Definition df-termc 49718

Description: Definition of the proper class (termcnex 49821) of terminal categories, or final categories, i.e., categories with exactly one object and exactly one morphism, the latter of which is an identity morphism (termcid 49731). These are exactly the thin categories with a singleton base set. Example 3.3(4.c) of [Adamek] p. 24.

As the name indicates, TermCat is the class of all terminal objects in the category of small categories (termcterm3 49760). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 49765). See also dftermc3 49776 where TermCat is defined as categories with exactly one disjointified arrow.

Unlike https://ncatlab.org/nlab/show/terminal+category 49776, we reserve the term "trivial category" for (SetCat‘1_o), justified by setc1oterm 49736.

Followed directly from the definition, terminal categories are thin (termcthin 49722). The opposite category of a terminal category is "almost" itself (oppctermco 49750). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 49785).

Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 49746) and terminal objects (dftermo4 49747). The universal properties provide an alternate proof of initoeu1 17935, termoeu1 17942, initoeu2 17940, and termoeu2 49483. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 49761).

The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17611, 0thincg 49703, func0g 49334, 0funcg 49330, and initc 49336.

(Contributed by Zhi Wang, 16-Oct-2025.)

Assertion

Ref	Expression
df-termc	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

Distinct variable group:   𝑥,𝑐

Detailed syntax breakdown of Definition df-termc

Step	Hyp	Ref	Expression
1		ctermc 49717	. 2 class TermCat
2		vc	. . . . . . 7 setvar 𝑐
3	2	cv 1540	. . . . . 6 class 𝑐
4		cbs 17136	. . . . . 6 class Base
5	3, 4	cfv 6492	. . . . 5 class (Base‘𝑐)
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1540	. . . . . 6 class 𝑥
8	7	csn 4580	. . . . 5 class {𝑥}
9	5, 8	wceq 1541	. . . 4 wff (Base‘𝑐) = {𝑥}
10	9, 6	wex 1780	. . 3 wff ∃𝑥(Base‘𝑐) = {𝑥}
11		cthinc 49662	. . 3 class ThinCat
12	10, 2, 11	crab 3399	. 2 class {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
13	1, 12	wceq 1541	1 wff TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

This definition is referenced by:  istermc  49719