$FinInj$ (or $FinSet_inj$ or often just $FI$) refers to the wide subcategory of FinSet with its morphisms restricted to monomorphisms (injective functions).
It may be characterised as:
the free (symmetric) semicocartesian category on a single object
the free symmetric monoidal category on a pointed object $I \to X$ (where $I$ is the monoidal unit)
the free monoidal category on an object $X$ equipped with an involution $s : X \otimes X \to X \otimes X$ satisfying the braid relations, and a morphism $i : I \to X$ satisfying $s \circ i \otimes X = s \circ X \otimes i$
A functor from $FinSet_{inj}$ to Vect (or more generally to some $R$Mod) is also known as an FI-representation, a kind of accumulated form of $Sym(n)$-representations as $n \in \mathbb{N}$ ranges (cf. also representation stability).
Introducing the notion of FI-modules ($FinSet_{inj}$-representations):
Thomas Church, Jordan S. Ellenberg, Benson Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 9 (2015) 1833-1910 [arXiv:1204.4533, doi:10.1215/00127094-3120274]
Thomas Church, Jordan S. Ellenberg, Benson Farb, Rohit Nagpal, FI-modules over Noetherian rings, Geom. Topol. 18 (2014) 2951-2984 [arXiv:1210.1854, doi:10.2140/gt.2014.18.2951]
Last revised on May 9, 2023 at 14:16:25. See the history of this page for a list of all contributions to it.