operators? I assume that one candidate for triangulation is the double dual space. What is a fact is that this ...
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25 views

Does the lax Gray tensor product preserve fully faithful 2-functors?

The question is in the title. By fully faithful 2-functor, I mean 2-functors such that the maps on the hom categories are isomorphism, and by preserve, I mean in each variable. I have an argument ...
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107 views

Relation between existential and universal quantificator in category theory

Let $\mathscr C$ be a cartesian (i.e. with finite limits) category with subobject classifier $\Omega$ and generic subobject $\tau:I\to\Omega$ (here $I$ denote the terminal object). Let $f:X\to Y$ and ...