Questions tagged [category-theory]

Categories are structures containing objects and arrows between them. Many mathematical structures can be viewed as objects of a category, with structure morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seeming different areas of mathematics.

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Natural morphism of sheaves $pr_1^{-1} F \otimes pr_2^{-1}G \rightarrow j_*(F\otimes G)$

I am reading a book, and the book said there is a natural map, which I don't know how. Can someone help me please? I can try to define the map, but it will be complicated, using sheafification many ...
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Construction of colimit via semi-final lift

Let $U:J\rightarrow C$ be a diagram and $V:C\rightarrow D$ a functor such that there exists a colimit $(Y,\{g_\alpha:V(U(I_\alpha))\rightarrow Y\}_\alpha)$ of $V\circ U$ in $D$ and a semi-final lift $(...
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“Absolute retracts” in arbitrary category

Is there a standard notion of something like "absolute retract" in arbitrary categories that generalizes absolute retracts in topology? I am mostly interested in categorical approach to Hausdorff ...
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Biproduct in category

Let $(X\oplus X', \pi_X, \pi_{X'}, \iota_X,\iota_{X'})$ and $(Y \oplus Y', p_{Y'},p_Y, j_Y,j_{Y'})$ biproducts in a category $\mathcal{C}$. In MacLane's book, he defines using the structure of product,...
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Subobjects Equivalent iff Isomorphic Domains?

Regarding subobjects as monics (not as equivalence classes of monics), I seem to have proven that subobjects are equivalent iff their domains are isomorphic as objects of the category in question. ...
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Naturality of Transformations

When we say some arrow $\eta _A$ is natural in $A$ ($A$ being an object of the category in question, $\mathsf C$), we mean it is a component of a natural transformation. I have consistently stumbled ...
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Simple problem about morphism in abelian categories

$f$ : $X\to$ $Y$ and $g$ : $Y\to$$Z$ a sequence in abelian categories. Show that if $gf$=$0$ if and only if exist a monomorphism $h$:$Im(f)$ $\to$ $Ker(g)$ such $kh$=$j$, where $j$:$Im(f)$$\to$ $Y$ ...
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If $f= \mathrm{ker}\,g$, then $g = \mathrm{coker}\,f$?

I didn't understand a step in the proof of Proposition 5.92 from Rotman's Introduction to Homological Algebra (2nd Ed.) where he says: "there is a morphism $g: B\to C$ [in a given abelian category $\...
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Contravariant functor properties

What does F as an exact contravariant additive functor preserves or changes over an abelian category? (i.e kernels, cokernels, images, etc) Thanks
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Catsters Video Question

The first Catsters video on adjunctions has just finished, at this time, describing adjunctions in 2-categorical terms. Basically, the idea is to whisker the adjoint functors and the (co)unit of ...
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References about “algebras over monoids”

Please, could someone point me any reference (with a bit of details) about "algebras over monoids" (in the sense of Schwede & Shipley, Algebras and modules in monoidal model categories)? Thank you ...
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Instructive sources for arguing without elements

There is a trend in mathematics towards reasoning without elements if possible (coming from category theory, I presume). I see the appeal and want to learn how to argue avoiding the use of elements, ...
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Direct limit with non-injective maps

Suppose I take a direct limit in the category of groups, or the category of R-modules, or similar. Let $I$ denote the index set, $A_i$ the objects and $f_{ij}: A_i \to A_j$ the structure maps of the ...
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A contravariant functor taking colimits to limits is representable.

If $F$ is a contravariant functor from $Sets$ to $Sets$. And for any functor $H: I \to Sets$ that has a colimit $C$ we have $F(C)$ is a limit for $F \circ H: I ^{op} \to Sets $. Show that $F$ is ...
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“Identity-free” definition of an isomorphism in a semigroupoid / semicategory

I am looking for a way to define "Isomorphism" in a semigroupoid (or semicategory), that is a "category", which does not necessarily have identities. To be more specific I am looking for a way to ...
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Categorically expressed property in $\textbf{Set}$ for $P(A)=\{17\}$

I'm reading Fokkinga's Gentle Introduction to Category Theory, of which page 12 asks to give categorically expressed properties (i.e., using the language of basic CT and existensial qualifiers) in $\...
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Algebraic K-theory: Categories of modules and their equivalences

I'm currently preparing a presentation on the second chapter, called "Categories of modules and their equivalences", in Algebraic K-theory by Hyman Bass. I have a VERY elementary understanding of ...
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In an abelian category,every morphism can be written as composition of epi and mono. [duplicate]

Following Weibel's book on homological algebra, he states without proof that every morphism $f\colon A \to B$ can be written as composition of an epimorphism followed by a monomorphism. After many ...
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example of algebraic theory,free product completion,graphs

Let us denote by $\def\Graph{{\sf Graph}}\Graph$ the category of directed graphs $G$ with multiple edges: they are given by a set $G_v$ of vertices, a set $G_e$ of edges, and two functions from $G_e$ ...
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surjection between sets which are defined through a functor

I'm facing the following problem and have no idea how to deal with it. We consider a functor $T:\underline{Set}\rightarrow\underline{Set}$ and two sets $X,Y$. We can build the product $X\times Y$ ...
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Polynomial Functor is ω-Continuous

Suppose $P:Set\rightarrow Set$ is given by $X \mapsto \sum_{i=0}^{n} C_{i}\times X^{i}$. I want to prove that $P$ is ω-continuous. Three questions: 1). Is my following argument correct? 2). Is there ...
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A term for category where every loop of morphisms is an identity

"A category where composition of every loop of morphisms is an identity." Moreover, in the case I am thinking about, morphisms are bijective functions. Is there a name for this concept?
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A question regarding $Hom(X,X)$ (or $Mor(X,X)$)

I refer to Rankeya's answer on this question. Shouldn't $Mor(X,X)$ consist of monomorphisms only, for each morphism to have an inverse?
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Is there a term for an endomorphism defined up to conjugation by an automorphism?

Is there a standard term to designate the equivalence class of endomorphisms where two endomorphisms $\phi$ and $\psi$ are considered equivalent if there exists an automorphism $\alpha$ such that $\...
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Check that this is a category

Assume we have a fixed field $F$. We define objects as homomorphisms $\phi:F\rightarrow G$. Then we define morphisms between $\phi:F\rightarrow G$ and $\psi:F\rightarrow L$ as ring homomorphism from $...
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exercise involving exactness

I have been stuck on this exercise for a little while. We are in abelian category. How do I show that $0 \rightarrow A \rightarrow B$ is exact if and only if $f: A \rightarrow B$ is a monomorphism? ...
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functors on Zero-Object in $_RMod$-category

If I have a functor $F:_RMod\rightarrow _SMod$ that is between the category of $R$-modules to the category of $S$-modules. Can I show that it must be the case that $F(0)=0$. I know that $_RMod$ and $...
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$\mathbf{Cat}$ the category of the categories is a category

I'm studying this book and I'm trying to prove this assertion the author made: The identity functor is the identity in this category, i.e., for each category $C$, $Id_C:C\to C$ is the identity ...
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Dependent Product Functor [duplicate]

I am trying to finish the proof that in category C with Cartesian closed slices, the dependent product functor is the right adjoint of the pullback, so that C is locally Cartesian closed. The proof is ...
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$\mathbf{Vec}_k$ is not autodual

I am looking to prove that the category of vector spaces is not autodual, i.e., equivalent to its opposite category, in the simplest way. Here are my ideas. We have an equivalence between finite-...
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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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Determination of a connected groupoid by its objects and by a set of automorphisms. [duplicate]

One may readily show that a connected groupoid $G$ is determined up to isomorphism by a group (one of the groups $\hom_G(x,x)$) and by a set (the set of all objects). This is the nature of the problem ...
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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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Comodules as a functor category

Let C be a comonoid in some preadditive monoidal category $\mathfrak{C}$, then how can we express the category of C-comodules, in terms of some sort of functor category? I mean is there a similar ...
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Subset/subgroups/etc as Equalizers using Pushouts

I am trying to show that in a given category $\mathcal{C}$ which has pushouts, we can show that any morphism $f:X\rightarrow Y$ is an equalizer of some pair of morphisms. My solution to this is by ...
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Commutativity with cotensor

If C is a cocommutative R-coalgebra, R is some commutative semi-simple artinian ring and A and B are C-bicomodules, then is $A\square_C B \cong B \square_C A$? If not what other conditions are ...
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Groupoid-valued presheaf as a colimit of representables

Is there a specific way to see a presheaf of groupoids as a colimit of representables ? As you can understand I'm looking for a similar result to the well-known fact that presheaves of sets are ...
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Double categories

So, I wanted to ask a question about double groupoids until I find myself having the answer, meanwhile I wrote a lot of stuff about double categories, so I decided to create a question and answer my ...
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How to define this function on arrows?

I want to solve the following exercise: Let $U: \mathcal{C}\rightarrow\mathcal{D}$ be a functor, $F:\operatorname{Obj}(\mathcal{D})\rightarrow \operatorname{Obj}(\mathcal{C})$ be a function. Assume ...
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Is the category of chain complexes complete and cocomplete in small?

Does the category of chain complexes (let's say of modules over some ring) have all small limits and colimits? What I understand is that the category of chain complexes is certainly finitely ...
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Sheaffication using a $\lambda$-transfinite colimit

I was reading this article http://ncatlab.org/nlab/files/cech.pdf and I could not understand the construction of the left adjoint of the inclusion $\mathbf{PSh}(C, J) \rightarrowtail \mathbf{Sh}(C, J)$...
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Exactness of functors as “iff”; conjecture about bifunctors

The definition of (right-/left-) exact functors is that they preserve (right-/left-) exactness of SESs. However, for some certain nice functors, as $\def\Hom{\text{Hom}\,}\Hom (A,-)$ and $A\otimes-$ ...
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Two equivalent categories

Let $f: A\to B$ be an appropriate functor of small categories. It induces a functor $f^*: Psh(A)\to Psh(B)$, whose left adjoint is $f_!$. Is it true that $$ Psh(A)\to Psh(B)/f_!(cst) $$ an ...
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Embedding vs restriction

Embedding is the morphism $( A ; B ; \operatorname{id}_A)$ of the category $\mathbf{Rel}$ for sets $A \subseteq B$. I call restriction the morphism $( A ; B ; \operatorname{id}_B)$ for sets $A \...
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Proving that it is an equalizer

Having given a category, a pair of parallel morphisms $f$ and $g$ and a morphism $e$, what is the simplest way to prove that $e$ is an equalizer of $f$ and $g$? Can equalizers be defined purely ...
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explicit proof of pullback stability of epics in a topos

As in Explicit construction of a initial object in a topos I'm looking an elementary proof of the fact that, in a topos, epimorphisms are stable under pullback or, equivalently, that images are ...
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Direct products in a partially ordered category

Consider a category, whose set of objects is a poset. Let $f=(f_0,f_1)$ is an indexed family of objects of this category. (Thus $f$ is a 2-indexed family, we can extend it to any index set in an ...
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When does a pushout of monics induces a monic arrow?

There are lots of arguments in the homotopy theory of simplicial sets (I refer bascially to the first chapter of Goerss-Jardine) which exploit the fact that in certain cases the map induced in a ...
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Long Exact Sequence on Homology in an Abelian Category

Let $\mathcal A$ be an abelian category and let $0 \xrightarrow{} X \xrightarrow f Y \xrightarrow g Z \xrightarrow{} 0$ be an exact sequence of chain complexes in $\mathcal A$. I am using the ...
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images stable under precomposing with epis

Suppose that $C$ is a category admitting images. Given an arrow $f:a\to b$ and an epi $e:a'\to a$ is there a common name for a category where (any) of the following properties hold(s)? 1) any image ...