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I don't believe there are non-trivial examples of this concept. Assume that $e: X \to X$ is a constant map, then $e$ is idempotent. Any category can be embedded fully faithfully into a Cauchy complete …

I was talking about the following tentative argument. The 2-category of distributors (also called profunctors) $\mathrm{Dist}$ has (small) categories for objects. For $C,D:\mathrm{Dist}$ the 2-categor …

Tom Leinster's book is very old. For higher category theory 2003 is like a previous epoch. In those times there were many competing definitions of higher category theory and higher algebra, for most o …

Ok, I have no idea how to fix the invariant of string diagrams, but I have an explicit counterexample. Consider the 2-category of distributors $Dist$: its objects are small categories, its morphisms f …

N.B.: I have reread your question and it occured to me that you a probably asking something entirely different. However since I'm unclear what exactly is your question and since I don't want to delete …

As per Qiaochu Yuan's comment we need to only understand the space of based maps between $K(A,n)$ with a chosen base point. The loop-deloop pair of functors establish an equivalence between the categ …

As requested, making comment into the answer. I think Helemskii's book "Lectures and Exercises on Functional Analysis" contains a very nice intro into category theory. It may be a bit light on the alg …

In a nutshell, the question is: is it true that any explicit (not involving axiom of choice) pointwise transformation between sufficiently complicated functors is natural almost everywhere? Let $C$ …

As already noted above, any category can be considered concrete after a base change to a suitably large universe. However, doing so would be completely missing the point of concreteness. The underlyin …

Consider a theory which has no models in $\mathrm{Set}$, but has a model in $\mathrm{Sh}(L)$ for some locale $L$. For example, the theory $\mathcal{CLF}$ of complete linearly ordered fields with more …

Firstly, note that it is enough to construct an isomorphism $$\mathcal{C}(L(A,B),C) \simeq \mathcal{B}(B, R_2(A,C))$$ The third natural isomorphism then follows automatically. Secondly, standard appli …

The definition of object being initial is obviously equivalent to the statement "the inclusion of one-point category is final". Here by final functor I mean $p: I\to C$, such that for any $F: C\to D$ …

We have an equivalence of categories $Aff\simeq Ring^{op}$ and a pair of adjoint functors $$\mathcal{O}:Sch\rightleftharpoons Ring^{op} : Spec$$ $$\mathcal{O} \dashv Spec$$ The category of affine sche …

It is a general fact that if you consider an abelian group $A$ in topos $T$, then equivalence classes of $A$-torsors in $T$ are classified by cohomology group $H^1(T;A)=Ext^1_T(\mathbb{Z},A)$. Here $\ …

I am not really familiar with homotopical category theory, so please forgive me if I make rude mistakes. I know quite a bit of common category theory, as well as familiar with algebraic topology. How …