semicategory (changes) in nLab
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Context
Category theory
Higher category theory
Basic concepts
Basic theorems
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homotopy hypothesis-theorem
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delooping hypothesis-theorem
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stabilization hypothesis-theorem
Applications
Models
- (n,r)-category
- Theta-space
- ∞-category/∞-category
- (∞,n)-category
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
- (∞,Z)-category
- n-category = (n,n)-category
- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category
- categorification/decategorification
- geometric definition of higher category
- algebraic definition of higher category
- stable homotopy theory
Morphisms
Functors
Universal constructions
Extra properties and structure
1-categorical presentations
Contents
Idea
The notion of semicategory or non-unital category is like that of category but omitting the requirement of identity-morphisms.
This generalizes the notions of semigroup, semiring, etc:
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a semigroup is (the hom-set of) a semicategory with a single object;
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a semiring is (the hom-set of) a semicategory enriched in Ab with a single object.
Semicategories, like categories, appear as semipresheaves on the category with two objects and two morphisms.
Definition
Definition
A (small) semicategory or non-unital category 𝒞\mathcal{C} consists of
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a set 𝒞 1\mathcal{C}_1 of morphisms (or arrows);
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two functions s,t:𝒞 1→𝒞 0s, t : \mathcal{C}_1 \to \mathcal{C}_0 called source (or domain) and target (or codomain);
- one writes f:x→yf : x \to y if s(f)=xs(f) = x and t(f)=yt(f) = y;
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a function ∘:𝒞 1× t,s𝒞 1→𝒞 1\circ \colon \mathcal{C}_1 \times_{t,s} \mathcal{C}_1 \to \mathcal{C}_1 (composition) from the set of pairs of morphisms such that the target of the first is the source of the second;
such that the following properties are satisfied:
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source and target are respected by composition: s(g∘f)=s(f)s(g \circ f) = s(f) and t(g∘f)=t(g)t(g\circ f) = t(g);
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composition is associative: (h∘g)∘f=h∘(g∘f)(h \circ g)\circ f = h\circ (g \circ f) whenever t(f)=s(g)t(f) = s(g) and t(g)=s(h)t(g) = s(h).
Definition
For 𝒞,𝒟\mathcal{C}, \mathcal{D} two semicategories, a semi-functor F:𝒞→𝒟F \colon \mathcal{C} \to \mathcal{D} is a pair of functions F 0:𝒞 0→𝒟 0F_0 \colon \mathcal{C}_0 \to \mathcal{D}_0, F 1:𝒞 1→𝒟 1F_1 \colon \mathcal{C}_1 \to \mathcal{D}_1 that respects all the given structure in the obvious way.
Write SemiCatSemiCat for the (large) category whose objects are semicategories, and whose morphisms are semifunctors.
Properties
Relation to categories
We discuss the relation of semicategories to categories. (See for instance the beginning of (Harpaz) for a quick review of basics, with an eye towards their generalization to the relation between complete Segal spaces and complete semi-Segal spaces.)
Definition
There is an evident forgetful functor
U:Cat→SemiCat U \colon Cat \to SemiCat
from the category Cat of categories to that of semicategories, def. 2, given simply by forgetting the identity-assigning map i:𝒞 0→𝒞 1i \colon \mathcal{C}_0 \to \mathcal{C}_1 in a category.
Definition
For 𝒞\mathcal{C} a semi-category, def. 1, write
Id(𝒞 1)↪𝒞 1 Id(\mathcal{C}_1) \hookrightarrow \mathcal{C}_1
for the subset on those morphisms which are endomorphisms on some object x∈𝒞 0x \in \mathcal{C}_0 and such that they are neutral elements with respect to composition in their endomorphismssemimonoids𝒞\mathcal{C} End 𝒞(x)End_{\mathcal{C}}(x).
Proposition
A semicategory is the semicategory underlying a category, hence is in the image of the functor UU of def. 3, precisely if every object has a neutral endomorphism, hence precisely if the composite diagonal function in
Id(𝒞 1) ↪ 𝒞 1 ≃↘ ↓ s 𝒞 0 \array{ Id(\mathcal{C}_1) &\hookrightarrow& \mathcal{C}_1 \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{s}} \\ && \mathcal{C}_0 }
is an isomorphism, where the horizontal function is that of def. 4.
Moreover, if a semicategory lifts to a category, it does so in a unique way: the functor U:Cat→SemiCatU \colon Cat \to SemiCat is an injection on isomorphism classes.
Transitive relations
A transitive relation is a semicategory enriched on truth values, or a semicategory CC where there is at most one morphism from every object aa to object bb in CC
Nerves and semi-simplicial sets
The nerve of a semicategory is a semi-simplicial set which satisfies the Segal conditions.
Examples
Start with the category of metric spaces and short maps. An occasionally useful semicategory can be formed from it by considering the nonempty spaces and strictly contractive functions.
This is a semicategory, since:
- the composition of two strictly contractive functions is strictly contractive
- identity maps are not contractive (they are trivial isometries)
The interest in this semicategory arises from the fact that all morphisms f:A→Af : A \to A have unique fixed points, by Banach’s fixed point theorem.
In higher category theory
The concept of semicategory has more or less evident analogs and generalizations in higher category theory.
For models of higher categories by simplicial sets, i.e. presheaves on the simplex category (such as Kan complexes, quasi-categories, weak complicial sets) the corresponding semi-category notion is obtained by discarding the degeneracy maps (which are the identity-assigning maps in the simplicial framework), i.e. by considering just presheaves on the subcategory Δ +⊂Δ\Delta_+ \subset \Delta on injective morphisms (see the discuss of Δ +\Delta_+ at Reedy model structure for more details).
Accordingly, there is the semi-category analog of a Segal space, called a semi-Segal space.
Simpson's conjecture says that every ∞\infty-category has a model where all composition operations are strict and only the unit laws hold up to coherent homotopies. This would mean that the ∞\infty-semicategory underlying any ∞\infty-category can always be chosen to be strict.
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dagger semicategory?
References
Semicategories were introduced in
- Barry Mitchell, The dominion of Isbell, Transactions of the American Mathematical Society 167 (1972) 319-331 [[doi:10.2307/1996142](https://doi.org/10.2307/1996142)]
Enriched semicategory theory is developed in
- M.-A. Moens, U. Bernani-Canani, F. Borceux, On regular presheaves and regular semi-categories , Cah. Top. Géom. Diff. Cat. XLIII no.3 (2002) pp.163-190. (numdam)
This is turned one notch further in
- Isar Stubbe, Categorical structures enriched in a quantaloid : regular presheaves, regular semicategories , Cah. Top. Géom. Diff. Cat. XLVI no.2 (2005) pp.99-121. (numdam)
Semicategories and semigroups are mentioned in section 2 in
- W. Dale Garraway, Sheaves for an involutive quantaloid, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 46 no. 4 (2005), p. 243-274 (numdam)
Semicategories with an eye towards their generalization to semi-Segal spaces are briefly discussed at the beginning of
- Yonatan Harpaz, Quasi-unital ∞\infty-Categories (arXiv:1210.0212)
Structures obtained by further relaxing also the associativity law are discussed in
- Salvatore Tringali, Plots and Their Applications - Part I: Foundations (arXiv:1311.3524)
Topologically enriched semicategories are used for studying some aspects of concurrency theory in computer science. It is necessary to work with semicategories to have functorial definitions of the branching and merging homologies of a concurrent system. A starting point for reading the theory can be the paper
- Philippe Gaucher, Flows revisited: the model category structure and its left determinedness, Cahiers de Topologie et Géométrie Différentielle Catégoriques, vol LXI-2 (2020) (published, arXiv:1806.08197)
Semi-categories, semi-adjunctions and semi-cartesian closed categories have been used to study the lambda calculus since
- Susumu Hayashi, Adjunction of semifunctors: Categorical structures in nonextensional lambda calculus, Theoretical Computer Science Volume 41, 1985, Pages 95-104 doi:10.1016/0304-3975(85)90062-3
Last revised on June 5, 2023 at 10:38:38. See the history of this page for a list of all contributions to it.