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A Coalgebra $C$ is called metrizable if there is a base $B$ for $C$(as a vector space) and a metric $d:B \times B \to \mathbb{R}$ on $B$ such that the linear extension $\tilde{d}: C\otimes C ... 0answers 119 views ### Injectivity criterion for surjective coalgebra maps: does it hold in full generality? Let$k$be a commutative ring. Let$C$be a filtered$k$-coalgebra. This means a$k$-coalgebra equipped with an increasing$k$-module filtration$C^0 \subseteq C^1 \subseteq C^2 \subseteq ...$... 0answers 68 views ### Coinduction and corestriction are quasi-inverse equivalences for comodules? I'm reading http://arxiv.org/abs/math/0310337. There the following statement is given without proof: Let$k$be a field. Let$C$be a counitary coaugmented coalgebra, i.e. there is$\eta: C\to k$... 1answer 235 views ### (Co)Universal Property of Quotients/Subs I'm not completely sure if this bunch of questions is the appropriate Level of MO. However at the same time I think that it is at least slightly above the level of stackex. ... The tensor algebra ... 1answer 241 views ### Reference for the fact that a coderivation of the (non reduced) tensor coalgebra is determined by its corestrictions If$V$is a vector space, let us consider the tensor coalgebra$TV=\bigoplus\limits_{k=0}^\infty V^{\otimes^k}$with coproduct given by$$\Delta (x_1\otimes \dots \otimes x_n):= ... 1answer 136 views ### Linear functional kills all primitives of a connected filtered coalgebra => it lies in m^2? Let$C$be a connected filtered coalgebra over a field$k$. Maybe$k$has characteristic$0$(though I don't know where this can be of use). Let$1$denote the unique element of$C_0$mapping to$1\in ...
Let $R$ be a ring, and $R\text{-Mod}$ its category of all left modules. There is a "forgetful" functor $\operatorname{Forget}: R\text{-Mod} \to \text{AbGp}$, which is additive, continuous, and ...