# Tagged Questions

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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### How to construct a G-extension of a category C?

Note: I'm a physicist so this will be phrased somewhat in physics language. Suppose we have a unitary modular tensor category $\mathcal{C}$. In physics language, we can think of $\mathcal{C}$ as ...
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### Can these definitions of the words “problem” and “solution” be formalized, and if so, has this been done? If so, where can I learn more about it?

I had a thought. Define that: Vague Definition 0. A problem consists of: a set $X$ a set $Y$ a function $f : X \rightarrow Y$ a way $\overline{X}$ of representing the elements of $X$ ...
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### What would be an arrow in category of Hilbert space?

Let $H,K$ be Hilbert spaces. Let $S$ be a Hilbert basis for $H$. (Which means that $S$ is orthogonal and the span of $S$ is dense in $H$.) For each arrow $S\rightarrow K$, there exists a unique ...
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Here it is said that a natural transformation $\varphi:F\Rightarrow G$ is the same as a function $\varphi_0:\mathrm{Ob}\mathcal A\rightarrow \mathrm{Mor}\mathcal B$ satisfying ${\rm dom}\circ\varphi=F,... 1answer 31 views ### Adjunction with reversed elements There is an adjunction between$L$and$R$when: $$\text{Hom}(LA,B) \approx \text{Hom}(A,RB)$$ Is there something related we can say when instead we have: $$\text{Hom}(B,LA) \approx \text{Hom}(A,... 2answers 109 views ### What are the group objects in the category of finite sets and bijections, and its functor category? An object G in a category \mathcal{C} is called a group object if, given any object X in \mathcal{C}, there is a group structure on the morphisms \operatorname{hom}\left(X,G\right) such that ... 1answer 35 views ### A linear category is a Vect-module I would like to know how to show that any linear category is a \mathrm{Vec}-module. Here \mathrm{Vect} denotes a category of finite dimensional vector spaces. More general statement can be found ... 1answer 37 views ### Existence of adjoints with commutativity condition Are there (non-trivial) examples of adjunctions F \operatorname{\dashv} G with unit and counit \eta, \epsilon such that$$\eta GF = GF \eta$$and$$FG \epsilon = \epsilon FG,$$and if so, what ... 0answers 63 views ### Deducing the existence of particular functions \mathbb{N}\longrightarrow\mathbb{Q} in the context of Tom Leinster's “Rethinking Set Theory” This question concerns the set theory given by Tom Leinster in his paper "Rethinking Set Theory," available here: http://arxiv.org/abs/1212.6543 In this paper, axioms for set theory are given in the ... 1answer 88 views ### Orthogonal Factorization Systems and functoriality In definition 1.1 of these notes on factorization systems, (III) calls the factorization functorial if given the solid diagram described, there's a unique horizontal arrow making both squares commute. ... 1answer 68 views ### What limits/colimits are preserved by the Yoneda embedding? I know that the contravariant Yoneda embedding X\mapsto \mathcal{C}(-,X) preserves all small limits that exist in \mathcal{C}. I guess it follows that the covariant Yoneda embedding preserves all ... 1answer 56 views ### Associativity of the tensor product of bimodules Let A_0,\dots,A_n be algebras over some fixed commutative ring k (you may assume k=\mathbb{Z} for simplicity). Let M_i be an (A_{i-1},A_i)-bimodule for i=1,\dots,n. A multilinear map from ... 0answers 39 views ### An asymmetry in the Galois connection between topologies and sequential convergences? On a set X consider a relation c \subseteq X^{\mathbb{N}} \times X and for ((x_n), x) \in c write x_n \to_c x. Such a relation c is a sequential convergence if (i) x \to_c x for all x \in ... 1answer 39 views ### Simplicial homotopy's exponential law The following was cited from Simplicial Homotopy Theory by John F. Jardine and Paul Gregory Goerss. We have an adjunction$$\hom_{\mathbf{S}}(X\times Y,Z)\simeq \hom_{\mathbf{S}}(Y,\mathbf{Hom}(X,Z))$$... 0answers 44 views ### Functorial construction of the convolution algebra of measures on a group Let G be a lcoally compact group and C_c(G) = \lim_{K \subset G} C(K) its space of continuous functions with compact support endowed with the topology of the limit of banach spaces C(K) with K ... 1answer 35 views ### Action groupoid as G\rightrightarrows \textrm{Bij}(X)? Let G be a group and X a set. A left action of G on X can be thought either as a map G\times X\longrightarrow X, (g, x)\longmapsto g\cdot x, satisfying: (i) g\cdot (h\cdot x)=(gh)\cdot ... 2answers 53 views ### Epimorphisms and faithful functors in a rigid abelian tensor category Let \mathsf{C} be a rigid abelian tensor category in the sense of Deligne and Milne's notes (p.9). Let \mathbf{1} denote the identity object in \mathsf{C} with respect to \otimes. The abelian ... 2answers 42 views ### Subcategory of category of Module satisfies SSA? Let \mathcal{D}=Mod-A be a category of module over an Algebra A. and \mathcal{C} be a subcategory of \mathcal{D}, I have the following questions:- Is C complete category? Is \mathcal{C} ... 4answers 126 views ### Composing functors with natural transformations So I'm doing a project in Category Theory. I fully understand natural transformations and functors, but what does it mean to compose them, for example in the monad axioms where you have something like ... 0answers 25 views ### Question Janelidze and Tholen's 'Beyond Barr Exactness: Effective Descent Morphisms' I have some questions about Janelidze and Tholen's paper Beyond Barr Exactness: Effective Descent Morphisms. First of all, they remark at the end of the introduction: Throughout this chapter \... 1answer 41 views ### Polynomial functors preserving ω-colimits I'm working my way through Steve Awodey's Category Theory book and on p.271, Proposition 10.12 says: If the category S has an initial object 0 and colimits of diagrams of type \omega (call them ... 2answers 71 views ### Is there a category in which, between any two objects, there is a unique morphism? I am interested in knowing information about such a category (if it is well-defined). Does there exist a category \mathcal{C}, in which there is a unique morphism between any two objects in it? 1answer 46 views ### adjoint functor and subcategories First at all i have to say I am very new to categories (just basic definition). My Question:- We have two categories ( \mathcal{C},\mathcal{R}) object in both of these two categories are module ( ... 0answers 33 views ### Extension Operator. I am working on my thesis about completion and extensions from an algebraic point of view. We have the closure operator which takes subsets to subsets with 3 criterias to meet X\subseteq C(X) X\... 1answer 46 views ### Presheaf and copresheaf categories on finite sets Briefly: do they agree? In more detail: denote by \mathbf{Finset} the category of finite sets, and by \mathbf{Set} the category of sets. I want to know what the functor categories [\mathbf{... 2answers 63 views ### Non-monadic adjunction Could someone give some examples of a non-monadic adjunctions please? Possibly explaining why they are not monadic and how they contradict the monadicity theorem? Thanks! 1answer 42 views ### Establish canonical isomorphism in Set category Using notation A^B := \mathsf{Mor}_{\mathcal{SET}}(B, A) establish canonical isomorphisms for any sets X, Y and Z:$$ (Z^Y)^X \cong Z^{Y \times X} \; , \;(Z \times Y)^X \cong Z^X \times Y^X.$$... 1answer 71 views ### Formal Proof that f^{-1} \circ f = id_x \ , \forall f Given f as an invertible function with domain X and codomain Y, then we can say$$f^{-1}(f(x)) = x $$Or since they are both logically equivalent$$ f(f^{-1}(x)) = x$$This can also be ... 1answer 25 views ### Adjointness of internal contravariant Hom in symmetric monoidal categories. Let consider a closed symmetric monoidal cateogry,$\mathscr C,\otimes$, with adjunction$(X\otimes-)\dashv\mathrm{Hom}(X,-)$for all objects$X$. The following isomorphis, valid in categories such ... 0answers 31 views ### Functor between Category and its Free Strict Monoidal Category Let$C$be a category and let$\sum(C)$denote the free strict monoidal category over$C$. According to Wikipedia, the operation$\sum: C\rightarrow\sum(C)$extends to a 2-monad on$Cat$. Can someone ... 1answer 61 views ### Terminal object in the category of sheaves? Let$\mathsf{C}$denote one of the categories$\mathsf{Set}$,$\mathsf{Group}$,$\mathsf{Ring}$, or$\mathsf{Mod}_R$, or some other sensible category in which we would like sheaves to take their ... 0answers 37 views ### split object as coproduct Let$X$be an object and$f : Y \to X$a monomorphism in some category with coproducts. Is there a categorical characterization for objects$Z$such that$Y + Z \cong X$and left injection composed ... 1answer 38 views ### If monoid satisifes universal mapping property over$X$, then$X$generates the monoid A monoid$M$satisfies the universal mapping property (UMP) over$X$, if$X \subseteq M$and for every map$\varphi : X \to N$, where$N$is another monoid, there exists a unique homomorphism$\varphi ...
Let $F$ be a presheaf of sets on the usual topological site. Let $\left\{U_i\rightarrow U\right\}$ be covering of $U$. On the second page of Hypercovers and Simplicial Presheaves the authors write the ...
Let $\mathbf{C}$ be any category. Is there a way to adjoin a terminal object $\ast$ to $\mathbf{C}$ such that each constant morphism factors through any terminal object, i.e., for each constant \$f:A\...