# 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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### 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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### $\mathbf{Cat}$ 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 $A\otimes-$ ...
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 ...