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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Iterated endofunctors

Suppose $F : \mathbb{C} \rightarrow \mathbb{C}$, with the following constraints: $F^{n+1}(\mathbb{C})$ is a subcategory of $F^{n}(\mathbb{C})$ for all objects $X \in \mathbb{C}$ there exists a ...
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54 views

Covariant functors represented by schemes

Let $S$ be a nice base scheme and let $F : \mathsf{Sch}/S \to \mathsf{Set}$ be a functor. Are there necessary and sufficient conditions that $F$ is represented by some scheme $X$, i.e. $F \cong ...
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34 views

morphism of monoids and group

let $B,C$ be commutative monoids, $A$ be an abelian group and $B \stackrel{b}{\leftarrow} C \stackrel{a}{\rightarrow} A$ morphisms of monoids. Let $P$ be the push-out of $A$ and $B$ w.r.t. $C$ in the ...
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192 views

Reflection of Exact Sequences

Consider the category of $R$-modules. I am trying to see how i can express a short exact sequence in terms of kernels and cockerels, and how this description can be used to prove that a conservative ...
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125 views

Which (endo)functors of the category of finite-dimensional real vector spaces induce continuous maps between Hom-sets?

Let $\operatorname{Vect-fin}$ be a category of finite-dimensional vector spaces over $\mathbb{R}$. In this category Hom-sets $\operatorname{Hom}(V,W)$ are themselves finite-dimensional vector spaces ...
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Functors $f^*$ and $f_*$ on the category of sheafs of modules (Hartshorne)

Let $f: (X, O_X) \rightarrow (Y, O_Y)$ be a scheme morphism, $F$ - module over $O_X$, $G$ - module over $O_Y$. How to prove, that $$ Hom_{O_X}(f^*G, F) = Hom_{O_Y}(G, f_* F). $$ Please give the most ...
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How to express, say, the Hausdorff property of a topological space using categories. (And a request for general advise in such practices)

I am new to Category theory and for the sake of the practice, I am interested in revisiting and expressing those concepts that I am familiar with --however basic--, in the language of categories. ...
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173 views

Karoubi envelope and factorization of a functor

I am trying to understand the solution to the following exercise. Let $\mathcal{C}$ be a category with idempotents $e:A\to A$ and $d:B\to B$, and a morphism $f: A \to B$. The Karoubi envelope ...
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63 views

What categories have all the products?

What is the best characterization of categories which have all the products? Is this question related to Axiom of choice in any way?
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283 views

On the bounded derived category of a finite dimensional algebra with finite global dimension

Let $A$ be a finite dimensional $k$-algebra with finite global dimension. How can I prove that the category $D^b(A)$ (bounded derived category of the category of left finitely generated $A$-modules) ...
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129 views

Has the notion of a unique factorization category been defined and studied?

I am using a this notion in a paper that I am writing. The notion is that $(\mathcal{C},\otimes, I)$ (which we will just call $\mathcal{C}$) is a symmetric tensor category, with a unit $I$. Then a ...
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The category of adjoint functors

Can a category structure be defined on the collection $Adj(\mathbf C,\mathbf D)$ of all pairs of adjoint functors $$ (F\colon\mathbf C\to \mathbf D)\dashv (G\colon \mathbf D\to \mathbf C)$$ in such a ...
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73 views

Cartesian monoidal functors

Let $\mathcal{C}$ and $\mathcal{D}$ be categories with finite products, and consider them as monoidal categories in the obvious way. Every functor $\mathcal{C} \to \mathcal{D}$ can be canonically ...
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86 views

A finite diagram in an abelian category which may not be locally small

This question is motivated by this. I will use the notations of my answer to this. We say a category $\mathcal C$ is locally small if Hom($X, Y$) is small for any $X, Y \in$ Ob($C$). Let $\mathcal ...
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99 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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184 views

Does the restriction functor (big to small) commute with the inverse functor?

I would like to know whether the restriction functor commutes with the inverse image functor. (I basically follow the terminology in "Topological and Smooth Stacks".) Let $X$ be a topological space. ...
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180 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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Can this statement about factorization of measurable mapping be described in category language?

Problem 13.3 of Probability and Measure by Billingsley states: $(\Omega, \mathcal{F})$ and $(\Omega', \mathcal{F}')$ are two measurable spaces. Suppose that $f: \Omega \rightarrow ...
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179 views

Constructing a semigroup from a small category

The following was given as an example for a semigroup without an identity: Finite sets of matrices of varying dimensions, where the product $A*B=\{PQ \mid P \in A, Q \in B \text{ and } ...
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451 views

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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106 views

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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137 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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41 views

How to compute (co)limits of enriched categories?

Let $\mathscr{V}$ be a monoidal category. Let $\mathbf{Cat}_{\mathscr{V}}$ be the category of (small) categories. I would like to know how to compute (co)limits in $\mathbf{Cat}_{\mathscr{V}}$. This ...
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Pullbacks in filtered categories?

A sufficient condition for the inclusion of a full subcategory $\mathsf C\hookrightarrow \mathsf D$ to be cofinal is that: Every object of $\mathsf D$ has an arrow into some object of $\mathsf C$. ...
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Unorthodox definition of semi-abelian category

I recently stumbled upon the book Derived Functors in Functional Analysis by Wengenroth. In it, he defines semi-abelian categories quite differently from the nlab: An $\mathsf{Ab}$-category ...
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65 views

Are subvarieties just full subcategories that happen to be algebraic categories?

Suppose $T$ is a Lawvere theory. Suppose furthermore that $\mathbf{C}$ is a replete full subcategory of $\mathbf{Mod}(T)$ that is equivalent (as a category) to $\mathbf{Mod}(S)$ for some Lawvere ...
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27 views

Topological reflection of pretopological closure operator

Given a pretopological space $(X,\mbox{cl})$ where $\mbox{cl}$ is a pretopological closure operator. How does one find the topological reflection of $(X,\mbox{cl})$? I know of a way namely by ...
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40 views

Algebra structure on dual to coalgebra

I'm trying to prove the following theorem using braided diagrams: Let $(C,\Delta,\varepsilon)$ be a finite-dimensional coalgebra. There is an algebra structure on $C^*$ given by multiplication ...
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60 views

Interaction of functors and homology in abelian categories

I'm working on exercise 1.6.H.a) of Ravi Vakil's algebraic geometry course notes. I'm aware that a question was posted on the same topic before (Prove the FHHF theorem using as much abstract non-sense ...
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when will homology and direct limit commute?

Question: Let a sequence of maps between topological spaces $$ X_1\to^{f_1}X_2\to^{f_2}X_3\to^{f_3}\cdots $$ The mapping telescope is denoted by $T$. Under what conditions will $H_*(T)$, the ...
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Given bifunctor $F$, what is the name of the functor with switched arguments?

Sorry for the unspecific title. Here the actual question: Given categories $\mathcal{A},\mathcal{B}$, let $S$ be the canonical functor $\mathcal{B} \times \mathcal{A} \to \mathcal{A} \times ...
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53 views

What's the largest universe we use?

I know that the notion of a Grothendieck universe is used to deal with the fact that sometimes the categories of category theory are "too large". In general, how large of a universe is worked in? If ...
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77 views

Is this a better way to think about Groups as Categories?

I asked a bit ago how to reconcile the category theoretic and set theoretic definitions of groups (groupoid which is a monoid vs the set theoretic definition), and I got the answer I was looking ...
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properties of pullback diagrams

Suppose you have a commutative diagram: $\require{AMScd}$ $\begin{CD} A @>>> B\\ @VVV @VVV \\ C @>>> D \\ @VVV @VVV \\ E @>>> F \end{CD}$ Let $T$ be the top "square", $B$ ...
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How does a symmetric graph indexing category work?

In Category theory for the sciences, section 4.2.1.20 it is explained how a graph is a functor from an indexing category to a set. I think I understand the basic concept: Ar is mapped to a set of ...
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Homotopy colimits preserve weak equivalences

It is well known that homotopy colimits of diagrams are constructed so that if one has weak equivalences between all objects of two diagrams (under the same indexing category) the induced map between ...
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What should this definition be?

This is from Advanced Linear Algebra by S.Roman. Definition (on page 356). Referring to the figure below, let A be a set and let $\mathcal S$ be a family of sets. Let $$ \mathcal F = \{g\colon ...
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51 views

Monomorphism preservation by pullback

I'm working through Category theory for the sciences (by David Spivak) and I'm a bit stuck in section 2 - my problem is with Proposition 2.7.5.5 and Exercise 2.7.5.6, I hope someone can help :) The ...
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Representation of functions in the simplex category

I am using the following characterization of the simplex category $\Delta$: The objects are finite ordinals and the arrows are weakly monotone functions $f:n\rightarrow m$. $0$ is initial and $1$ is ...
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56 views

Why are morphisms of monads lax and not oplax natural transformations

Monads in a bicategory $\mathscr B$ correspond to lax functors $* → \mathscr B$, so one expects morphisms of monads should correspond to nautral transformations between them. A natural ...
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Proof of theorems in the field of banach-and $c^*$-algebras in a categorial language

At the moment I'm studying the basics in the theory of banach- and $c^*$-algebras. There are many results in the theory of $c^*$-algebra which you first prove in the unital case and then in the ...
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32 views

Does the wedge product of bundle-valued forms induce a universal object?

Given a smooth manifold $M$ and a vector bundle $E$ over $M$, the $C^\infty(M)$-module of $E$-valued $p$-forms on $M$ is defined to be $$\Omega^p(M; E) := \Gamma_M\left( \bigwedge^p T^*M \otimes E ...
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31 views

Is Tambara-Yamagami category admits a braiding when G is a nonabelian group?

Tambara-Yamagami category is a fusion category which its simple objects are elements of a group and one element out of group. i.e : $$simple\;objects = G \cup \{m\}$$ The fusion rule of this ...
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Homotopy groups of mapping spaces

If I have an $\infty$-category $\mathcal{C}$ (AKA quasi-category), can I say anything about the homotopy groups of the mapping spaces $\mathrm{Hom}_\mathcal{C}(X,Y)$ for two objects $X$ and $Y$? ...
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Characterization of epic morphisms in the category of rings.

Let $C$ denotes the category of rings (with identities). It's easy to prove that for any $R\in \text{ob }C$, any multiplicative set $S\subset R$ and any ideal $I\leqslant R$, both $R\to S^{-1}R$ and ...
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Definition of Schur Functors on morphisms

I've been learning about Schur functors on nLab: http://ncatlab.org/nlab/show/Schur+functor A definition is given, for $R$ some finite dimensional representation of $S_n$, by the formula $S_R(-)=R ...
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Question on the coskeleton functor $\mathbf{coskel}'$ for pointed simplicial sets

There is an abstract construction of the coskeleton functor for simplicial set as follows: Fix an integer $n\geq 0$ and take the inclusion of the full subcategory $i_n\colon\Delta_{\leq ...
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$\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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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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About the construction of Quot-Schemes

I am reading in the paper of Nitsure (link: http://arxiv.org/pdf/math/0504590v1.pdf) about the construction of the Quot-scheme $\mathrm{Quot}_{E/X/S}^{\Phi,L}$. After Lemma 5.4. they reduce to the ...